OPERADS, ALGEBRAS, MODULES, AND MOTIVES IGOR KRIZ AND J. P. MAY Submitted January 27, 1994 Contents Introduction 3 Part I. Definitions and examples of operads and operad actions 10 1. Operads 10 2. Algebras over operads 13 3. Monadic reinterpretation of algebras 16 4. Modules over C-algebras 19 5. Algebraic operads associated to topological operads 23 6. Operads, loop spaces, n-Lie algebras, and n-braid algebras 26 7. Homology operations in characteristic p 30 Part II. Partial algebraic structures and conversion theorems 33 1. Statements of the conversion theorems 33 2. Partial algebras and modules 35 3. Monadic reinterpretation of partial algebras and modules 41 4. The two-sided bar construction and the conversion theorems 44 5. Totalization and diagonal functors; proofs 47 6. Higher Chow complexes 51 Part III.Derived categories from a topological point of view 55 1. Cell A-modules 55 __________ 1991 Mathematics Subject Classification. 14A20, 18F25, 18G99, 19D99, 19E99, 5* *5U99. Both authors were partially supported by the NSF, and the second author was p* *artially supported by the Sloan Foundation. 1 2 IGOR KRIZ AND J. P. MAY 2. Whitehead's theorem and the derived category 60 3. Brown's representability theorem 63 4. Derived tensor product and Hom functors: Tor and Ext 65 5. Commutative DGA's and duality 69 6. Relative and unital cell A-modules 71 Part IV. Rational derived categories and mixed Tate motives 73 1. Statements of results 73 2. Minimal algebras, 1-minimal models, and co-Lie algebras 77 3. Minimal A-modules 80 4. The t-structure on DA 83 5. Twisting matrices and representations of co-Lie algebras 85 6. The bar construction and the Hopf algebra OA 88 7. The derived category of the heart and the 1-minimal model 91 Part V. Derived categories of modules over E1 algebras 98 1. The category of C-modules and the product 100 2. Unital C-modules and the products C, B, and 105 3. A new description of A1 and E1 algebras and modules 108 4. Cell A-modules and the derived category of A-modules 112 5. The tensor product of A-modules 115 6. The Hom functor on A-modules; unital A-modules 119 7. Generalized Eilenberg-Moore spectral sequences 122 8. E1 algebras and duality 127 9. The linear isometries operad; change of operads 130 References 135 OPERADS, ALGEBRAS, MODULES, AND MOTIVES 3 Introduction There are many different types of algebra: associative, associative and commu* *tative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module and * *thus with an associated theory of representations. Moreover, as is becoming more and* * more important in a variety of fields, including algebraic topology, algebraic geome* *try, dif- ferential geometry, and string theory, it is very often necessary to deal with * *"algebras up to homotopy" and with "partial algebras". The associated theories of modules* * have not yet been developed in the published literature, but these notions too are b* *ecoming increasingly important. We shall study various aspects of the theory of such ge* *neralized algebras and modules in this paper. We shall also develop some related algebra * *in the classical context of modules over DGA's. While much of our motivation comes fro* *m the theory of mixed Tate motives in algebraic geometry, there are pre-existing and * *potential applications in all of the other fields mentioned above. The development of abstract frameworks in which to study such algebras has a * *long history. It now seems to be widely accepted that, for most purposes, the most c* *onvenient setting is that given by operads and their actions [46]. While the notion was * *first written up in a purely topological framework, due in large part to the resistan* *ce of topologists to abstract nonsense at that period, it was already understood by 1* *971 that the basic definitions apply equally well in any underlying symmetric monoi* *dal (= tensor) category [35]. In fact, certain chain level concepts, the PROP's and PA* *CT's of Adams and MacLane [42], were important precursors of operads. From a topological point of view, the switch from algebraic to topological PROP's, which was made * *by Boardman and Vogt [11], was a major step forwards. Perhaps for this reason, a c* *hain level algebraic version of the definition of an operad did not appear in print * *until the 1987 paper of Hinich and Schechtman [31]. Applications of such algebraic operad* *s and their actions have appeared in a variety of contexts in other recent papers, fo* *r example [27, 28, 29, 32, 34, 33, 56]. In the algebraic setting, an operad C consists of suitably related chain comp* *lexes C (j) with actions by the symmetric groups j. An action of C on a chain complex* * A is specified by suitably related j-equivariant chain maps C (j) Aj! A; where Aj is the j-fold tensor power of A. The C (j) are thought of as parameter complexes for j-ary operations. When the differentials on the C (j) are zero, w* *e think of C as purely algebraic, and it then determines an appropriate class of (diffe* *rential) algebras. When the differentials on the C (j) are non-zero, C determines a cla* *ss of (differential) algebras "up to homotopy", where the homotopies are determined b* *y the 4 IGOR KRIZ AND J. P. MAY homological properties of the C (j). For example, we say that C is an E1 operad* * if each C (j) is j-free and acyclic, and we then say that A is an E1 algebra. An E1 alg* *ebra A has a product for each degree zero cycle of C (2). Each such product is unit* *al, associative, and commutative up to all possible coherence homotopies, and all s* *uch products are homotopic. There is a long history in topology and category theory* * that makes precise what these "coherence homotopies" are. However, since the homotop* *ies are all encoded in the operad action, there is no need to be explicit. There is* * a class of operads that is related to Lie algebras as E1 operads are related to commuta* *tive algebras, and there is a concomitant notion of a "strong homotopy Lie algebra".* * In fact, any type of algebra that is defined in terms of suitable identities admits an a* *nalogous "strong homotopy" generalization expressed in terms of actions by appropriate o* *perads. We shall give an exposition of the basic theory of operads and their algebra* *s and modules in Part I. While we shall give many examples, the deeper aspects of the* * theory that are geared towards particular applications will be left to later Parts. In* * view of its importance to string theory and other areas of current interest, we shall i* *llustrate ideas by describing the relationship between the little n-cubes operads of iter* *ated loop space theory on the one hand and n-Lie algebras and n-braid algebras on the oth* *er. An operad S of topological spaces gives rise to an operad C#(S ) of chain complexe* *s by passage to singular chains. On passage to homology with field coefficients, the* *re results a purely algebraic operad H*(S ). There is a particular operad of topological s* *paces, denoted Cn, that acts naturally on n-fold loop spaces. For n 2, the algebras d* *efined by H*(Cn; Q) are exactly the (n - 1)-braid algebras. Even before doing any calc* *ulation, one sees from a purely homotopical theorem of [46] that, for any path connected* * space X, H*(nnX; Q) is the free H*(Cn; Q)-algebra generated by H*(X; Q). This allows a topological proof, based on the Serre spectral sequence, of the algebraic fac* *t that the free n-braid algebra generated by a graded vector space V is the free commu* *tative algebra generated by the free n-Lie algebra generated by V . Actually, the resu* *lts just summarized are the easy characteristic zero case of Cohen's much deeper calcula* *tions in arbitrary characteristic [15, 16], now over twenty years old. Operads and their actions are specified in terms of maps that are defined on* * tensor products of chain complexes. In practice, one often encounters structures that * *behave much like algebras and modules, except that the relevant maps are only defined * *on suitable submodules of tensor products. For geometric intuition, think of inter* *section products that are only defined between elements that are in general position. * *Such partial algebras have been used in topology since the 1970's, for example in [4* *8] and in unpublished work of Boardman and Segal. In Part II, we shall generalize the not* *ions of algebras over operads and of modules over algebras over operads to the conte* *xt of OPERADS, ALGEBRAS, MODULES, AND MOTIVES 5 partially defined structures. Such partially defined structures are awkward to* * study algebraically, and it is important to know when they can be replaced by suitabl* *y equiv- alent globally defined structures. We shall show in favorable cases that partia* *l algebras can be replaced by quasi-isomorphic genuine algebras over operads, and similarl* *y for modules. When k is a field of characteristic zero, we shall show further that E* *1 algebras and modules can be replaced by quasi-isomorphic commutative algebras and modules and, similarly, that strong homotopy Lie algebras and modules can be replaced by quasi-isomorphic genuine Lie algebras and modules. The arguments work equally w* *ell for other kinds of algebras. One of the main features of the definition of an operad is that an operad det* *ermines an associated monad that has precisely the same algebras. This interpretation i* *s vital to the use of operads in topology. The proofs of the results of Part II are bas* *ed on this feature. The key tool is the categorical "two-sided monadic bar construction" t* *hat was introduced in the same paper that first introduced operads [46]. This construct* *ion has also been used to prove topological analogs of many of the present algebraic re* *sults, along with various other results that are suggestive of further algebraic analo* *gs [47, 49, 26, 52]. In particular, the proofs in Part II are exactly analogous to a to* *pological comparison between Segal's -spaces [55] and spaces with operad actions that is * *given in [26]. While these results can be expected to have other applications, the motivatio* *n came from algebraic geometry. For a variety X, Bloch [7] defined the Chow complex Z(* *X). This is a simplicial abelian group whose homology groups are the Chow groups of X. It has a partially defined intersection product, and we show in Part II tha* *t it gives rise to a quasi-isomorphic E1 algebra, denoted N (X). After tensoring wit* *h the rationals, we obtain a commutative differential graded algebra (DGA) NQ(X) that* * is quasi-isomorphic to N (X) Q. The construction of these algebras answers questi* *ons of Deligne [20] that were the starting point of the present work. His motivatio* *n was the intuition that, when X = Spec(F ) for a field F , the associated derived ca* *tegories of modules ought to be the appropriate homes for categories of integral and rat* *ional mixed Tate motives over F . This raises several immediate problems. On the rational level, it is necessa* *ry to connect this approach to mixed Tate motives with others. On the integral level* *, in order to take the intuition seriously, one must first construct the derived cat* *egory of modules over an E1 algebra. As a preliminary to the solution of these problems,* * in Part III we shall give a new, topologically motivated, treatment of the classic* *al derived category of modules over a DGA. We shall give a theory of "cell modules" that i* *s just like the theory of "CW spectra" in stable homotopy theory, and we shall prove d* *irect 6 IGOR KRIZ AND J. P. MAY algebraic analogs of such standard and elementary topological results as the ho* *motopy extension and lifting property, the Whitehead theorem, and Brown's representabi* *lity theorem. One point is that there is not the slightest difficulty in handling un* *bounded algebras and modules: except that the details are far simpler, our substitute f* *or the usual approximation of differential modules by projective resolutions works in * *exactly the same way as the approximation of arbitrary spectra by (infinite) CW spectra* * with cells of arbitrarily small dimension, which has long been understood. Similarly* *, derived tensor products of modules work in the same way as smash products of spectra. In Part IV, we shall specialize this theory to study the derived category DA* * of coho- mologically bounded below A-modules, where A is a cohomologically connected com- mutative DGA over a field of characteristic zero. In the language of [3], we sh* *all give the triangulated category DA a t-structure. Its heart HA will be the Abelian su* *bcate- gory of modules whose indecomposable elements have homology concentrated in deg* *ree zero. In the language of [21], we shall show that the full subcategory F HA of * *finite dimensional modules in HA is a neutral Tannakian category. It is therefore the * *cate- gory of representations of an affine group scheme or, equivalently, of finite d* *imensional comodules over a Hopf algebra. In fact, without using Tannakian theory, we shall prove directly that HA is * *equivalent to the category of comodules over the explicit commutative Hopf algebra OA = H0* *B (A). The "cobracket" associated to the coproduct on OA induces a structure of "co-Lie algebra" on its vector space flA of indecomposable elements, and we shall see t* *hat HA is also equivalent to the category of generalized nilpotent representations of * *the co-Lie algebra flA. Part IV is really a chapter in rational homotopy theory, and it may well hav* *e appli- cations to that subject. As was observed by Sullivan [58], a co-Lie algebra fl * *determines a structure of DGA on the exterior algebra ^(fl[-1]), where fl[-1] is a copy of* * fl con- centrated in degree one. For a cohomologically connected DGA A, ^(flA[-1]) is * *the 1-minimal model of A. We shall prove the rather surprising result that the der* *ived category of modules over the DGA ^(flA[-1]) is equivalent to the derived catego* *ry of the Abelian category HA. Curiously, although the theory of minimal rational DGA* *'s has been widely studied since Sullivan's work, the analogous theory of minimal * *modules does not appear in the literature. That theory will be central to our work in P* *art IV. In view of the relationship between Chow groups and K-groups, the Beilinson- Soule conjecture for the field F is equivalent to the assertion that the DGA NQ* * = NQ(Spec(F )) is cohomologically connected. When the conjecture holds, the resu* *lts just summarized apply to A = NQ. Assuming the Beilinson-Soule conjecture (and a* *s- suming our construction of the DGA A), Deligne [20, 17] proposed F HA as a cand* *idate OPERADS, ALGEBRAS, MODULES, AND MOTIVES 7 for the Abelian category M T M (F ) of mixed Tate motives over F . He (in [18])* * and Bloch and the first author (in [6]) proposed the category of finite dimensional* * comodules over OA as a candidate for M T M (F ), and [6] proves realization theorems inet* *ale and Hodge theory starting from this definition. Our work shows that these two categ* *ories are equivalent, and it gives a fairly concrete and explicit description of them* *. When A is a K(ss; 1), in the sense that A is quasi-isomorphic to its 1-minimal model, * *we shall have the relation ExtpMT M(F)(Q; Q(r)) ~=grrflK2r-p(F ) Q between the Abelian category M T M (F ) and the algebraic K-theory of F . (Unde* *fined notations are explained in the introduction to Part IV.) Finally, in Part V, we shall construct the derived category of modules over a* *n A1 or E1 k-algebra A, where k is a commutative ground ring. Here A1 algebras are DGA* *'s up to homotopy (without commutativity). There are a number of subtleties. From Part I, we know that A-modules are equivalent to modules over an associative, b* *ut not commutative, universal enveloping DGA U(A). In particular, U(k) = C (1). In ear* *lier Parts, all E1 operads were on the same footing. In Part V, we work with a parti* *cular E1 operad C that enjoys special properties, but we show that restriction to th* *is choice results in no loss of generality. Remarkably, with this choice, the category o* *f E1 k- modules, alias the category of C (1)-modules, admits a commutative and associat* *ive "tensor product" . This product is not unital on the module level, although the* *re is a natural unit map k M ! M that becomes an isomorphism in the derived category. This fact leads us to introduce certain modified versions of the product M N t* *hat are applicable when one or both of M and N is unital, in the sense that it has a gi* *ven map k ! M. The product "" that applies when both M and N are unital is commutative, associative, and unital up to coherent natural isomorphism; that is, the catego* *ry of unital E1 k-modules is symmetric monoidal under . Conceptually, we now change ground categories from the category of k-modules * *to the category of E1 k-modules. It turns out that A1 and E1 algebras can be described* * very simply in terms of products A A ! A. In fact, an A1 k-algebra is exactly a mon* *oid in the symmetric monoidal category of unital E1 k-modules, and an E1 k-algebra * *is a commutative monoid. There is a similar conceptual description of modules over A* *1 and E1 algebras. From here, the development of the triangulated derived category D* *A of modules over an A1 algebra A proceeds exactly as in the case of an actual DGA i* *n Part III. When A is an E1 algebra, the category of A-modules admits a commutative and associative tensor product A and a concomitant internal Hom functor Hom A. Agai* *n, there is a natural unit map A A M ! M that becomes an isomorphism on passage to 8 IGOR KRIZ AND J. P. MAY derived categories. There are Eilenberg-Moore, or hyperhomology, spectral seque* *nces L for the computation of the homology of the derived tensor product M AN and the derived Hom functor R Hom A(M; N) in terms of the classical Tor and Ext groups *(A) * * * * * TorH* (H (M); H (N)) and ExtH*(A)(H (M); H (N)): Thus our new derived categories of modules over A1 and E1 algebras enjoy all of* * the basic properties of the derived categories of modules over DGA's and commutative DGA's. In view of the unfamiliarity of the constructions in Part V, we should perha* *ps say something about our philosophy. In algebraic topology, it has long been standa* *rd practice to work in the stable homotopy category. This category is hard to cons* *truct rigorously, and its objects are hard to think about on the point-set level. (Al* *though the definitional framework in algebraic geometry is notoriously abstract, the o* *bjects that algebraic geometers usually deal with are much more concrete than the spec* *tra of algebraic topology.) However, once the machinery is in place, the stable hom* *otopy category gives an enormously powerful framework in which to perform explicit ca* *lcula- tions. It may be hoped that our new algebraic derived categories will eventuall* *y serve something of the same purpose. Actually, the analogy with topology is more far-reaching. There are analogs* * of E1 algebras in stable homotopy theory, namely the E1 ring spectra that were i* *n- troduced in [47]. With Elmendorf [25], we have worked out a theory of module sp* *ectra over A1 and E1 ring spectra that is precisely parallel to the algebraic theory * *of Part V. Although it is much more difficult, its constructive and calculational power ar* *e already evident. Basic spectra that previously could only be constructed by the Baas-Su* *llivan theory of manifolds with singularities are easily obtained from the theory of m* *odules over the E1 ring spectrum MU that represents complex cobordism. Spectral sequen* *ces that are the precise analogs of the Eilenberg-Moore (or hyperhomology) spectral* * se- quences in Part V include K"unneth and universal coefficient spectral sequences* * that are of clear utility in the study of generalized homology and cohomology. Some * *other applications were announced in [24], and many more are now in place. An exposit* *ion of the analogy between the algebraic and topological theories is given in [51]. Parts II and V constitute a revision and expansion of material in the prepri* *nt [37], which had a rather different perspective. That draft was intended to lay founda* *tions for work in both algebra and topology, but it has since become apparent that, d* *espite the remarkably close analogy between the two theories and the resulting exposit* *ory duplication, the technical differences dictate separate and self-contained trea* *tments. Some of the present results were announced in [38]. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 9 LOGICAL DEPENDENCIES OF THE PARTS: Each Part has its own introduction, and we have tried to make them readable independently of one another. Except th* *at the examples that motivate Parts IV and V are constructed in Part II and that P* *art V makes peripheral use of some general definitions in Part II, the logical depenc* *ies are I @ IIIE "" @@@ --- EEE """ @@ --- EEE """ @ -- E II V IV: In particular, Parts III and IV have nothing to do with operads. REFERENCES: A reference of the form "II.m.n" is to statement m.n in Part II; within Part II, the reference would be to "m.n". THE GROUND RING: We shall work over a fixed commutative ground ring k. There are no restrictions on k in Parts I, III, and V; k will be assumed to be a Dede* *kind ring in Part II and to be a field of characteristic zero in Part IV. CONVENTIONS ON k-MODULES: Except in Part II, a "k-module" will mean a differential Z-graded chain complex over k. In Part II, we use the term k-compl* *ex for this notion and we use the term k-module in its classical ungraded sense. In P* *arts I and II, the grading is homological and differentials will lower degree. In P* *arts III- V, the grading is cohomological and differentials will raise degree. In Parts I* *I-V, we usually assume given an auxiliary "Adams grading" on k-modules and k-complexes; differentials will preserve this grading. We wish to thank many people who have taken an interest in this work. Part I * *can serve as an introduction not only to this paper, but also to the closely relate* *d papers of Ginzburg and Kapranov [29], Getzler and Jones [27, 28], and Hinich and Schechtm* *an [31, 32]. Some of the more interesting insights in Part I are due to these auth* *ors, and we are grateful to them for sharing their ideas with us. The second author wish* *es to take this opportunity to offer his belated thanks to Max Kelly and Saunders Mac* *Lane for conversations in 1970-71. Discussions then about operads in symmetric monoi* *dal categories are paying off now. We are also very grateful to Jim Stasheff, who a* *lerted us to how seriously operads are being used in mathematical string theory, urged* * us to give the general exposition of Parts I and II, and offered helpful criticism of* * preliminary versions. We also thank Spencer Bloch for detecting an error in the first vers* *ion of Part II and for spirited discussions about motives. We are especially grateful* * to our collaborator Tony Elmendorf; the original version of the theory in Part V was f* *ar more complicated, and this material has been reshaped by the insights developed in o* *ur parallel topological work with him. It is a pleasure to thank Deligne for his l* *etters that led to this paper and for his suggestions for improving its exposition. 10 IGOR KRIZ AND J. P. MAY Part I. Definitions and examples of operads and operad actions We define operads in Section 1, algebras over operads in Section 2, and modu* *les over algebras over operads in Section 4, giving a number of variants and examples. T* *he term "operad" is meant to bring to mind suitably compatible collections of j-ary pro* *duct operations. It was coined in order to go well with the older term "monad" (= tr* *iple), which specifies a closely related mathematical structure that has a single prod* *uct. As we explain in Section 3, operads determine associated monads in such a way that* * an algebra over an operad is the same thing as an algebra over the associated mona* *d. While not at all difficult, this equivalence of definitions is central to the t* *heory and its applications. Section 4 includes a precisely analogous description of modules a* *s algebras over a suitable monad, together with a quite different, and more familiar, desc* *ription as ordinary modules over universal enveloping algebras. Both points of view are es* *sential. In Section 5, we discuss the passage from topological operads and monads to * *algebraic operads and monads via chain complexes and homology. We speculate that similar * *ideas will have applications to other situations, for example in algebraic geometry, * *where one may encounter operads in a category that has a suitable homology theory defined* * on it. In Section 6, we specialize to the little n-cubes operads Cn. These arose i* *n iterated loop space theory and are now understood to be relevant to the mathematics of s* *tring theory. We show that H*(Cn) contains a suboperad which, when translated to degr* *ee zero, is isomorphic to the operad that defines Lie algebras, and we observe tha* *t work in Cohen's 1972 thesis [15, 16] implies that the full operad H*(Cn) defines n-brai* *d algebras. While current interest focuses on characteristic zero information, we shall giv* *e some indications of the deeper mod p theory. In particular, in Section 7, we shall d* *escribe the Dyer-Lashof operations that are present on the mod p homologies of E1 algeb* *ras. Such operations are central to infinite loop space theory, and our later work w* *ill indicate that they are also relevant to the mod p higher Chow groups in algebraic geomet* *ry. 1. Operads We work in the tensor category of differential Z-graded modules over our gro* *und ring k, with differential decreasing degree by 1. Thus will always mean k. Rea* *ders who prefer the opposite grading convention may reindex chain complexes C* by se* *tting Cn = C-n. While homological grading is most convenient in Parts I and II, we sh* *all find it convenient to switch to cohomological grading in the later Parts. We refer t* *o Z-graded chain complexes over k as "k-modules" in this Part. As usual, we consider grad* *ed k-modules without differential to be k-modules with differential zero, and we v* *iew ungraded k-modules as graded k-modules concentrated in degree 0. These conventi* *ons OPERADS, ALGEBRAS, MODULES, AND MOTIVES 11 allow us to view the theory of generalized algebras as a special case of the th* *eory of differential graded generalized algebras. The differentials play no significant* * role in the theory of the first four sections. As will become relevant in Part II, everythi* *ng in these sections works just as well in the more general context of simplicial k-complex* *es. We begin with the definition of an operad of k-modules. While there are perh* *aps more elegant equivalent ways of writing the definition, the original explicit v* *ersion of [46] still seems to be the most convenient, especially for concrete calculation* *al purposes. Whenever we deal with permutations of k-modules, we implicitly use the standard convention that a sign (-1)pq is to be inserted whenever an element of degree p* * is permuted past an element of degree q. Definition 1.1.An operad C consists of k-modules C (j), j 0, together with a u* *nit map j : k ! C (1), a right action by the symmetric group j on C (j) for each j,* * and maps fl : C (k) C (j1) . . .C (jk) ! C (j) P for k 1 and js 0, where js= j. The fl are required to be associative, unital* *, and equivariant in the following senses. P P (a)The following associativity diagrams commute, where js= j and it= i; * *we set gs= j1+ . .+.js, and hs= igs-1+1+ . .+.igsfor 1 s k: Ok Oj flId Oj C (k) ( C (js)) ( C (ir))_______//C (j) ( C (ir)) s=1 r=1 r=1 | |fl | | | fflffl| | shuffle| C (i) | OO | | | |fl fflffl| | Ok Ojs Ok C (k) ( (C (js) ( C (igs-1+q)))___//_C (k) ( C (hs)): s=1 q=1 Id(sfl) s=1 (b)The following unit diagrams commute: ~= ~= C (k) (k)k____//C8(k)8 k C (j)______//C (j) qqq qq88q Idjk || qflqqqq jId || qflqqqq fflffl|qqq fflffl|qqq C (k) C (1)k C (1) C (j): (c)The following equivariance diagrams commute, where oe 2 k; os 2 js, the permutation oe(j1; : :;:jk) 2 k permutes k blocks of letter as oe permute* *s k 12 IGOR KRIZ AND J. P. MAY letters, and o1 . . .ok 2 k is the block sum: oeoe-1 C (k) C (j1) . . .C (jk)_//C (k) C (joe(1)) . . .C (joe(k)) fl|| |fl| |fflffl oe(joe(1);:::;joe(kfflffl|)) C (j)________________________//C (j) and Ido1...ok C (k) C (j1) . . .C (jk)______//C (k) C (j1) . . .C (jk) fl|| fl|| fflffl| o1...ok fflffl| C (j)___________________________//C (j): The C (j) are to be thought of as modules of parameters for "j-ary operation* *s" that accept j inputs and produce one output. Thinking of elements as operations, we * *think of fl(c d1 . . .dk) as the composite of the operation c with the tensor produ* *ct of the operations ds. We emphasize that the definition makes sense in any symme* *tric monoidal ground category, with product and unit object k. In the present algeb* *raic context, the unit map j is specified by a degree zero cycle 1 2 C (1). The defi* *nition admits several minor variants and particular types. Recall that a map of k-modu* *les is said to be a quasi-isomorphism if it induces an isomorphism of homology groups. Variants 1.2. (i) Non- operads. When modelling non-commutative algebras, it is often useful to omit the permutations from the definition, giving the notion of* * a non- operad. However, one may also keep the permutations in such contexts, using the* *m to record the order in which products are taken. An operad is a non- operad by neg* *lect of structure. (ii) Unital operads. By convention, the 0thtensor power of a k-module A is i* *nterpreted to be k (concentrated in degree 0). The module C (0) parametrizes "0-ary operat* *ions" k ! A. In practice, one is most often concerned with unital algebras, and one t* *hinks of the unit element 1 2 A as specifying a map k ! A. In such contexts, it is se* *nsible to insist that C (0) = k, and we then say that C is a unital operad. For types of * *algebras without units, such as Lie algebras, it is natural to set C (0) = 0. (iii) Augmentations. If C is unital, the C (j) have the augmentations ffl = fl : C (j) ~=C (j) C (0)j! C (0) = k: Definition 1.3.Let C be a unital operad. We say that C is acyclic if its augmen* *tations are quasi-isomorphisms. We say that C is -free (or -projective) if C (j) is k[j* *]-free (or k[j]-projective) for each j. We say that C is an E1 operad if it is both ac* *yclic and -free; C (j) is then a k[j]-free resolution of k. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 13 Example 1.4. An explicit example of an E1 operad C can be obtained as follows. There is a standard product-preserving functor D* from sets to contractible sim* *plicial sets [46, x10]. The set Dq(X) of q-simplices of D*(X) is the (q +1)-fold Cartes* *ian power Xq+1; the faces and degeneracies are given by projections and diagonal maps. Fo* *r a group G, D*(G) is a simplicial free G-set (and a simplicial group) , and its no* *rmalized k-chain complex is the classical homogeneous bar resolution for the group ring * *k[G] (e.g. [14, p. 190]). Letting C (j) be the normalized k-chain complex of D*(j), * *we can use functoriality to construct structural maps fl making C an E1 operad. Passage to normalized singular k-chain complexes from E1 operads of spaces gi* *ves other examples; see Section 5. 2. Algebras over operads Let Xj denote the j-fold tensor power of a k-module X, with j acting on the l* *eft. Again, X0 = k. (We shall never use Cartesian powers in the algebraic context.) Definition 2.1.Let C be an operad. A C -algebra is a k-module A together with m* *aps : C (j) Aj! A for j 0 that are associative, unital, and equivariant in the following senses. P (a)The following associativity diagrams commute, where j = js: flId j C (k) C (j1) . . .C (jk) Aj_____//_C (j) A | | | | | | fflffl| shuffle|| AOO | | | | fflffl| Idk | C (k) C (j1) Aj1 . . .C (jk) Ajk___//C (k) Ak: (b)The following unit diagram commutes: ~= k A _____//_A::u uuu jId || uuuu fflffl|uu C (1) A: (c)The following equivariance diagrams commute, where oe 2 j: oeoe-1 j C (j) Aj___________//C (j) A JJ tt JJJ ttt flJJJJ ttflt %%Jyytt A: 14 IGOR KRIZ AND J. P. MAY One way to motivate the precise data in the definition is to define the endo* *morphism operad End(X) of a k-module X. For k-modules X and Y , let Hom(X; Y ) be the k-module whose elements of degree n are the homomorphisms f : X ! Y of graded k-modules (not commuting with differential) that raise degree by n. The differe* *ntial is specified by (df)(x) = d(f(x)) - (-1)nf(d(x)): If Mk denotes the category of k-modules and maps of degree 0, then Mk(X Y; Z) ~=Mk(X; Hom(Y; Z)): Now define End (X)(j) = Hom (Xj; X): The unit is given by the identity map X ! X, the right actions by symmetric gro* *ups are given by their left actions on tensor powers, and the maps fl are given by the * *following P composites, where js= j: Hom (Xk; X) Hom (Xj1; X) . . .Hom(Xjk; X) Id(k-fold|tensor|product of maps) fflffl| Hom (Xk; X) Hom (Xj; Xk) composition|| fflffl| Hom (Xj; X): Conditions (a)-(c) of Definition 1.1 are then forced by direct calculation. An * *action of C on A can be redefined in adjoint form as a morphism of operads C ! End(A), and conditions (a)-(c) of Definition 2.1 are then also forced by direct calculation. Examples 2.2. (i) The unital operad M has M (j) = k[j] as a right k[j]-module (concentrated in degree 0). The unit map j is the identity and the maps fl are * *dictated by the equivariance formulas of Definition 1.1(c). Explicitly, for oe 2 k and o* *s2 js, fl(oe; o1; : :;:ok) = oe(j1; : :;:jk)(o1 . . .ok): An M -algebra A is the same thing as a "DGA", that is, a unital and associative differential graded algebra. The action on a DGA A is given by the explicit fo* *rmula (oe a1 . . .aj) = aoe(1) . . .aoe(j); where oe 2 j and ai2 A. (The sign is given by our standing convention.) (ii) The unital operad N has N (j) = k for all j. The j-actions are trivial,* * the unit map j is the identity, and the maps fl are the evident identifications. An N -a* *lgebra is the same thing as a commutative DGA. If we regard N as a non- operad and delete OPERADS, ALGEBRAS, MODULES, AND MOTIVES 15 the equivariance diagram from Definition 2.1, then the resulting notion of an N* * -algebra is again a not necessarily commutative DGA. (iii) For a unital operad C , the augmentations ffl : C (j) ! k give a map ff* *l : C ! N of operads. Therefore, by pullback along ffl, an N -algebra may be viewed as a C -* *algebra. (iv) We define an E1 algebra to be a C -algebra for any E1 operad C . We do n* *ot insist on a particular choice of C . Hinich and Schechtman [31] studied algebra* *s of this type, which they called "May algebras". One can treat operads as algebraic systems to which one can apply versions of* * classical algebraic constructions. An ideal I in an operad C consists of a sequence of su* *b k[j]- modules I (j) of C (j) such that fl(c d1 . . .dk) is in I if either c or any o* *f the ds is in I . There is then a quotient operad C =I with jthk-module C (j)=I (j).* * As observed by Ginzburg and Kapranov [29], one can adapt work of Boardman and Vogt [11, 11x2] to construct the free operad F G generated by any sequence G = {G (j* *)} of k[j]-modules, and one can then construct an operad that describes a particul* *ar type of algebra by quotienting out by the ideal generated by an appropriate seq* *uence R = {R(j)} of defining relations, where R(j) is a sub k[j]-module of (F G )(j). Actually, there are two variants of the construction, one unital and one non-un* *ital. In many familiar examples, called quadratic operads in [29], G (j) = 0 for j * *6= 2 and R(j) = 0 for j 6= 3. Here, if G (2) is k[2] and R(3) = 0, this reconstructs M * *. If G (2) = k with trivial 2-action and R(3) = 0, this reconstructs N . In these ca* *ses, we use the unital variant. If k is a field of characteristic other than 2 or 3, we* * can use the non-unital variant to construct an operad L whose algebras are the Lie algebras* * over k. To do this, we take G (2) = k, with the transposition in 2 acting as -1, and* * take R(3) to be the space (F G )(3)3 of invariants, which is one dimensional. Basis * *elements of G (2) and R(3) correspond to the bracket operation and the Jacobi identity. * *As we explain in Section 6, L can be realized homologically by the topological little* * n-cubes operads for any n > 1. Various other examples of quadratic operads are describ* *ed in [29]. Note that, in these "purely algebraic" examples, all C (j) are concent* *rated in degree zero, with zero differential. The definition of a Lie algebra over a field k requires the additional relati* *ons [x; x] = 0 if char(k) = 2 and [x; [x; x]] = 0 if char(k) = 3. Purely algebraic operads are* * not well adapted to codify such relations with repeated variables, still less such nonli* *near oper- ations as the restriction (or pthpower operation) of restricted Lie algebras in* * charac- teristic p. The point is simply that the elements of an operad specify operatio* *ns, and operations by their nature cannot know about special properties (such as repeti* *tion) of the variables to which they are applied. 16 IGOR KRIZ AND J. P. MAY As an aside, since in the absence of diagonals it is unclear that there is a* * workable algebraic analog, we note that a topological theory of E1 ring spaces has been * *developed in [49]. The sum and product, with the appropriate version of the distributive * *law, are codified in actions by two suitably interrelated operads. Remarks 2.3.(i) A k-module X also has a "co-endomorphism operad" Co-End(X); its jth k-module is Hom(X; Xj), and its structural maps are given in an evident* * way by composition and tensor products. We define a coaction of an operad C on a k- module X to be a map of operads C ! Co-End(X); such an action is given by suita* *bly interrelated maps C (j) X ! Xj. (ii) We have defined operads in terms of maps. If we reverse the direction o* *f every arrow in Definition 1.1, we obtain the dual notion of "co-operad". Similarly, i* *f we reverse the direction of every arrow in Definition 2.1, we obtain the notion of a coalg* *ebra over a co-operad. Again, if we reverse the direction of every arrow in Definition 4.* *1 below, we obtain the notion of a comodule over such a coalgebra. 3.Monadic reinterpretation of algebras We recall some standard categorical definitions. Definition 3.1.Let G be any category. A monad in G is a functor C : G ! G together with natural transformations : CC ! C and j : Id ! C such that the following diagrams commute: jC Cj C C ____//CCCoo_C and CCC ____//CC CC | --- | | CCC | --- C | | Id C!!Cfflffl|""Id-- fflffl| fflffl| C CC _____//_C: A C-algebra is an object A of G together with a map : CA ! A such that the following diagrams commute: j C A ____//CCA and CCA ____//CA CC | | | CCC | | | IdCC!!fflffl| fflffl| fflffl| A CA _____//_A: Taking = , we see that CX is a C-algebra for any X 2 G . It is the free C-a* *lgebra generated by X. That is, for C-algebras A, restriction along j : X ! CX gives * *an adjunction isomorphism (3.2) C[G ](CX; A) ~=G (X; A); where C[G ] is the category of C-algebras. The inverse isomorphism assigns the * *com- posite O Cf : CX ! A to a map f : X ! A. Formally, we are viewing C as a funct* *or OPERADS, ALGEBRAS, MODULES, AND MOTIVES 17 G ! C[G ], and our original monad is given by its composite with the forgetful * *functor C[G ] ! G . Thus the monad C is determined by its algebras. Quite generally, ev* *ery pair L : G ! H and R : H ! G of left and right adjoints determines a monad RL on G , but many different pairs of adjoint functors can define the same monad. Returning to the category of k-modules, we have the following simple construc* *tion of the monad of free algebras over an operad C . Definition 3.3.Define the monad C associated to an operad C by letting M CX = C (j) k[j]Xj: j0 The unit j : X ! CX is j Id : X = k X ! C (1)X and the map : CCX ! CX P is induced by the maps (j = js) C (k) C (j1) Xj1 . . .C (jk) Xjk shuffle|| fflffl| C (k) C (j1) . . .C (jk) Xj flId|| fflffl| C (j) Xj: Proposition 3.4.A C -algebra structure on a k-module A determines and is deter- mined by a C-algebra structure on A. Formally, the identity functor on the cate* *gory of k-modules restricts to give an isomorphism between the categories of C -algebra* *s and of C-algebras. Proof.Maps j : C (j) j Aj ! A that together specify an action of C on A are the same as a map : CA ! A that specifies an action of C on A. __|_| Not all monads come from operads. Rather, operads single out a particularly c* *onve- nient, algebraically manageable, collection of monads. For the operad M , the free algebra MX is just the free associative k-algebra* * gener- ated by X, with the differential induced from that of X. Similarly, for the ope* *rad N , the free algebra NX is the free associative and commutative algebra generated b* *y X, with its induced differential. Again, for the operad L , we obtain the free Lie* * algebra functor L. While these observations can be checked by observation, they are als* *o formal consequences of the freeness adjunction (3.2). Some less obvious examples are d* *iscussed in Section 6 and are generalized to situations of particular interest in string* * theory in [27, 28]. In the rest of this section, we suppose that C is a unital operad. In this c* *ase, there is a monad that is different from that defined above but that nevertheles* *s has 18 IGOR KRIZ AND J. P. MAY essentially the same algebras. Since C is unital, a C -algebra A comes with a * *unit j 0 : k ! A. Thinking of the unit as preassigned, it is natural to change grou* *nd categories to the category of unital k-modules and unit-preserving maps. Workin* *g in this ground category, we obtain a reduced monad "C. This monad is so defined th* *at the units of algebras that are built in by the 0 component of operad actions coinci* *de with the preassigned units j. In detail, note that we have "degeneracy maps" oei: C (j) ! C (j - 1) specif* *ied by (3.5) oei(c) = fl(c 1i-1 * 1j-i) for 1 i j, where 1 denotes j(1) in C (1) and * denotes the identity element in k = C (0). For a unital k-module X with unit 1, define "CX to be the quotient o* *f CX obtained by the identifications (3.6) c x1 . . .xi-1 1 xi+1 . . .xj= oei(c) x1 . . .xi-1 xi+1 . . .xj for 1 i j. With unit map j and product map induced from those of the monad C, "Cis a monad in the category of unital k-modules. Proposition 3.7. Let C be a unital operad. Then a C -algebra structure satisfy* *ing j = 0 on a unital k-module A determines and is determined by a "C-algebra struc* *ture on A. The proof is immediate from Proposition 3.4 and the definitions. With a sli* *ght restriction, the monads C and "Cdetermine each other. Define an augmentation of* * a unital k-module X to be a map ffl : X ! k whose composite with the unit is the * *identity. Proposition 3.8. (i) For a k-module X, let X+ be the unital k-module X k. Then CX ~="C(X+) as C -algebras. (ii) For an augmented k-module X, let "Xbe the k-module Ker(ffl). Then "CX ~=CX* *" as C -algebras. Proof. Part (i) can be viewed as a special case of part (ii). For (ii), the com* *posite of the map CX" ! CX induced by the inclusion "X! X and the quotient map CX ! "CX gives the required isomorphism, and the following diagrams commute: ~= "X_____//_X CCX" ____//CC"X____//"C"CX j || j|| || || fflffl|~fflffl|= fflffl| ~= fflffl| __ CX" ____//"CX CX" ______________//_"CX: |_| OPERADS, ALGEBRAS, MODULES, AND MOTIVES 19 There is an obvious analogy with the adjunction of a disjoint basepoint to a * *space X to obtain a space X+ such that H*(X) ~=H"*(X+). In the original topological the* *ory of [46], all operads were unital and the reduced topological monad "Cassociated* * to an operad C was denoted C. In that context, as we shall recall in Sections 5 and 6* *, there is a great difference in homotopy types between "Cand C, with "Cbeing by far th* *e more interesting construction. While there is an evident topological analog of the f* *irst part of the previous proposition, there is no analog of the second part: topological* *ly, the reduced construction is strictly more general. In the preprint version of this * *paper [37], C"was denoted by C. We have followed a suggestion of Deligne in placing the emp* *hasis on the simpler construction C in the present algebraic context. 4.Modules over C -algebras Fix an operad C and a C -algebra A. Definition 4.1.An A-module is a k-module M together with maps : C (j) Aj-1 M ! M for j 1 that are associative, unital, and equivariant in the following sense. P (a)The following associativity diagrams commute, where j = js: Ok flId (C (k) ( C (js))) Aj-1 M________________//C (j) Aj-1 M s=1 | | | | | | | fflffl| shuffle|| MOO | | | | fflffl| | k-1O | C (k) ( (C (js) Ajs)) (C (jk) Ajk-1_M)k-1___//C (k) Ak-1 M: s=1 Id (b)The following unit diagram commutes: ~= k M _____//_M99t ttt jId|| tttt fflffl|tt C (1) M: (c)The following equivariance diagram commutes, where oe 2 j-1 j: oeoe-1Id j-1 C (j) Aj-1OM ___________//_C (j) A M OOOO ooooo OOOO oooo OO''O wwooo M: 20 IGOR KRIZ AND J. P. MAY A map f : M ! N of k-modules between A-modules M and N is a map of A-modules if the following diagram commutes for each j 1: C (j) Aj-1 M ___//_M | IdIdf || |f| fflffl| fflffl| C (j) Aj-1 N ____//_N: We think of these as left modules. However, motivated by the first of the fo* *llowing examples, one can also think of them as bimodules [29]. Examples 4.2. (i) For an M -algebra A, an A-module M in our sense is the same * *as an A-bimodule in the classical sense. Precisely, given the maps , we define am = (e a m) and ma = (oe a m); where e and oe are the identity and transposition in 2. Conversely, just as in * *Example 2.2(i), given an A-bimodule M, we define (oe a1 . . .aj) = aoe(1).a.o.e(j); where oe 2 j, ai2 A for 1 i < j and aj2 M. (ii) For an N -algebra A, an A-module in our sense is the same as an A-modul* *e in the classical sense. If we use N regarded as a non- operad to define non-commut* *ative algebras and delete part (c) of the definition, then a module over an N -algebr* *a A is a classical left A-module. (iii) For an L -algebra L, an L-module in our sense is the same as a Lie alg* *ebra module in the classical sense. Just as for algebras, modules admit a monadic reinterpretation. Definition 4.3.For k-modules X and Y , define M C(X; Y ) = C (j) k[j-1]Xj-1 Y: j1 Define j : Y ! C(X; Y ) to be j Id: Y = k Y ! C (1) Y and define : C(CX; C(X; Y )) ! C(X; Y ) OPERADS, ALGEBRAS, MODULES, AND MOTIVES 21 P to be the map induced by the following composites (j = js): C (k) C (j1) Xj1 . . .C (jk-1) Xjk-1 C (jk) Xjk-1 Y shuffle|| fflffl| C (k) C (j1) . . .C (jk) Xj-1 Y flId|| fflffl| C (j) Xj-1 Y: Define a monad C[1] in the category of pairs (X; Y ) by letting C[1](X; Y ) = (CX; C(X; Y )): The unit j and product of C[1] are given by the evident pairs (j; j) and (; ). Proposition 4.4.A C -algebra structure on a k-module A together with an A-module structure on a k-module M determine and are determined by a C[1]-algebra struct* *ure on the pair (A; M). Formally, the identity functor on the category of pairs of * *k-modules restricts to an isomorphism between the evident category of C -algebras togethe* *r with modules and the category of C[1]-algebras. When C is unital, there is a similar reduced monad "C[1] in the category of p* *airs (X; Y ), where X is a unital k-module and Y is an arbitrary k-module. Explici* *tly, define "C(X; Y ) to be the quotient of C(X; Y ) obtained by the identifications (4.5) c x1 . . .xi-1 1 xi+1 . . .xj= oei(c) x1 . . .xi-1 xi+1 . . .xj for 1 i < j, where xi2 X if i < j and xj2 Y . Then define "C[1](X; Y ) = (C"X; "C(X; Y )): The unit map j and product map are induced from those of C[1]. Proposition 4.6.Let C be a unital operad. A C -algebra structure satisfying j =* * 0on a unital k-module A together with an A-module structure on a k-module M determi* *nes and is determined by a "C[1]-algebra structure on the pair (A; M). Proposition 4.7.(i) For k-modules X and Y , C(X; Y ) ~="C(X+; Y ). (ii) For an augmented k-module X and a k-module Y , "C(X; Y ) ~=C(X"; Y ). Observe that free objects in our monadic context are pairs (CX; C(X; Y )), wh* *ere C(X; Y ) is a module over CX. Formally, we can rewrite the present instance of * *the freeness adjunction (3.2) in the form C[1][Mk2]((CX; C(X; Y )); (A; M)) ~=Mk2((X; Y ); (A; M)); 22 IGOR KRIZ AND J. P. MAY where Mk2denotes the category of pairs of k-modules. Of course, this is quite different from fixing an algebra A and constructing* * free A- modules F Y = F (A; Y ). Such a free module functor F is characterized by an ad* *junction Hom A(F Y; M) ~=Hom (Y; M) relating maps of A-modules and maps of k-modules. We shall construct the free * *A- module functor F (A; ?) for an algebra A over an operad C in a moment, and we w* *ill then have the following formal comparison of definitions. Proposition 4.8. For any operad C and any k-modules X and Y , C(X; Y ) is isomo* *r- phic to the free CX-module F (CX; Y ) generated by Y . Proof. The forgetful functor C[1][Mk2] ! Mk2factors through the category of pai* *rs (A; Y ), where A is a C -algebra and Y is a k-module. That is, we can first for* *get the module structure on the second coordinate and then forget the algebra structure* * on the first coordinate. These two forgetful functors have left adjoints (Id; F (I* *d; ?)) and (C; Id). Their composite must coincide with C[1] by the uniqueness of adjoints.* * __|_| With the morphisms of Definition 4.1, it is clear that the category of A-mod* *ules is abelian. In fact, as was observed in [29] and [32], it is equivalent to the ca* *tegory of modules over the universal enveloping algebra U(A) of A. Of course, at our pre* *sent level of generality, U(A) must be a DGA. This gives us the free A-module functo* *r F just asked for as the ordinary free U(A)-module functor. The definition of U(A) is f* *orced by Definition 4.1. Definition 4.9.Let A be a C -algebra. The action maps : C (j) Aj-1 M ! M of an A-module M together define an action map : C(A; k) M = C(A; M) ! M: Thus C(A; k) may be viewed as a k-module of operators on A-modules. The free DGA M(C(A; k)) generated by C(A; k) therefore acts iteratively on all A-modules. De* *fine the universal enveloping algebra U(A) to be the quotient of M(C(A; k)) by the i* *deal of universal relations. Explicitly, reading off from Definition 4.1, the elemen* *t 1 2 C (1) must be identified with the unit element of the algebra and the element fl(d c1 . . .ck) a1 . . .aj-12 C (j) Aj-1 must be identified with the product [d (c1; b1) . . .(ck-1; bk-1)][ck bk] 2 [C (k) Ak-1][C (jk) Ajk-1]; OPERADS, ALGEBRAS, MODULES, AND MOTIVES 23 where d 2 C (k), cs 2 C (js), ai2 A, and bs is the tensor product of the sth bl* *ock of a's; bs has js tensor factors if s < k and jk- 1 factors if s = k. Taking ci= 1* * for i < k and changing notation, we obtain the relation [c a1 . . .aj][d a01 . . .a0k] = fl(c 1j d) a1 . . .ak a01 . . .a0j for c 2 C (j +1) and d 2 C (k +1). Reinterpreting this formula as a product on * *C(A; k), we see that U(A) can be described more economically as the quotient of the alge* *bra C(A; k) by the relations originally specified. The following result is immediate from the definition. Proposition 4.10.The category of A-modules is isomorphic to the category of U(A* *)- modules. It is an illuminating and not quite trivial exercise to check the first of th* *e following examples from the explicit relations just specified. Examples 4.11. (i) For an M -algebra A, U(A) is isomorphic to A Aop. (ii) For an N -algebra A, U(A) is isomorphic to A. (iii) For an L -algebra L, U(L) is isomorphic to the classical universal enve* *loping algebra of L. In Part V, we shall construct a derived tensor product on modules over an E1 * *algebra A. From the universal enveloping algebra point of view, this should look most i* *mplau- sible: a U(A)-module is just a left module, and, since U(A) is far from being c* *ommuta- tive, there is no obvious way to define a tensor product of A-modules, let alon* *e a tensor product that is again a module. 5.Algebraic operads associated to topological operads Recall that operads can be defined in any symmetric monoidal category, such a* *s the category of topological spaces under Cartesian product. Thus an operad S of spa* *ces consists of spaces S (j) with right actions of j, a unit element 1 2 S (1), and* * maps fl : S (k) x S (j1) x . .x.S (jk) ! S (j) such that associativity, unity, and equivariance diagrams precisely like those * *in Defini- tion 1.1 commute. For definiteness, we assume that S (0) is a point. Via the singular complex functor, an operad of topological spaces gives rise * *to an operad of simplicial sets. Via the free k-module functor, an operad of simplici* *al sets gives rise to an operad of simplicial ungraded k-modules. By passage to normal* *ized chains, which we denote by C#, an operad of simplicial ungraded k-modules gives* * rise to an operad of k-modules in our original sense of chain complexes over k. The * *proof 24 IGOR KRIZ AND J. P. MAY of the last assertion depends on the associativity and commutativity of the sta* *ndard shuffle quasi-isomorphism (e.g. [44, x29], or [30, Appendix]) C#(X) C#(Y ) ! C#(X x Y ): Therefore the normalized singular k-chain functor restricts to a functor from o* *perads of spaces to operads of k-modules. We write C#(S ) for the operad of k-modules associated to an operad S of spaces. The operad S is said to an E1 operad if each space S (j) is j-free and contr* *actible (a universal j-bundle), and C#(S ) is then an E1 operad in the sense of Definit* *ion 1.3. Similarly, the chain functor C# carries S -algebras (= S -spaces) to C#(S )-alg* *ebras and carries modules over an S -algebra to modules over the associated C#(S )-al* *gebra. Following [29, 27, 28] and others, we can go further and define homology ope* *rads. We take k to be a field in the rest of this section, and all homology groups ar* *e to be taken with coefficients in k. Definition 5.1.Let S be an operad of spaces. Define H*(S ) to be the unital ope* *rad whose jth k-module is the graded k-module H*(S (j)), with algebraic structure m* *aps fl induced by the topological structure maps. For n 0, define Hn(S ) to be the suboperad of H*(S (j)) whose jth k-module is Hn(j-1)(S (j)) for j 0; in partic* *ular, the 0th k-module is zero unless n = 0. The degrees are so arranged that the def* *inition makes sense. We retain the grading that comes naturally, so that the jth term * *of Hn(S ) is concentrated in degree n(j - 1). We obtain a "degree zero translate" * *operad associated to Hn(S ) by regrading so that all terms are concentrated in degree * *zero. If the spaces S (j) are all connected, then H0(S ) = N and H*(X) is a commut* *ative algebra for any S -space X. If the spaces S (j) are all contractible, for exam* *ple if S is an E1 operad, then H*(S ) = N . Thus, on passage to homology, E1 operads record only the algebra structure on the homology of S -spaces, although the ch* *ain level operad action gives rise to the homology operations discussed in Section * *7. It is for this reason that topologists did not formally introduce homology operads de* *cades ago. In fact, there is a sharp dichotomy between the calculational behavior of op* *erads in characteristic zero and in positive characteristic. The depth of the original t* *opological theory lies in positive characteristic, where passage to homology operads jetti* *sons most of the interesting structure. In characteristic zero, in contrast, the homology* * operads completely determine the homology of the monads S and "Sassociated to an operad* * S . Here, for a space X, a SX = S (j) xj Xj: OPERADS, ALGEBRAS, MODULES, AND MOTIVES 25 For a based space X, "SX is the quotient of SX obtained by basepoint identifica* *tions, exactly as in (3.6). The space "SX has a natural filtration with successive quo* *tients S (j)+ ^j X(j); where X(j)denotes the j-fold smash power of X. (X ^ Y is the quotient of the pr* *oduct X x Y obtained by identifying the wedge X _ Y to a point.) The calculational difference comes from a simple general fact: if a finite gr* *oup ss acts on a space X, then, with coefficients in a field of characteristic zero, H*(X=s* *s) is natu- rally isomorphic to H*(X)=ss. (We are assuming that our spaces are not patholog* *ical; for example, they may be ss-CW complexes.) In fact, H*(X=ss) is a homology theo* *ry on X _ this being true in any characteristic _ and H*(x)=ss is a homology theor* *y on X since the functor M=ss = M k[ss]k on k[ss]-modules M is exact (e.g. because k* * is a direct summand of k[ss]). It is obvious that these theories agree on orbits s* *s=ae, and it follows exactly as in nonequivariant algebraic topology that they are isomor* *phic. In the cases of interest to us, the shuffle map induces a chain map (*) C#(S (j)) j C#(X)j! C#(S (j) xj Xj); from which we obtain an instance of our general isomorphism on passage to homol* *ogy, and similarly for S (j)+ ^j X(j). This leads to the following result. Theorem 5.2. Let S be any operad of spaces. Let S denote both the monad in the category of spaces associated to S and the monad in the category of k-modul* *es associated to H*(S ). Similarly, let "Sdenote both the monad in the category of* * based spaces associated to S and the monad in the category of unital k-modules associ* *ated to H*(S ). If k is a field of characteristic zero, then H*(SX) ~=SH*(X) and H*(S"X) ~="S(H*(X)) as H*(S )-algebras for all spaces X (based spaces in the reduced case). Proof.On passage to homology, the unit X ! SX and the action of S on SX induce the composite map ff : S(H*(X)) ! S(H*(SX)) ! H*(SX) of H*(S )-algebras. Similarly, in the reduced case we have a composite "ff: "S(H*(X)) ! "S(H*(S"X)) ! H*(S"X): In the unreduced case, ff is the direct sum of isomorphisms induced by the chai* *n maps (*). For the reduced case, observe that if V is an augmented k-module, then th* *e k- module "SV has an evident filtration with successive quotients H*(S (j)) j V"j.* * The map "ffis filtration-preserving, and its successive quotients are isomorphisms 26 IGOR KRIZ AND J. P. MAY H*(S (j)) j H"*(X)j~= H*(S (j)+ ^j X(j)) induced by the chain maps (*). Therefore "ffis an isomorphism by induction up * *the filtration and passage to colimits. __|_| This allows us to realize free algebras topologically. For example, we have * *the obvious topological (actually, discrete) versions of the operads M and N , with M (j) =* * jand N (j) a point. For a based space X, M"X is the James construction (or free topo* *logical monoid) on X, and it is homotopy equivalent to X if X is connected. Similarly, * *"NX is the infinite symmetric product (or free commutative topological monoid) on X* *, and it is homotopy equivalent to the product over n 1 of the Eilenberg-MacLane spa* *ces K(Hn(X); n) if X is connected. Note that the unreduced constructions MX and NX are just disjoint unions of Cartesian powers and symmetric Cartesian powers and* * are therefore much less interesting. At least in characteristic zero, we conclude t* *hat H*(M"X) ~=M"(H*(X)) and H*(N"X) ~="N(H*(X)): By Proposition 3.5, these are the free and free commutative algebras generated * *by H"*(X). Note that any positively graded k-module can be realized as "H*(X) by t* *aking X to be a suitable wedge of spheres. 6.Operads, loop spaces, n-Lie algebras, and n-braid algebras We here specialize to the operads that come from the study of iterated loop * *spaces. These operads turn out to encode notions of n-Lie algebra and n-braid algebra. * *Im- plicitly or explicitly, the case n = 1 has received a great deal of attention i* *n the recent literature of string theory. See, e.g. [27, 28, 56], and the references therein. For each n > 0, there is a little n-cubes operad Cn. It was invented (befor* *e the introduction of operads) by Boardman and Vogt [11]; see also [46]. Its jth spac* *e Cn(j) consists of j-tuples of little n-cubes embedded with parallel axes and disjoint* * interiors in the standard n-cube. There is an analogous little n-disks operad defined in * *terms of embeddings of little disks in the unit disk via radial contraction and translat* *ion. These are better suited to considerations of group actions and of geometry, but they * *do not stabilize over n. There is a more sophisticated variant, due to Steiner [57], t* *hat enjoys the good properties of both the little n-cubes and the little n-disks operads. * *Each of these operads comes with a canonical equivalence from its jth space to the conf* *iguration space F (Rn; j) of j-tuples of distinct points of Rn. The little n-cubes operad* * (and any of its variants) acts naturally on all n-fold loop spaces nY . OPERADS, ALGEBRAS, MODULES, AND MOTIVES 27 Since C1 maps by a homotopy equivalence to M , we concentrate on the case n >* * 1. When k is a field of characteristic p > 0, the homology of a Cn-space, such as * *nY , has an extremely rich and complicated algebraic structure, carrying Browder operati* *ons and some of the Dyer-Lashof operations that are present in the homology of E1 algeb* *ras (see the next section). For a detailed description, see Cohen [16, IIx1]. (Minor cor* *rections are given in Wellington [61, I,x1].) We will here describe the characteristic zero * *information and a portion of the mod p information in Cohen's exhaustive mod p calculations* *. We take k to be a field throughout this section. Cohen's calculations have two essential starting points. One is his complete* * and explicit calculation of the integral homology of F (Rn; j), with its action by * *j, for all n and j [16, II xx6-7]. He used this to define homology operations. The other i* *s the "approximation theorem" of [46]. It asserts that, for a based space X, the redu* *ced free Cn-space "CnX maps to nnX via a natural map of Cn-spaces that is an equivalence when X is connected. This allowed Cohen to combine the homology operations with the Serre spectral sequence to compute simultaneously both H*(C"nX) and H*(nnX) for any X. In characteristic zero, the calculations simplify drastically since Theorem 5* *.2 shows that calculation of the homology operads H*(Cn) suffices to determine H*(C"nX).* * Cohen showed that each space F (Rn; j) has the same integral homology as a certain pr* *oduct of wedges of (n-1)-spheres. Therefore, with the notations of Definition 5.1, th* *e operad H*(Cn) can be written additively as the reduced sum N "Hn-1(Cn) of its subopera* *ds N and Hn-1(Cn), where the reduced sum is obtained from the direct sum by identi* *fying the unit elements in N (1) and H0(Cn(1)). When char(k) = 0 and n = 1, the follo* *wing result was implicit in [4] and was made explicit by Schechtman and Ginzburg. It* * was observed by Getzler and Jones [28] that the general case was already implicit i* *n Cohen's thesis [15]. Theorem 6.1. If char(k) 6= 2 or 3, then, for all n 1, the degree zero transla* *te of the operad Hn(Cn+1) is isomorphic to the operad L that defines Lie algebras over k. We are more interested in the algebras defined by the untranslated operads and by the full homology operads. If char(k) 6= 2 or 3, these turn out to be the n* *-Lie algebras and n-braid algebras. (A 1-braid algebra is also called a braid algeb* *ra or a Gerstenhaber algebra.) Recall our standing convention that k-modules are Z-gra* *ded and have differentials. Definition 6.2.An n-Lie algebra is a k-module L together with a map of k-modules [ ; ]n : L L ! L that raises degrees by n and satisfies the following identiti* *es, where deg(x) = q - n, deg(y) = r - n, and deg(z) = s - n. 28 IGOR KRIZ AND J. P. MAY (i)(Anti-symmetry) [x; y]n = -(-1)qr[y; x]n: (ii)(Jacobi identity) (-1)qs[x; [y; z]n]n + (-1)qr[y; [z; x]n]n + (-1)rs[z; [x; y]n]n =* * 0: (iii)[x; x]n = 0 if char(k) = 2 and [x; [x; x]n]n = 0 if char(k) = 3. Of course, a 0-Lie algebra is just a Lie algebra. For a k-module Y and an in* *teger n, define the n-fold suspension nY by (nY )q = Yq-n, with differential (-1)nd. * *(The sign depends on conventions: see IIIx1.) Proposition 6.3. The category of n-Lie algebras is isomorphic to the category o* *f Lie algebras. There is an operad Ln whose algebras are the n-Lie algebras, and its * *degree zero translate is isomorphic to L . Proof. For an n-Lie algebra L, nL is a Lie algebra with bracket [nx; ny] = n[x; y]n: Similarly, for a Lie algebra L, -nL is an n-Lie algebra. This gives the first s* *tatement. For the second, Ln can be constructed by a precisely similar use of suspensions* *, and the isomorphism with L is then obvious. __|_| Definition 6.4.An n-braid algebra is a k-module A that is an n-Lie algebra and a commutative DGA such that the bracket and product satisfy the following identit* *y, where deg(x) = q - n and deg(y) = r - n. (i)(Poisson formula) [x; yz]n = [x; y]nz + (-1)q(r-n)y[x; z]n: The Poisson formula implies and is implied by the following identities, where d* *eg(x) = q - n, deg(y) = r - n, deg(z) = s - n, and deg(w) = t - n. (ii)[1; x]n = 0, where 1 is the unit for the product. (iii)[xy; zw]n = x[y; z]nw + (-1)(r-n)s[x; z]nw + (-1)(q+r-n)(s-n)zx[y; w]n +(-1)q(s-n)+(r-n)(s+t-n)z[x; w]ny: The Poisson formula asserts that the map dx = [x; ?]n is a graded derivation* *, in the sense that dx(yz) = dx(y)z + (-1)deg(y)deg(dx)ydx(z): Batalin-Vilkovisky algebras are examples of 1-braid algebras [27], hence the ge* *neral case, with non-zero differentials, is relevant to string theory. However, our c* *oncern here is with structures that have zero differential. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 29 Theorem 6.5. The homology H*(X) is an n-braid algebra for any Cn+1-space X and any field of coefficients. The n-bracket is denoted n and called a Browder operation in [45, x6], and [1* *6, II], where the theorem is proven. The first appearance of n was in [12], in characte* *ristic 2. We have displayed (iii) since that is the version of the Poisson formula gi* *ven in [16] (where signs are garbled on page 216 but correct on page 317). Identity (* *iii) of Definition 6.2 is of conceptual interest: it cannot be visible in the operad Hn* *(Cn+1), but it follows directly from the chain level definition of n. For a k-module V , let LnV be the free n-Lie algebra generated by V ; as in P* *roposition 6.3, LnV = -nLnV . For the moment, let n denote the monad on k-modules associ- ated to the operad Hn(Cn+1), and recall the duplicative use of the notation "Cn* *+1from Theorem 5.2. For a Cn+1-space X, the action of Cn+1 induces an action of Hn(Cn+* *1) on H*(C"n+1X). It is clear from the decomposition of H*(Cn+1) as a reduced dire* *ct sum that all of the iterated n-bracket operations must be codified as part of this * *action. Theorem 6.6 (Cohen). Assume that char(k) = 0. For any based space X, C"n+1H*(X) ~=H*(C"n+1X) ~=NLnH"*(X) is the free commutative algebra generated by the free n-Lie algebra generated b* *y "H*(X). Moreover, the image of nH"*(X) in H*(C"n+1X) under the composite nH"*(X) ! nH"*(C"n+1X)) ! H*(C"n+1X) induced by the unit X ! "Cn+1X and the action of Hn(Cn+1) coincides with the n-* *Lie algebra LnH"*(X). The first isomorphism is given by Theorem 5.2 and the second by Cohen's calcu* *la- tions. With char(k) = 0, the deduction of Theorem 6.1 from Proposition 6.3 and * *The- orem 6.6 is a conceptual exercise. The fact that Hn(Cn+1) induces the n-Lie bra* *cket on the homology of Cn+1-spaces implies that there is a map of operads Ln ! Hn(Cn+1* *). Any positively graded k-module V is the homology of some space. Therefore this * *map of operads induces an isomorphism LnV ! nV for all such V . This is enough to c* *on- clude that Ln ! Hn(Cn+1) is an isomorphism. A similar exercise gives the char(k* *) = 0 case of the following further consequence of Theorem 6.6. The second statement * *can be proven algebraically, but it is more amusing to deduce it from the topology. Theorem 6.7. If char(k) 6= 2 or 3, then, for all n 1, the algebras over the o* *perad H*(Cn+1) are exactly the n-braid algebras. The free n-braid algebra generated * *by a k-module V is isomorphic to NLnV . 30 IGOR KRIZ AND J. P. MAY It remains to say something about the proofs of Theorem 6.1 and 6.7 in posit* *ive characteristic. Here we still have a natural map C"n+1H*(X) ! H*(C"n+1X); but it is no longer an isomorphism. Cohen's complete calculation of the target * *shows that it contains NLnH"*(X), and one again sees that all iterated Browder operat* *ions are determined by the action of elements of Hn(Cn+1). Now the dimension of the k-mo* *dule Ln(j) is independent of the characteristic by Proposition 6.3 and the correspon* *ding fact for Lie algebras, while the dimension of Hn(Cn+1(j)) is independent of the char* *acteristic by Cohen's integral calculations. By the characteristic zero result, these dim* *ensions must be equal for all characteristics. We deduce that the displayed map must be* * an isomorphism onto NLnH"*(X), and the rest of the argument goes as before. 7.Homology operations in characteristic p When C is an E1 operad, an action of C on A builds in the kinds of higher ho* *mo- topies for the multiplication of A that are the source, for example, of the Dye* *r-Lashof operations in the homology of infinite loop spaces and the Steenrod operations * *in the cohomology of general spaces. We describe the form that these operations take i* *n the homology of general E1 algebras A in this section. When we connect up partial a* *lge- bras and E1 algebras in Part II, this will give new homological invariants on t* *he mod p higher Chow groups. Many other examples are known to topologists, such as the Steenrod operations in the Ext groups of cocommutative Hopf algebras (e.g. [45,* * x11]) and in the cohomology of simplicial restricted Lie algebras (e.g. [53], [45, x8* *]). We begin with the trivial observation that, in characteristic zero, E1 opera* *ds carry no more homological information than the operad N . Lemma 7.1. Let ffl : C ! P be a quasi-isomorphism of operads over a field k * *of characteristic zero, such as the augmentation ffl : C ! N of an acyclic operad* *. Then the maps CX ! P X and C(X; Y ) ! P (X; Y ) induced by ffl are quasi-isomorphisms for all k-modules X and Y . Proof. This is an easy consequence of the definitions and the fact that all mod* *ules over the group ring k[G] of a finite group G are projective. __|_| Taking P = N and P = L , we will see in Part II that this leads to a proof * *that, when k is a field of characteristic zero, E1 algebras are quasi-isomorphic to c* *ommuta- tive DGA's and strong homotopy Lie algebras are quasi-isomorphic to differentia* *l Lie algebras, and similarly for modules. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 31 We take k = Z and consider algebras A over an integral E1 operad C in the res* *t of this section. Let Zp = Z=pZ and consider the mod p homology H*(A; Zp). Theorem 7.2. For s 0, there exist natural homology operations Qs: Hq(A; Z2) ! Hq+s(A; Z2) and Qs: Hq(A; Zp) ! Hq+2s(p-1)(A; Zp) if p > 2. These operations satisfy the following properties (1)Qs(x) = 0 if p = 2 and s < q or if p > 2 and 2s < q. (2)Qs(x) = xp if p = 2 and s = q or if p > 2 and 2s = q. (3)Qs(1) = 0 if s > 0, where 1 2 H0(A; Zp) is the identity element. P t s-t (4)(Cartan formula) Qs(xy) = Q (x)Q (y). (5)(Adem relations) If p 2 and t > ps, then X QtQs= (-1)t+i(pi - t; t - (p - 1)s - i)Qs+t-i-1Qi; i if p > 2, t ps, and fi denotes the mod p Bockstein, then X QtfiQs= (-1)t+i(pi - t; t - (p - 1)s - i)fiQs+t-iQi i X - (-1)t+i(pi - t - 1; t - (p - 1)s - i)Qs+t-ifiQi; i (i + j)! here (i; j) = ______ if i 0 and j 0 (where 0! = 1), and (i; j) = 0 if i* * or j is i!j! negative; the sums run over i 0. The proof is the same as in [16, Ix1]; as there, one simply checks that one i* *s in the general algebraic framework of [45], which does the relevant homological al* *gebra once and for all. (Actually, [45] should be read as a paper about operad actio* *ns. Unfortunately, it was written shortly before operads were invented.) The point * *is that C (p) is a p-free resolution of Z, so that the homology of C (p) p Ap is readi* *ly computed, and computation of * : H*(C (p) p Ap; Zp) ! H*(A; Zp) allows one to read off the operations. The Cartan formula and the Adem relations are derived * *from special cases of the diagrams in Definition 2.1(a) via calculations in the homo* *logy of groups. Notice the grading. The first non-zero operation is the pth power, and there* * can be infinitely many non-zero operations on a given element. This is in marked c* *on- trast with Steenrod operations in the cohomology of spaces, where the last non-* *zero operation is the pth power. In fact, Steenrod operations are defined on cohomo* *logi- cally graded E1 algebras that are concentrated in positive degrees, where the c* *ochain 32 IGOR KRIZ AND J. P. MAY complexes C (j) of the relevant E1 operad are concentrated in negative degrees.* * If we systematically regrade homologically, then Dyer-Lashof and Steenrod operations * *both fit into the general context of the theorem, except that the adjective "Dyer-La* *shof" is to be used when the underlying chain complexes are positively graded and the ad* *jective "Steenrod" is to be used when the underlying chain complexes are negatively gra* *ded. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 33 Part II. Partial algebraic structures and conversion theorems In this Part, and this Part only, we refer to a Z-graded chain complex over k* * as a k-complex, and we reserve the term k-module for an ungraded k-module. We assume that k is a Dedekind ring. Let C be an operad of simplicial k-modules, or of k-complexes, or, generaliz* *ing both, of simplicial k-complexes. In Part I, we defined C -algebras and modules * *over C - algebras, and we showed how to interpret these notions in terms of actions of m* *onads associated to C . We gave the definitions in terms of k-complexes, but we noted* * that they apply verbatim in the generality of simplicial k-complexes. We here gener* *alize these ideas further by specifying partial C -algebras and their modules in Sect* *ion 2 and then expressing these notions in terms of monads in Section 3. Again, these def* *initions apply in the generality of simplicial k-complexes. The main point of this Part is to study the conversion of such partial C -alg* *ebras and modules to quasi-isomorphic genuine C -algebras and modules. As we shall se* *e in Part V, we can construct derived categories of modules over E1 algebras that en* *joy all of the standard properties of derived categories of modules over commutative DG* *A's. One might instead try to develop a theory of derived categories of partial modu* *les over partial algebras. However, modules over E1 algebras are much more tractable for* * this purpose since they are defined entirely in terms of actual iterated tensor prod* *ucts rather than the tensor products up to quasi-isomorphism that are intrinsic to the defi* *nition of partial algebras and modules. For subtle technical reasons, explained in Section 5, our conversion theorems* * do not work in the full generality of our definitions. In fact, rather than workin* *g in the category of simplicial k-complexes, we must work in its subcategory of simplici* *al k- modules. Fortunately, this is the situation that occurs in the motivating examp* *les that arise in algebraic geometry. We discuss these examples briefly in Section 6. We* * explain the proofs of our conversion theorems in Sections 4 and 5. 1.Statements of the conversion theorems Modulo precise definitions, our conversion theorems read as follows. To start* * with, we work in the ground category of simplicial (ungraded) k-modules. Theorem 1.1. Let C be a -projective operad of simplicial k-modules. Then there* * is a functor V that assigns a quasi-isomorphic C -algebra V A to a partial C -alge* *bra A. There is also a functor V that assigns a quasi-isomorphic V A-module V M to a p* *artial A-module M. 34 IGOR KRIZ AND J. P. MAY When k is a field of characteristic zero, every operad C is -projective and * *we have the following complement. Theorem 1.2. Let k be a field of characteristic zero and let ffl : C ! P be a * *quasi- isomorphism of operads of simplicial k-modules. Then there is a functor W that * *assigns a quasi-isomorphic P-algebra W A to a partial C -algebra A. There is also a fun* *ctor W that assigns a quasi-isomorphic W A-module W M to a partial A-module M. An acyclic operad C is one that maps by a quasi-isomorphism to the operad N * *that defines commutative simplicial k-algebras, hence the following result is a spec* *ial case. Corollary 1.3. Let k be a field of characteristic zero and let C be an acyclic * *operad of simplicial k-modules. Then there is a functor W that assigns a quasi-isomor* *phic simplicial commutative k-algebra W A to a partial C -algebra A. There is also a* * functor W that assigns a quasi-isomorphic W A-module W M to a partial A-module M. As usual, we apply the normalized chain complex functor to pass from simplic* *ial k-modules to k-complexes, and a map of simplicial k-modules is said to be a qua* *si- isomorphism if the associated map of k-complexes is a quasi-isomorphism (induce* *s an isomorphism on homology). The passage from simplicial k-modules to k-complexes carries an operad of simplicial k-modules to an operad of k-complexes. Similarl* *y, this passage preserves algebras and modules. However, it does not preserve partial a* *lgebras and modules. For essentially the same technical reason, we do not have an analo* *g of Theorem 1.1 when working in the ground category of k-complexes. Therefore, alth* *ough our motivation and applications concern k-complexes, we are forced to work on t* *he level of simplicial k-modules as long as possible, only passing to associated k* *-complexes after the conversion of partial algebras and modules to genuine algebras and mo* *dules. Of course, in view of Theorem 1.1, Theorem 1.2 is only needed when A is alre* *ady a genuine C -algebra. There is a version of this case of Theorem 1.2 that does wo* *rk in the context of k-complexes. Theorem 1.4. Let k be a field of characteristic zero and let ffl : C ! P be a * *quasi- isomorphism of operads of k-complexes. Then there is a functor W that assigns a quasi-isomorphic P-algebra W A to a C -algebra A. There is also a functor W th* *at assigns a quasi-isomorphic W A-module W M to an A-module M. Corollary 1.5. Let k be a field of characteristic zero and let C be an acyclic * *operad of k-complexes. Then there is a functor W that assigns a quasi-isomorphic commutat* *ive DGA W A to a C -algebra A. There is also a functor W that assigns a quasi-isomo* *rphic W A-module W M to an A-module M. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 35 While our original motivation came from algebraic geometry, these results may* * also be of interest in other subjects. For example, operads of k-complexes whose alg* *ebras are "strong homotopy Lie algebras" are becoming increasingly important in string th* *eory (see [32, 56] and the references therein). The defining property of such an ope* *rad J is that it must admit a quasi-isomorphism ffl : J ! L , where L is the operad that* * defines Lie algebras over k. We then say that J is a strong homotopy Lie operad. Theore* *m 1.4 applies directly to replace strong homotopy Lie algebras by quasi-isomorphic ge* *nuine differential graded Lie algebras. A version of this result is known to the expe* *rts, via an entirely different proof, but the corresponding result for modules is new. Corollary 1.6.Let k be a field of characteristic zero and let J be a strong hom* *otopy Lie operad of k-complexes. Then there is a functor W that assigns a quasi-isomo* *rphic differential graded Lie algebra W L to a J -algebra L. There is also a functor * *W that assigns a quasi-isomorphic W L-module W M to an L-module M. Similarly, modulo the appropriate definitions, Theorems 1.1 and 1.2 apply to * *con- vert partial simplicial strong homotopy Lie algebras first to genuine simplicia* *l strong homotopy Lie algebras and then, when k is a field of characteristic zero, to si* *mplicial Lie algebras. 2.Partial algebras and modules One often encounters k-modules A that come with products that are only define* *d on appropriate submodules of A A. We first define the ground categories for such * *partial algebras and their modules, then specify partial commutative DGA's and their pa* *rtial modules, and finally generalize to define partial structures defined by operad * *actions. We begin by being precise about the categories in which we shall work. For the motivic applications, we must allow "Adams graded" objects, such as k- complexes or simplicial k-modules. An Adams graded object X = {X(r)|r 2 Z} is j* *ust a sequence of objects X(r). A map X ! Y is just a sequence of maps X(r) ! Y (r). For Adams graded objects X and Y in a tensored category, we define X (X Y )(r) = X(s) Y (r - s); and the category of Adams graded objects is again tensored; its unit is the uni* *t of the given category viewed as concentrated in Adams grading zero. We think of t* *he Adams grading as if it were concentrated in even degrees: it will not contribu* *te to signs under permutations. We use the term "Adams grading" to avoid confusion wi* *th any other grading that we may have. This grading will be important in the geome* *tric context, where it is closely related to the grading of rational algebraic K-the* *ory by 36 IGOR KRIZ AND J. P. MAY the eigenvectors of Adams operations, but it will carry through the theory of t* *his Part without introducing any complications. With operads understood to be concentrat* *ed in Adams degree zero, the conversion theorems all apply verbatim to Adams graded objects. While the examples of Section 6 are concentrated in positive Adams gra* *ding, a satisfactory theory of modules over algebras requires us to allow negative degr* *ees. We assume from now on that all of our objects are Adams graded. Let K f denote the category of Adams graded flat k-modules and let Mkfdenote the category of Adams graded k-complexes of flat k-modules. For any category T * *, let S T denote the category of simplicial objects in T . Recall that, if T is tenso* *red, then the tensor product of simplicial objects X and Y in T has q-simplices Xq Yq, w* *ith faces and degeneracies @i @iand si si. For definiteness and generality, we shal* *l work in the category S Mkfof simplicial Adams graded flat k-complexes, but we are ma* *inly interested in its subcategories S K f of simplicial Adams graded flat k-modules* * and Mkf, where we regard a k-complex X as the constant simplicial k-complex X_with * *each X___q= X and each face and degeneracy the identity map. In the following definition, "domain" should be thought of as shorthand for * *"domain of definition". We let Xj denote the j-fold tensor power of an object X 2 S Mkf* *, with X0 = k_. As we recall in Section 5, X has an associated total k-complex C#X 2 M* *kf, and we say that OE : X ! Y is a quasi-isomorphism if C#OE is a quasi-isomorphis* *m (in each Adams grading). Definition 2.1.Let X 2 S Mkf. A domain X* in X is a sequence of j-invariant subobjects Xj of Xj such that the given inclusions ffij : Xj ! Xj satisfy the f* *ollowing properties. (a) ffio and ffi1 are identity maps and each ffij is a quasi-isomorphism. (b) For each partition j = j1+. .+.jk, ffijfactors through Xj1. .X.jk, as in* *dicated in the following commutative diagram: ~= Xj1 . .O.XjkO ____//_XjOO ffij1...ffijk|| ffij|| | | Xj1 . . .Xjk oo_?Xj:___ Our standing assumption that k is a Dedekind ring and our requirement that X* * be flat ensure that the tensor product ffij1 . . .ffijkin the diagram just given i* *s both an inclusion and a quasi-isomorphism. This is a consequence of the following lemma* * and the fact that, over any commutative ring k, a tensor product of flat k-modules * *is flat and a tensor product of inclusions of flat modules is an inclusion. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 37 Lemma 2.2. Assume that k is a Dedekind ring. Then submodules of flat k-modules are flat. Let X be a k-complex and f : Y ! Y 0be a quasi-isomorphism of k-compl* *exes, where either X or both Y and Y 0are flat. Then 1 f : X Y ! X Y 0is a quasi-isomorphism. Proof.The first part is standard, and it implies that the cycles and boundaries* * of flat k-complexes are flat. In turn, this implies that a flat k-complex X is the unio* *n of its bounded below flat k-complexes X[n], where X[n]q = 0 for q < n, X[n]n = ZnX (the cycles of degree n), and X[n]q = Xq for q > n. For positively graded k-complexe* *s X and Y , one of which is flat, there is a K"unneth spectral sequence that conver* *ges from Tork*;*(H*(X); H*(Y )) to H*(X Y ) [41, XII.12.1]. By generalizing from posit* *ive to bounded below k-complexes and passing to direct limits, we obtain a natural con* *vergent K"unneth spectral sequence for any two k-complexes X and Y , one of which is fl* *at. The conclusion follows. __|_| A map f* : X* ! X0*between domains in X and X0is a sequence of maps fj: Xj! X0jsuch that fj is the restriction of fj, where f = f1. The map f* is said to * *be a quasi-isomorphism if f : X ! X0 is a quasi-isomorphism. It follows from the lem* *ma and the definitions that each fj is then also a quasi-isomorphism. Let D be the category of domains in objects of S Mkf. Let L : D ! S Mkfbe the functor that sends X*to X = X1and let R : S Mkf! D be the functor that sends X * *to {Xj}. Then LR = Idand the inclusions ffijdefine a natural map ffi : X* ! RX = R* *LX* such that Lffi = Id. We therefore have an adjunction (2.3) S Mkf(LX*; Y ) ~=D(X*; RY ): Informally, given any type of algebraic structure that is defined in terms of* * maps Aj! A, we define a partial structure on A to be a domain A* in A together with * *maps Aj! A that satisfy the same formal properties as the given type of structure. W* *e shall shortly formalize this with a general definition of a partial operad action. For motivation, and because it is the type of structure that we are most inte* *rested in, we first consider commutative DGA's explicitly. Such an algebra A has a j-* *fold product j: Aj! A, with 0 = j : k ! A and 1 = Id. For oe 2 j, j 2, jO oe = j. For any partition j = j1 + . .+.jk with ji 0, the following associativity and * *unity diagram commutes: ~= Aj1 . . .Ajk ____//Aj ... || || fflffl| fflffl| Ak _________//_A: 38 IGOR KRIZ AND J. P. MAY Recall that 2x2 = 0 if x has odd degree. It is standard in topology to say tha* *t a commutative DGA is "strictly commutative" if x2 = 0 when x has odd degree. Unle* *ss A is strictly commutative (or the ring k is of characteristic two), A will not * *be a flat k-module. We have the concomitant notion of a partial commutative DGA A*. The only poi* *nt that might require clarification is the partial version of the previous diagram* *, which now takes the form Aj1 . . .Ajk oo________?_Aj__ "" | ... || """" | fflffl| _ """" fflffl| Ak oo______?Ak_____//A: That is, the restriction to Aj of . . . : Aj1 . . .Ajk! Ak factors through Ak, and the two resulting maps from Aj to A coincide. More generally, we have the following direct generalization of I.2.1. The o* *perads that we are interested in are in either S K f or Mkf; in the latter case, we re* *gard them as operads of constant simplicial objects in S Mkf. We assume that operads* * are concentrated in Adams grading zero. Definition 2.4.Let C be an operad in S Mkfand let A 2 S Mkf. A partial C -algeb* *ra is a domain A* in A together with j-equivariant maps j: C (j) Aj! A; j 0; that satisfy the following properties. (a) 1(1 a) = a. (b) The map C (j1) . . .C (jk) Aj! Ak that is obtained by including Aj in Aj1 . . .Ajk, shuffling, and applyin* *g k P factors through Ak, where j = js. (c) The following associativity diagrams commute: flId C (k) C (j1) . . .C (jk) Aj________//C (j) Aj "|_ WWWWW III WWWW III | WWWWWWWW II fflffl| WWWW++W II$$I C (k) C (j1) . . .C (jk) Aj1 . . .Ajk C (k) Ak____//_A |" _ |shuffle| | fflffl| fflffl| C (k) C (j1) Aj1 . . .C (jk) Ajk_____//_C (k) Ak: Idk Observe that RA = {Aj} may be viewed as a partial C -algebra if A is a C -al* *gebra. In our formal theory, we generally write A*for a partial C -algebra. Informally* *, however, OPERADS, ALGEBRAS, MODULES, AND MOTIVES 39 as in the statements of the results in the previous section, we think of the su* *bmodules Aj of the Aj as implicitly given and simply write A. The following examples gen* *eralize I.2.2. Examples 2.5. Restrict attention to the category Mkf. (i) A partial M -algebra is a partial DGA. (ii) A partial N -algebra is a partial commutative DGA, as defined above. (iii) By pullback along ffl : C ! N , a partial commutative DGA is a partial * *C -algebra for any unital operad C . (iv) We define a partial E1 algebra to be a partial C -algebra, where C is * *any E1 operad. Remark 2.6.We noted in I.7.2 that the mod p homology of an integral E1 algebra has homology operations. The cited result is true precisely as stated with E1 a* *lgebras replaced by partial E1 algebras. In fact, one can construct the operations by p* *assing to mod p homology from the diagram C (p) p Ap - C (p) p Ap -! A: The first arrow is induced by the inclusion Ap Ap, and this arrow induces an i* *so- morphism on mod p homology by Lemma 3.1 below. The second arrow is p. If we start with a partial commutative DGA A*, then p = fflp p. Use of the augmentati* *on fflp : C (p) ! Z may make it appear that the resulting operations ought to be t* *rivial. However, as is explained in [26], nontriviality is allowed by the fact that the* * inclusion Ap Ap need not be a p-equivariant homotopy equivalence. We have a precisely parallel definition of a partial module over a partial al* *gebra. Definition 2.7.Define a domain (X*; Y*) in a pair (X; Y ) of objects of S Mkfto* * be a domain X* in X together with a sequence of j-1-invariant subobjects Yj of Xj-1 * *Y , j 1, such that the given inclusions ffij: Yj! Xj-1Y satisfy the following prop* *erties. (a)ffi1 = Idand each ffij is a quasi-isomorphism. (b)For j = j1+ . .+.jk, ffij factors through Xj1 . . .Xjk-1 Yjk, as indicate* *d in the following commutative diagram: ~= Xj1 . . .Xjk-1OXjk-1OY ___//_Xj-1OYO ffij1...ffijk|| |ffij| | | Xj1 . . .Xjk-1 Yjk oo________?Yj:__ A map (f*; g*) : (X*; Y*) ! (X0*; Y*0) consists of a map of domains f* : X* !* * X0*and a sequence of maps gj: Yj! Yj0such that gjis the restriction of fj-1g, where g * *= g1. 40 IGOR KRIZ AND J. P. MAY The map (f*; g*) is said to be a quasi-isomorphism if f : X ! X0 and g : Y ! Y * *0are quasi-isomorphisms, and then each fj and gj is also a quasi-isomorphism. Let (S Mkf)2 be the category of pairs of objects of S Mkfand let D2 be the c* *ategory of domains in such pairs. Let L : D2 ! (S Mkf)2 send (X*; Y*) to (X; Y ) and R* * : (S Mkf)2 ! D2 send (X; Y ) to ({Xj}; {Xj-1 Y }). Again, LR =Id and the ffij spe* *cify a natural map ffi : X* ! RX = RLX* such that Lffi =Id, hence we have an adjunct* *ion (2.8) (S Mkf)2(L(X*; Y*); (X0; Y 0)) ~=D2((X*; Y*); R(X0; Y 0)): We shall often abbreviate (X*; Y*) to Y* when X* is implicit from the context. Let A* be a partial commutative DGA. An A*-module (informally, a partial A- module) is a domain M* together with maps of k-modules j : Mj ! M such that 1 =Id, jO oe = j for oe 2 j-1, and the following diagrams commute, where ji 0 for i < k; jk 1, and j = j1+ . .+.jk; Aj1 . . .Ajk-1 Mjk oo_________?Mj__ __ ... || ____ || fflffl| _ ""__ fflffl| Ak-1 M oo_______?Mk_ ____//M: That is, the restriction to Mj of . . . factors through Mk, and the two resulting maps from Mjto M coincide. This is the special case C = N of the foll* *owing definition, which generalizes I.4.1. We return to our general ground category (* *S Mkf)2. Definition 2.9.Let C be an operad and A*be a partial C -algebra in A. An A*-mod* *ule M* in M is a domain (A*; M*) in (A; M) together with j-1-equivariant maps j: C (j) Mj! M; j 1; that satisfy the following properties. (a) 1(1 m) = m. (b) The map C (j1) . . .C (jk) Mj! Mk that is obtained by including Mjin Aj1. .A.jk-1Mjk, shuffling, and apply* *ing P k-1 factors through Ak, where j = js. (c) The following associativity diagrams commute: OPERADS, ALGEBRAS, MODULES, AND MOTIVES 41 C (k) (k-1s=1C (js)) C (jk)__Mj_________//C (j) Mj |" _ XXXXXX NNNNN XXXXXXX NNN | XXXXXXX NNN fflffl| XXXXX++X NN'' C (k) (k-1s=1C (js)) C (jk) (k=1s=1Ajs) Mjk C (k) Mk _______//M "|_ |shuffle| || fflffl| fflffl| C (k) (k-1s=1C (js) Ajs) C (jk) _Mjk___//C (k) Ak-1 M: Idk-1 Remark 2.10.There is also a notion of a partial operad C , with structural maps* * fl defined on submodules C (k; j1; : :;:jk) of C (k) C (j1) . . .C (jk). In top* *ology, the little convex bodies partial operads of [47, VII x2], were the first exampl* *es, but Steiner [57] later showed how to replace these particular partial operads by eq* *uivalent genuine operads with all of the desired properties. Partial operads have arise* *n more recently, and more substantially, in work of Huang and Lepowski on vertex opera* *tor algebras [34, 33]. It is an easy matter to generalize the definitions above to* * specify partial algebras and modules over partial operads. However, the resulting conce* *pts are harder to work with algebraically since they seem not to admit equivalent monad* *ic descriptions. 3. Monadic reinterpretation of partial algebras and modules In this section, we assume given an operad C in S Mkfthat is -projective, in * *the sense that each C (j) is a projective k[j]-module in each simplicial degree. As* * before, we are interested in operads in either S K for Mkf. The projectivity condition * *holds automatically when k is a field of characteristic zero since every module over * *the group ring k[G] of a finite group G is then projective. It allows us to make use of t* *he following standard observation, which complements Lemma 2.2. Lemma 3.1. Let G be a group and P be a projective k[G]-module. Then a quasi- isomorphism X ! X0 of k[G]-modules induces a quasi-isomorphism P k[G]X ! P k[G]X0of k-modules. Proof.If we filter P k[G]X by the degrees in P , we obtain a natural spectral s* *equence that converges from H*(P k[G]H*(X)) to H*(P k[G]X). __|_| In Ixx3-4, we constructed monads C in Mk and C[1] in Mk2such that a C-algebra determines and is determined by a C -algebra and a C[1]-algebra determines and * *is de- termined by a C -algebra together with a module over it. In this Part, we are r* *estricting to flat k-modules and pairs, and our assumption on C ensures that C and C[1] ta* *ke flat modules and pairs to flat modules and pairs. We generalize these construct* *ions to the context of partial algebras and modules. 42 IGOR KRIZ AND J. P. MAY Definition 3.2.Define the monad C* in D associated to C as follows. Let X* be a domain in X. Define M CX* = C (j) k[j]Xj: j0 For k 0, define CkX* (CX*)k to be the direct sum of the images of the followi* *ng P composites (where js 0 and j = js): C (j1) . . .C (jk) k[j1x...xjk]Xj |" _ | fflffl| C (j1) . . .C (jk) k[j1x...xjk]Xj1 . . .Xjk |shuffle| fflffl| C (j1) k[j1]Xj1 . . .C (jk) k[jk]Xjk: This inclusion is a quasi-isomorphism by Lemma 3.1, hence the inclusion CkX* (* *CX)k is a quasi-isomorphism. The action of k on CkX* is induced from the action on (* *CX)k; more explicitly, it is obtained from permutations of the variables C (js) and a* *ction of the permutations oe(j1; : :;:jk) associated to oe 2 k on the factors Xj (see* * I.1.1). Condition (b) of Definition 2.1 is inherited from the corresponding condition f* *or X*. Let jk : Xk ! CkX* be induced by the map j . . .j Id: Xk = k . . .k Xk ! C (1) . . .C (1) Xk: P Similarly, let k : CkC*X* ! CkX* be induced by the following maps, where js =* * j, P it= i, gs= j1+ . .+.js, and hs= igs-1+1+ . .+.igsfor 1 s k: C (j1) . . .C (jk) C (i1) . . .C (ij) Xi | | O fflfflshuffle| (C (js) C (igs-1+1) . . .C (igs)) Xi s O |( fl) Id | fflffls| C (h1) . . .C (hk) Xi: It is easy to check that (C*; *; j*) is a monad in D. The following observation is immediate from Lemma 3.1. Lemma 3.3. If f : X* ! X0*is a quasi-isomorphism of domains, then so is C*f : C*X* ! C*X0*. We have the following generalizations of I.3.4. Recall (2.3). OPERADS, ALGEBRAS, MODULES, AND MOTIVES 43 Theorem 3.4. Let C be a -projective operad. (i) A partial C -algebra determines and is determined by a C*-algebra in D. For* *mally, the identity functor on D restricts to an isomorphism between the categories of* * partial C -algebras and of C*-algebras. (ii) RC = C*R, hence C = LC*R, and the unit j and product for C are given as follows in terms of the unit j* and product * of C*: j = Lj*R : Id= LR ! LC*R = C; and = L*R : CC = LC*RC = LC*C*R ! LC*R = C: Proof.If A* is a partial C -algebra, then the given maps : C (j) Aj ! A toget* *her induce a map : CA* ! A. For k 1, the maps C (j1) . . .C (jk) Aj! Ak that factor the evident map to Ak (as in Definition 2.4) together induce a map * *k : CkA* ! Ak. It is easily checked that (A*; *) is a C*-algebra. Conversely, if (A* **; *) is a C*-algebra, then the evident composites C (j1) . . .C (jk) Aj! CkA* ! Ak; k 1; give A* a structure of a partial C -algebra. Part (ii) is easily verified by a * *direct com- parison of definitions. __|_| The analogous theory for partial modules generalizes material in Ix4. Recall * *(2.8). Definition 3.5.Define the monad C*[1] in D2 associated to C as follows. Let (X** *; Y*) be a domain in (X; Y ). Define M CY* = C (j) k[j-1]Yj: For k 1, define CkY* (CX*)k-1 CY* to be the direct sum of the images of the P following composites (where js 0 and j = js): C (j1) . . .C (jk) k[j1x...xjk-1xjk-1]Yj "|_ | fflffl| C (j1) . . .C (jk) k[j1x...xjk-1xjk-1]Xj1 . . .Xjk-1 Yjk shuffle|| fflffl| C (j1) k[j1]Xj1 . . .C (jk-1) k[jk-1]Xjk-1 C (jk) k[jk-1]Yjk: This inclusion is a quasi-isomorphism by Lemma 3.1, hence the inclusion CkY* (CX*)k-1 CY* is a quasi-isomorphism. The action of k-1on CkY* is induced from 44 IGOR KRIZ AND J. P. MAY the action on (CX*)k-1 CY*. Condition (b) of Definition 2.7 is inherited from t* *he corresponding condition for Y*. Maps jk : Yk ! CkY* and k : CkC*Y* ! CkY* are defined as in Definition 3.2 and I.4.3. Taking C[1]*(X*; Y*) to be (C*X*; C*Y** *) and using the pairs of maps (j*; j*) and (*; *) as the unit and product, we obtain * *the desired monad in D2. Lemma 3.6. If (f*; g*) : (X*; Y*) ! (X0*; Y*0) is a quasi-isomorphism of doma* *ins, then the induced map C*(f*; g*) : C*Y* ! C*Y*0is a quasi-isomorphism. Theorem 3.7. Let C be a -projective operad. (i) A C[1]*-algebra structure on a domain (A*; M*) determines and is determined* * by a partial C -algebra structure on A* together with a partial A*-module structure * *on M*. (ii) RC[1] = C[1]*R, hence C[1] = LC[1]*R, and the unit j and product of C[1] * *are given in terms of the unit j* and product * of C[1]* by j = Lj*R and = L*R. Remark 3.8. For a unital operad C , there are generalizations to the partial co* *ntext of the reduced monads that we constructed in Ixx3-4. These were used in the prepri* *nt version [37] of this paper. Since the reduced monads are not essential to the * *theory and the details are fairly technical, we shall omit these constructions in the * *interests of brevity. 4. The two-sided bar construction and the conversion theorems We begin by recalling some categorical definitions from [46, xx2,9]. Their u* *se to prove the theorems stated in Section 1 will follow a conceptual pattern that is expla* *ined in detail in [49, x5]. Definition 4.1.Let (C; ; j) be a monad in a category T . A (right) C-functor in* * a category V is a functor F : T ! V together with a natural transformation : F C* * ! F such that the following diagrams commute: Fj C F C oo__F and F CC ____//F C -- | | ||---- F | | fflffl|""Id-- fflffl| fflffl| F F C______//F For a triple (F; C; A) consisting of a monad C in T , a C-algebra A, and a C-fu* *nctor F in V , define a simplicial object B*(F; C; A) in V by letting the q-simplices B* *q(F; C; A) be F CqA (where Cq denotes C composed with itself q times); the faces and degen* *eracies are given by @0 = Cq-1; @i= F Ci-1Cq-i-1for1 i < q; OPERADS, ALGEBRAS, MODULES, AND MOTIVES 45 @q = F Cq-1ffl; and si= F CijCq-i: In an evident sense, B*(F; C; A) is functorial in all three variables. Given a * *monad C0 in V and a left action : C0F ! F , we say that F is a (C0; C)-bifunctor if the* * following diagram commutes: _C__ C0F C //F C C0 || || fflffl| fflffl| C0F ______//F For such an F , B*(F; C; A) is a simplicial C0-algebra. Example 4.2. An obvious example of a (C; C)-bifunctor is C itself, with both le* *ft and right action . Thus we may regard C as a functor from T to the category C[T ] * *of C-algebras in T . This example gives a simplicial C-algebra B*(C; C; A) associa* *ted to a C-algebra A. Let A_denote A regarded as a constant simplicial object. Iterate* *s of and give a map * : B*(C; C; A) ! A_of simplicial C-algebras in T . Similarl* *y, iterates of j give a map j* : A_! B*(C; C; A) of simplicial objects in T (but * *not in C[T ]) such that *j* = Id. Moreover, there is a simplicial homotopy j* * ' Id[* *46, 9.8]. This is a generalized version of the classical bar resolution in homologi* *cal algebra, and we shall often abbreviate notation by setting B*(A) = B*(C; C; A): The following examples should be viewed as formal precursors of Theorems 1.1,* * 1.2, and 1.4. Fix a -projective operad C . Example 4.3. (i) As explained in [49, 5.5], part (ii) of Theorem 3.4 implies th* *at CL : D ! S Mkfis a (C; C*)-bifunctor with C*-action the composite L O CLC*ffi : CLC* ! CLC*RL = CCL ! CL and that C*ffi : C* ! C*RL = RCL is a map of (C*; C*)-bifunctors D ! D. Since f* *fi is a natural quasi-isomorphism, Lemma 3.3 allows us to view C*ffi as inducing a* * quasi- isomorphism of simplicial C*-algebras ffi* : B*(C*; C*; A*) ! B*(RCL; C*; A*) = RB*(CL; C*; A*) for any C*-algebra A*. Introduce the abbreviated notations B*A* = B*(C*; C*; A*) and V*A* = B*(CL; C*; A*): 46 IGOR KRIZ AND J. P. MAY Then B*A* is a simplicial C*-algebra, V*A* is a simplicial C-algebra, and * an* *d ffi* give a natural diagram of simplicial C*-algebras A_* B*A* ! RV*A*: (ii) Similarly, part (ii) of Theorem 3.7 implies that we may replace C by C[* *1] in (i) and obtain the analogous conclusions: C[1]L : D2 ! (S Mkf)2 is a (C[1]; C[1* *]*)- bifunctor with C[1]*-action given by L O C[1]LC[1]*ffi, and C[1]*ffi : C[1]* ! * *RC[1]L is a map of (C[1]*; C[1]*)-bifunctors D2 ! D2. By Lemma 3.6, we may view C[1]*ffi * *as inducing a quasi-isomorphism of simplicial C[1]*-algebras ffi* : B*(C[1]*; C[1]*; (A*; M*)) ! RB*(C[1]L; C[1]*; (A*; M*)) for any C[1]*-algebra (A*; M*); recall that A* is a C*-algebra and M* is an A*-* *module. We extend the abbreviated notations of (i) by setting (B*A*; B*M*) = B*(C[1]*; C[1]*; (A*; M*)) and (V*A*; V*M*) = B*(C[1]L; C[1]*; (A*; M*)): Then B*M* is a simplicial B*A*-algebra, V*M* is a simplicial V*A*-algebra, and * * * and ffi* give a natural diagram of simplicial C[1]*-algebras (A_*; M*) (B*A*; B*M*) ! R(V*A*; V*M*): Example 4.4. Let ffl : C ! P be a quasi-isomorphism of -projective operads and let ffl also denote the induced maps of monads C ! P and C[1] ! P [1]. If k is * *a field of characteristic zero, then ffl : CX ! P X and ffl : C(X; Y ) ! P (X; Y ) are * *quasi- isomorphisms for all k-modules X and Y . (This is I.7.1, and it also follows d* *irectly from Lemma 3.1.) In this case, the maps ffl* in the rest of this example are al* *l quasi- isomorphisms. (i) P L : D ! S Mkfis a (P; C*)-bifunctor with C*-action the composite L O P fflL O P LC*ffi : P LC* ! P LC*RL = P CL ! P P L ! P L; fflL : CL ! P L is a map of (C; C*)-bifunctors and therefore induces a map of s* *implicial C-algebras ffl* : V A* = B*(CL; C*; A*) ! B*(P L; C*; A*) W*A* for any C*-algebra A*, where W*A* is abbreviated notation for the simplicial P * *-algebra B*(P L; C*; A*). (ii) P [1]L : D2 ! (S Mkf)2 is a (P [1]; C[1]*)-bifunctor; fflL : C[1]L ! P * *[1]L is a map of (C[1]; C[1]*)-bifunctors and therefore induces a map of simplicial C[1]-alge* *bras ffl* : (V*A*; V*M*) ! (W*A*; W*M*); OPERADS, ALGEBRAS, MODULES, AND MOTIVES 47 where W*M* is abbreviated notation for the second coordinate of the simplicial * *P [1]- algebra B*(P [1]L; C[1]*; (A*; M*)). (iii) For a C-algebra A, define B*A = B*(C; C; A) and W*A = B*(P; C; A): Then W*A is a simplicial P -algebra and ffl* : B*A ! W*A is a map of simplicial* * C- algebras. Similarly, for a C[1]-algebra (A; M), define (B*A; B*M) = B*(C[1]; C[1]; (A; M)) and (W*A; W*M) = B*(P [1]; C[1]; (A; M)): Then W*M is a simplicial W*A-module and ffl* : (B*A; B*M) ! (W*A; W*M) is a map of simplicial C[1]-algebras. That is, ffl* : B*M ! W*M is a map of sim* *plicial B*A-modules, where W*M is a simplicial B*A-module by pullback along ffl* : B*A ! W*A. 5.Totalization and diagonal functors; proofs The constructions in the previous section are given in all-embracing generali* *ty, in the ambient ground category S Mkfof simplicial flat k-complexes. To deduce the theo* *rems of Section 1, we restrict the constructions either to the category S K fof simp* *licial flat k-modules or to the category Mkfof flat k-complexes, as appropriate. It remains* * only to transport the information that these constructions then give on the level of* * simplicial objects to the desired information on the level of objects. Consider the categories S K of simplicial k-modules, Mk of k-complexes, and t* *heir associated categories of simplicial objects S S K and S Mk. All objects are t* *o be Adams graded. A simplicial simplicial k-module is the same thing as a bisimpli* *cial k-module, and such objects arise naturally as simplicial bar constructions B*(F* *; C; A), where F takes values in S K . To understand our conversion theorems, we must understand the properties of t* *he normalized chain complex functor C# : S K ! Mk; its generalization to the totalization functor C# : S Mk ! Mk; and the diagonal functor : S S K ! S K : 48 IGOR KRIZ AND J. P. MAY (We are using the notation C# to avoid confusion with our use of the notation C* ** for the monads in domains associated to an operad C .) For a simplicial k-module X without Adams grading, C#(X) is just the chain c* *om- P i plex X=D(X) with differential d = (-1) @i, where D(X) is the subcomplex of X generated by the degenerate simplices (which is acyclic [44, 22.3]). Since D(X* *) is a direct summand of X [44, 22.2], C# preserves inclusions. When X is Adams graded, we define (C#X)(r) = C#(X(r)). The functor C# preserves algebraic structures that are defined in terms of t* *ensor products, but it does not carry partial algebras to partial algebras in general* *. To see this, recall that, for simplicial k-modules X and Y , we have the shuffle map g : C#(X) C#(Y ) ! C#(X Y ) and the Alexander-Whitney map f : C#(X Y ) ! C#(X) C#(Y ): These are inverse chain homotopy equivalences and, because we are working on the normalized level, f O g = Id [44, 29.10]. Thus g is a split inclusion. Moreov* *er, g is commutative, associative, and unital by [44, 29.9] and inspection. Given any kind of algebraic structure defined in terms of maps : X1. .X.j! X in S K , we obtain a similar kind of algebraic structure in Mk by composing the* * maps C# with iterates of g. Here, if we start with a structure defined in terms of a* *n operad C of simplicial k-modules, we end with a structure defined in terms of the operad* * C#(C ) of k-complexes. If A* is a partial algebra in a simplicial k-module A, so that Aj is a simpl* *icial submodule of Aj and the inclusion is a quasi-isomorphism, then the obvious way * *to try to define a domain C#(A*)* in the k-complex C#(A) is to set C#(A*)j= g-1(C#(Aj) \ g(C#(A)j)) = f(C#(Aj) \ g(C#(A)j)) C#(A)j: Thus the following diagram is a pullback, where g0is the restriction of g: " j C#(A*)j O___//C#(A) g0|| |g| fflffl|" fflffl| C#(Aj)_O___//C#(Aj) In general, the top inclusion need not be a quasi-isomorphism. It is a quasi-is* *omorphism if f restricts to a left inverse f0 of g0, that is, if f(C#(Aj)) f(C#(Aj) \ g(C#(A)j)) = C#(A)j; OPERADS, ALGEBRAS, MODULES, AND MOTIVES 49 since we then have compatible direct sum decompositions. While one can write do* *wn explicit conditions in terms of faces and degeneracies which ensure that these * *inclusions hold, this approach is not very satisfactory. Thus we accept that the functor C# fails to carry partial algebras of simplic* *ial k- modules to partial algebras of k-complexes in general. To get around this, we p* *rove our conversion functors for partial algebras on the simplicial level, as stated in * *Theorems 1.1 and 1.2, and only later apply C# to obtain the k-complexes that we are really i* *nterested in. Before getting to this, we briefly consider the generalization of C# to a fun* *ctor S Mk ! Mk. This is needed to prove Theorem 1.4 and make sense of the generality of Sections 2 and 3, and it will also be used in Parts IV and V. For X 2 S Mk, * *let Xp;q(r) denote the k-module of p-simplices of ordinary grading q and Adams grad* *ing P r. Then C#X is constructed by letting (C#X)n(r) be the quotient of p+q=nXp;q(* *r) by its subgroup of degenerate simplices. The differential on C#X is the sum of* * the P i p simplicial differential (-1) di and (-1) times the internal differential; se* *e [30, pp. 65-68] for details (some of which will be recalled in IV x6). By [30, A.2], the functor C# carries simplicial homotopies of the sort occurr* *ing in Example 4.2 to chain homotopies, and a standard spectral sequence argument shows that it carries simplicial quasi-isomorphisms to quasi-isomorphisms. The shuffl* *e product and Alexander-Whitney map are generalized and shown to continue to enjoy all of* * the properties that we mentioned above in [30, A.3]. Again, given an algebraic structure defined in terms of maps : X1 . . .Xj! X in S Mk, we obtain a similar algebraic structure in Mk by composing the maps C# with iterates of g. We are interested in simplicial C -algebras and their modul* *es, where C is an operad of k-complexes. These are the same things as C_-algebras and th* *eir modules, where C_is the assoaciated operad of constant simplicial k-complexes. * *Clearly, C#(C_) = C , and it follows that the functor C# carries simplicial C -algebras * *and modules to C -algebras and modules. Of course, this fails on the partial level. Proof of Theorem 1.4.With the hypotheses and notations of Theorem 1.4 and Examp* *le 4.4(iii), we define functors B = C#B* and W = C#W* on both C -algebras A and A- modules M. Noting that A = C#A_, we define = C# * : BA ! A and ffl = C#ffl* : BA ! W A: Then W A is a P-algebra and and ffl are quasi-isomorphisms and maps of C -alg* *ebras. Thus these maps give a natural quasi-isomorphism between the C -algebra A and t* *he P-algebra W A. The argument for modules is identical. __|_| 50 IGOR KRIZ AND J. P. MAY Now consider S S K . An object X = {Xp;q(r)} in this category has a "horizon* *tal" simplicial variable p and a "vertical" simplicial variable q, as well as the Ad* *ams grading r. We again have a total chain complex functor C# : S S K ! Mk, and we say that a map f : X ! Y is a quasi-isomorphism if C#f is a quasi-isomorphism. More generally, we must consider S S D, the category of bisimplicial domains of Adams graded k-modules. Such an object X* consists of inclusions ffij : Xj ! Xj of j- invariant subobjects, where ffi0 is the identity of k, ffi1 is the identity of * *X = X1, and each ffij is a quasi-isomorphism. Definition 5.1.The diagonal functor : S S K ! S K sends X = {(Xp;q)(r)} toX = {(Xq;q)(r)}; with the diagonal face and degeneracy operations and the obvious Adams grading. Extend to a functor * : S S D ! S D by setting jX* = Xj; the required inclusion jX (X)j is obtained by restriction of the given inclusions Xj Xj. To validate this definition, we need to check that the cited inclusions are * *quasi- isomorphisms, but this is immediate from the first statement of the following s* *tandard result. As usual, the horizontal and vertical simplicial structures of a bisim* *plicial k- module X give rise to corresponding iterated homology groups, and there are spe* *ctral sequences that converge from these iterated homology groups to the homology of * *the total chain complex of X. Lemma 5.2. ([22, Satz 2.9]) For a bisimplicial k-module X, the total chain co* *mplex of X is naturally quasi-isomorphic to the chain complex associated to X. There- fore there are spectral sequences converging to H*(X) from the vertical homolog* *y of the horizontal homology simplicial k-module and from the horizontal homology of* * the vertical homology simplicial k-module. We are concerned with actions by an operad C of simplicial k-modules. As usu* *al, we say that C is -free or -projective if each Cq(j) is -free or -projective, an* *d we say that C is acyclic if C#C is acyclic. We say that C is an E1 operad if it is* * -free and acyclic; C#C is then an E1 operad of k-complexes. As observed in Ix5, examp* *les arise naturally from operads of topological spaces. We think of the given simplicial structure on C as vertical, and we let C_ b* *e the associated horizontally constant bisimplicial operad. When the functor F takes * *values in simplicial C -algebras, B*(F; C; A) takes values in simplicial simplicial C * *-algebras, which are the same things as bisimplicial C_-algebras. Partial actions work sim* *ilarly. The crucial, if trivial, fact about the functor is that it manifestly preserve* *s any such OPERADS, ALGEBRAS, MODULES, AND MOTIVES 51 operad actions, even partial ones. This makes a valuable technical substitute * *for the total chain complex functor. Proofs of Theorems 1.1 and 1.2.With the hypotheses and notations of Theorem 1.1 and Example 4.3, define functors B = B* and V = V* on both partial C -algebras and their modules. Note that A_*= A* and define = * : BA* ! A*. While the horizontal homotopy j* * ' Id does not give rise to a homotopy on applicati* *on of , it does imply that * restricts to a horizontal equivalence on each fixed * *vertical degree, and is therefore a quasi-isomorphism. Define ffi = ffi* : BA* ! RV A* **. Since ffi* restricts to a vertical quasi-isomorphism on each fixed horizontal d* *egree, ffi is a quasi-isomorphism. Since and ffi are maps of partial C -algebras, they define* * a natural quasi-isomorphism A* BA* ! RV A* between A* and the genuine C -algebra V A*. Similarly, with the hypotheses and * *no- tations of Theorem 1.2 and Example 4.4, define functors W = W* on both algebras and modules and define ffl = ffl* : V A* ! W A*. Since ffl* restricts to a vert* *ical equiv- alence on each fixed horizontal degree, ffl is a quasi-isomorphism. Since ffl i* *s a map of C -algebras, it combines with and ffi to define a natural quasi-isomorphism b* *etween A* and the P-algebra W A*. The proofs for modules are identical. __|_| 6.Higher Chow complexes Before getting to our motivating examples, we insert some general remarks abo* *ut extension of scalars and about the specialization of the arguments just given t* *o partial commutative simplicial k-algebras. Remark 6.1.Let k be a subring of K such that K is a flat k-module. If C is an o* *perad over k, A is a C -algebra, and M is an A-module, then C k K is an operad over K, A k K is a (C k K)-algebra, and M k K is an (A k K)-module. This remains true of partial structures (in view of our flatness hypothesis). All of our mon* *ads, hence also our bar constructions, also commute with extension of scalars. Under the v* *arying hypotheses of Theorems 1.1, 1.2, and 1.4, there are natural isomorphisms (BA) k K ~=B(A k K); (V A) k K ~=V (A k K); and (W A) k K ~=W (A k K) 52 IGOR KRIZ AND J. P. MAY that preserve all structure in sight and are compatible with the various natura* *l quasi- isomorphisms that were used in the proofs of the cited results. The same conclu* *sions hold for modules. Remark 6.2. (i) (A shortcut). Consider a partial commutative simplicial k-algeb* *ra A*, where k is a field of characteristic zero. Then A*is a partial N -algebra, or, * *equivalently, an N*-algebra, and we can work directly with N and its monads to effect the con* *version of Theorems 1.1 and 1.2. That is, we set BA* = B*(N*; N*; A*) and V A* = B*(NL; N*; A*): Then V A* is a commutative simplicial k-algebra, and we have quasi-isomorphisms* * of N*-algebras = * : BA* ! A* andffi = ffi* : BA* ! RV A*: A similar shortcut converts partial A*-modules to genuine V A*-modules. If C is* * another acyclic operad, we have compatible quasi-isomorphisms B*(C*; C*; A*) ! B*(N*; N*; A*) and B*(NL; C*; A*) ! B*(NL; N*; A*): (ii) Now let A*be a partial commutative simplicial k-algebra over a general com* *mutative ring k. Here, to effect the conversion of Theorem 1.1, we choose an E1 operad* * C of simplicial k-modules and regard A* as a partial C -algebra by pullback along* * the augmentation ffl : C ! N . The point is that -projectivity is essential to the * *proof. Thus Theorem 1.1 eliminates the partialness of our structures at the expense of* * fattening up the operad. When k = Z, we obtain operations on mod p homology by passage to homology from the diagram C#(C (p)) p C#(A)p ! C#(C (p) p Ap) C#(C (p) p Ap) ! C#A: The left arrow is the shuffle map, and it and the middle arrow induce isomorphi* *sms on mod p homology. The rights arrow is C#(ffl ). A diagram chase shows that the resulting operations agree with those of the quasi-isomorphic C -algebra V * *A*, and Theorem I.7.2 is still valid as stated (compare [26]). The only point worth men* *tioning is that we have added an Adams grading, and the operations Qs and fiQs carry eleme* *nts of Adams grading r to elements of Adams grading pr. We now recall the motivating examples, as defined by Bloch [7]. These are pa* *rtial commutative simplicial rings, with ground ring k = Z. Example 6.3. Let X be a (smooth, quasi-projective) variety over a field F . B* *loch [7] has defined an Adams graded simplicial Abelian group Z(X). Its group Zr(X;* * q) OPERADS, ALGEBRAS, MODULES, AND MOTIVES 53 of q-simplices in Adams grading r is free Abelian on the set of those codimensi* *on r irreducible subvarieties of X x q which meet all faces properly, where X q = Spec(F [t0; : :;:tq]=( ti- 1)): There is a partially defined intersection product on this graded simplicial Abe* *lian group. In Adams grading r and simplicial degree q, the domain Z(X)j of the j-fold prod* *uct is the sum over all partitions {r1; : :;:rj} of r of the subgroups of Zr1(X; q) . . .Zrj(X; q) spanned by those j-tuples of simplices all intersections of subsets of which me* *et all faces properly. An "easy" moving lemma of Bloch, implicit in [7] and proven in detail* * when X = Spec(F ) in [8], gives that the inclusion Z(X)j ! Z(X)j is a quasi-isomorph* *ism. It is evident that the intersection product is commutative, associative, and un* *ital. If ss : X ! Y is a flat map, we obtain a map ss* : Z(Y ) ! Z(X) of Adams graded simplicial Abelian groups by pulling cycles in simplices back along the flat ma* *ps ss : Xxq ! Y xq. It extends to a map of partial rings. That is, the partial commutat* *ive simplicial ring Z(X)* is contravariantly functorial on flat maps. In particular* *, letting Z = Z(Spec(F )), we obtain a map ss* : Z* ! Z(X)* of partial commutative simpli* *cial rings for any X. The integral higher Chow groups of X are defined by CHr(X; q) = Hq(Zr(X; *); Z): By the previous remarks, if we define the mod p Chow groups by taking mod p hom* *ol- ogy, then these groups admit homology operations just like those familiar in al* *gebraic topology. A harder moving lemma of Bloch, first claimed in [7] and recently pro* *ven in [9], implies that CHr(X; q) Q ~=(Kq(X) Q)(r): Here the right side is the nr-eigenspace of the Adams operation n for any n > * *1 (which is independent of n); Kq(X) Q is the direct sum of these eigenspaces. Levine [* *39] has recently given a different proof of this isomorphism that avoids Bloch's hard m* *oving lemma. As in Remark 6.2(ii), we choose an E1 operad C of simplicial Abelian groups a* *nd regard our partial commutative simplicial rings as partial C -algebras. We appl* *y The- orem 1.1 to convert partial C -algebras to quasi-isomorphic genuine C -algebras* *, still in the category of simplicial Abelian groups. We then apply the functor C# to conv* *ert to algebras over the associated E1 operad C#C of chain complexes. It is these c* *hain complex level structures that really interest us. Recall that we defined an E1 * *algebra 54 IGOR KRIZ AND J. P. MAY to be an algebra over any E1 operad of chain complexes, such as C#C . Of course* *, we can proceed in the same way for modules. Definition 6.4.Fix a field F and consider varieties X over F . (i) Let A (X) be the E1 algebra obtained from Z(X)* by applying the functor * *C#V . Write A , or A =F , for A (Spec(F )). Write ss* : A (Y ) ! A (X) for the map of E1 algebras induced by a flat map ss : X ! Y ; A (X) is an A (Y )-module via s* *s*. In particular, A (X) is an A -module for every X. (ii) Let AQ(X) be the commutative DGA obtained from Z(X)* Q by applying the functor C#W . Write AQ, or AQ=F , for AQ(Spec(F )). Write ss* : AQ(Y ) ! AQ(X) * *for the map of DGA's induced by a flat map ss : Y ! X. Proposition 6.5. For varieties X, A (X) Q is an E1 algebra, and there is a qua* *si- isomorphism A (X) Q ! AQ(X) of E1 algebras. Proof. By Remark 6.1, A (X) Q ~=C#V (Z(X) Q). The map ffl : V ! W used in the proof of Theorem 1.2 gives the desired quasi-isomorphism. __|_| Remark 6.6. In order to relate these definitions to the usual cohomology theori* *es in algebraic geometry, it is appropriate to regrade by setting N 2r-p(X)(r) = Ap(X)(r): With this grading, it is reasonable to define (6.7) HiMot(X; Q(r)) Hi(NQ(X))(r): Write N , or N =F , for N (Spec(F )). The E1 algebras N (X) may be viewed as N -modules and thus as objects of the derived category DN . Deligne [20, 17] su* *ggested that this derived category should provide an appropriate site in which to defin* *e integral mixed Tate motives. If one accepts the integral analog of (6.7) as the definit* *ion of integral motivic cohomology, one can view DN as a "derived category of integra* *l mixed Tate modules". If there is a good Abelian category of integral mixed Tate moti* *ves, it should be an admissible Abelian subcategory [3, 1.2.5] of this triangulated * *derived category, and it would then necessarily be its heart with respect to a suitable* * t-structure [3, 1.3.13]. However, to take this idea seriously, we must first understand suc* *h derived categories of modules over E1 algebras: that is the subject of Part V. Most work on mixed Tate motives has concentrated on the rational theory, and* * our work gives a classical category of derived modules in which to think about the * *subject. We shall return to consideration of mixed Tate motives in Part IV, after develo* *ping a new approach to the study of classical derived categories in Part III. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 55 Part III.Derived categories from a topological point of view Let k be a commutative ring and let A be a differential graded associative and unital k-algebra (= DGA). As many topologists recognize, there is an extremely * *close analogy between the derived category DA of differential graded A-modules and the stable homotopy category of spectra. However, there is no published account of * *derived categories from this point of view. With the goal of developing an integral th* *eory of mixed Tate motives, we shall generalize the derived category DA to the case * *of E1 algebras A over k in Part V. Understanding of that more difficult theory re* *quires a prior knowledge of our treatment of the derived categories of ordinary DGA's.* * This elementary theory is adequate and illuminating for one approach to rational mix* *ed Tate motives ([20, 17, 6]), as we shall show in Part IV. Therefore, with the hope that our way of thinking about derived categories wi* *ll prove useful to others, we here give a topologically motivated, although purely algeb* *raic, ex- position of the classical derived categories of DGA's. These categories admit r* *emarkably simple and explicit descriptions in terms of "cell modules", which are the prec* *ise alge- braic analogs of cell spectra. Such familiar topological results as Whitehead's* * theorem and Brown's representability theorem transcribe directly into algebra. There is* * also a theory of CW modules, but these are less useful (at least in our motivating exa* *mples), due to the limitations of the cellular approximation theorem. Derived tensor pr* *oducts and Hom functors, together with differential Tor and Ext functors and Eilenberg* *-Moore (or hyperhomology) spectral sequences for their computation, drop out quite eas* *ily. Our methods can be abstracted and applied more generally, and some of what we do can be formalized in Quillen's context of closed model categories [54]. We p* *refer to be more concrete and less formal. We repeat that many topologists have long kno* *wn some of this material. For example, although the emphasis is quite different, o* *ur work overlaps that of [30] and [2]. On a technical note, we emphasize that, as in [3* *0], k is an arbitrary commutative ring and we nowhere impose boundedness or flatness hypoth* *eses. 1.Cell A-modules Returning to the conventions of Part I, but regrading cohomologically and add* *ing an Adams grading, we take k-modules to mean Z-bigraded chain complexes over k, with gradings written X = {Xq(r)}, throughout Parts III-V. We call q the ordina* *ry grading or degree and r the Adams grading or degree. The differential d sends X* *q(r) to Xq+1(r). The grading does and the Adams grading does not introduce signs und* *er permutations. These conventions are motivated by the motivic context. The reade* *r with other motivations may prefer to forget the Adams grading and to regrade homolog* *ically, 56 IGOR KRIZ AND J. P. MAY setting Xq = X-q; this makes the analogy with topology far more transparent. Ex* *cept where otherwise specified, a map f : X ! Y of k-modules means a map of bidegree (0; 0) that commutes with the differentials; f is a quasi-isomorphism if it ind* *uces an isomorphism on homology. We sometimes write x 2 (q; r) to indicate that an element of some module is * *of bidegree (q; r). We begin with some utterly trivial notions, expressed so as t* *o show the analogy with topology. Let I denote the "unit interval k-module". It is f* *ree on generators [0] 2 (0; 0), [1] 2 (0; 0), and [I] 2 (-1; 0), with d[I] = [0] - [1]* *. A homotopy is a map X I ! Y , where means k. Of course, M (X Y )q(r) = Xm (s) Y n(t); m+n=q s+t=r with d(x y) = dx y + (-1)deg(x)x dy. The cone CX is the quotient module X (I=k[1]) and the suspension X is X (I=@I), where @I has basis [0] and [1]. Additively, CX is the sum of copies of X and X, but with differential arranged * *so that H*(CX) = 0. The usual algebraic notation for the suspension is X = X[1], and(X)q = Xq+1. Since we have tensored the interval coordinate on the right, t* *he differential on X is the same as the differential on X, without the introductio* *n of a sign. The cofiber of a map f : X ! Y is the pushout of f along the inclusion X = X [0] ! CX. There results a short exact sequence 0 ! Y ! Cf ! X ! 0: Up to sign, the connecting homomorphism of the resulting long exact homology se- quence is f*. Explicitly, (Cf)q = Y q Xq+1, with differential d(y; x) = (dy + (-1)qfx; dx): The sequence X ! Y ! Cf ! X is called a cofiber sequence, or an exact triangle. Now assume given a DGA A over k; A is to be associative and unital, but not necessarily commutative, and A-modules will usually mean left A-modules. If X i* *s a k-module and M is an A-module, then M X is an A-module, hence the notion of a homotopy between maps of A-modules is defined. Since we defined cofiber sequenc* *es in terms of tensoring with k-modules, the cofiber sequence generated by a map of A-modules is clearly a sequence of A-modules. Let MA denote the category of A- modules and hMA its homotopy category. Then the derived category DA is obtained from hMA by adjoining formal inverses to the quasi-isomorphisms of A-modules. * *In OPERADS, ALGEBRAS, MODULES, AND MOTIVES 57 Construction 2.7, we shall give an explicit description that makes it clear tha* *t there are no set theoretic difficulties. (This point is typically ignored in algebraic ge* *ometry and obviated by concrete construction in algebraic topology.) The sequences isomorphic to cofiber sequences in the respective categories gi* *ve hMA and DA classes of exact triangles with respect to which they become triangulate* *d cate- gories in the sense of Verdier [60]. More precisely, they become so after the i* *ntroduction of graded maps or rather, in our context, bigraded maps. A map of bidegree (s; * *t) con- sists of maps Mq(r) ! Nq+s(r + t) that commute with the differentials and A-act* *ions. Such maps can be thought of as maps M ! s(t)N of bidegree (0; 0), where the suspension functor s(t) is specified by (s(t)M)q(r) = Mq+s(r + t); with differential and A-action inherited from M. Since we have allowed ourselve* *s Z- bigrading, each such functor is an automorphism of MA, and the introduction of * *bi- graded morphisms is in principle a notational device that can add nothing of su* *bstance to the mathematics. It becomes crucial when we define Hom modules of bigraded m* *or- phisms, but until then it is convenient to think solely in terms of maps of bid* *egree (0; 0). It is also convenient to think of the suspension functors in a different way.* * Let Ss(t) be the free k-module generated by a cycle is(t) 2 (s; t). Then our suspension f* *unctors are just s(t)M = M Ss(t): We think of the Ss(t) as sphere k-modules. We let F s(t) = A Ss(t) and think o* *f the F s(t) as sphere A-modules; they are free on the generating cycles is(t). Sinc* *e s and t run through Z, the analogy is with stable homotopy theory: that is where nega* *tive dimensional spheres live. In fact, the modern description of the stable homotopy category [40] translat* *es di- rectly into our new description of the derived category. (The preamble of [40] * *explains the relationship with earlier treatments of the stable homotopy category, which* * do not have the same flavor.) In brief, one sets up a category of spectra. In that cat* *egory, one defines a theory of cell and CW spectra that allows negative dimensional sphere* *s. One shows that a weak homotopy equivalence between cell spectra is a homotopy equiv* *a- lence and that every spectrum is weakly homotopy equivalent to a cell spectrum.* * The stable homotopy category is obtained from the homotopy category of spectra by f* *or- mally inverting the weak homotopy equivalences, and it is described more concre* *tely as the homotopy category of cell spectra. With spectra and weak homotopy equiv* *a- lences replaced by A-modules and quasi-isomorphisms, precisely the same pattern* * works 58 IGOR KRIZ AND J. P. MAY algebraically _ but of course far more simply. Definitions 1.1.(i) A cell A-module M is the union of an expanding sequence of * *sub A-modules Mn such that M0 = 0 and Mn+1is the cofiber of a map OEn : Fn ! Mn, wh* *ere Fn is a direct sum of sphere modules F s(t) (of varying bidegrees). The restric* *tion of OEn to a summand F s(t) is called an attaching map and is determined by the "attach* *ing cycle" OEn(is(t)). An attaching map F s(t) ! Mn induces a map CF s(t) = A CSs(t) ! Mn+1 M; and such a map is called an (s - 1; t)-cell. Thus Mn+1is obtained from Mn by ad* *ding a copy of F s-1(t) for each attaching map with domain F s(t), but giving the new * *generators js-1(t) = is(t) [I] the differentials d(js-1(t)) = (-1)sOEn(is(t)): We call such a copy of F s-1(t) in M an open cell; if we ignore the differentia* *l, then M is the direct sum of its open cells. (ii) A map f : M ! N between cell A-modules is cellular if f(Mn) Nn for all* * n. (iii) A submodule L of a cell A-module M is a cell submodule if L is a cell * *A-module such that Ln Mn and the composite of each attaching map F s(t) ! Ln of L with * *the inclusion Ln ! Mn is an attaching map of M. Thus every cell of L is a cell of M. We call {Mn} the sequential filtration of M. It is essential for inductive a* *rguments, but it should be regarded as flexible and subject to change whenever convenient* *. It merely records the order in which cells are attached and, as long as the cycles* * to which attachment are made are already present, it doesn't matter when we attach cells. Lemma 1.2. Let f : M ! N be an A-map between cell A-modules. Then M admits a new sequential filtration with respect to which f is cellular. Proof. Assume inductively that Mn has been filtered as a cell A-module Mn = [M0q such that f(M0q) Nq for all q. Let x 2 Mn be an attaching cycle for the constr* *uction of Mn+1 from Mn and let O : CF s(t) ! Mn+1 be the corresponding cell. Let q be minimal such that both x 2 M0qand f O O has image in Nq+1. Extend the filtratio* *n of Mn to Mn+1 by taking x to be a typical attaching cycle of a cell CF s(t) ! M0q+* *1. __|_| From a topological point of view, our cohomological grading has the effect t* *hat we are looking at things upside down: the bottom summand of a cone CF s(t) is the one * *that involves the unit interval. That may help explain the intuition behind the fol* *lowing definition. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 59 Definition 1.3.The dimension of a cell CF s(t) ! Mn+1 is s - 1. A cell A-module* * M is said to be a CW A-module if each cell is attached only to cells of higher di* *mension, in the sense that the defining cycles OEn(is(t)) are elements in the sum of the* * images of cells of dimension at least s. The n-skeleton Mn of a CW A-module is the sum of* * the images of its cells of dimension at least n, so that Mn Mn-1. We require of ce* *llular maps f : M ! N between CW A-modules that they be "bicellular", in the sense that both f(Mn) Nn and f(Mn) Nn for all n. By Lemma 1.2, the latter condition can be arranged by changing the order in which the cells of M are attached. Definition 1.4.A cell A-module is finite dimensional if it has cells in finitel* *y many dimensions. It is finite if it has finitely many cells. Just as finite cell spectra are central to the topological theory, so finite * *cell A-modules are central here, especially when we restrict to commutative DGA's and discuss * *duality. The collection of cell A-modules enjoys the following closure properties, which* * imply many others. Proposition 1.5.(i) A direct sum of cell A-modules is a cell A-module. (ii) If L is a cell submodule of a cell A-module M, N is a cell A-module, and f* * : L ! M is a cellular map, then the pushout N [fM is a cell A-module with sequential fi* *ltration {Nn [fMn}. It contains N as a cell submodule and has one cell for each cell of * *M not in L. (iii) If L is a cell submodule of a cell A-module M and X is a cell submodule o* *f a cell P k-module Y , then M Y is a cell A-module with sequential filtration { p(MpYn-p* *)}. It contains L Y + M X as a cell submodule and has a (q + s; r + t)-cell for e* *ach pair consisting of a (q; r)-cell of Mp and an (s; t)-cell of Yn-p, 0 p n. (iv) The mapping cylinder Mf = N [f (L I) of f : L ! N is the pushout defined by taking L = L k[0] L I. If f is a cellular map between cell A-modules, then Mf is a cell A-module, L = L k[1] is a cell submodule, the inclusion N ! Mf is* * a homotopy equivalence, and Cf = Mf=L. Proof.Parts (i) and (ii) are easy and (iv) follows from (ii) and (iii). For (ii* *i), observe that there are evident canonical isomorphisms Sq(r) Ss(t) ~=Sq+s(r + t) and F q(r) Ss(t) ~=F q+s(r + t): M Y has an open cell F q+s(r + t) for each open cell F q(r) of M and Ss(t) of * *Y ; the differential on its canonical basis element is the cycle d(jq(r)) js(t) + (-1)q(jq(r)) d(js(t)): __|_| 60 IGOR KRIZ AND J. P. MAY 2. Whitehead's theorem and the derived category A quick space level version of some of the results of this section may be fo* *und in [50], and the spectrum level model is given in [40, I x5]. We construct the derived c* *ategory explicitly in terms of cell modules. As in topology, the "homotopy extension an* *d lifting property" is pivotal. It is a direct consequence of the following trivial obser* *vation. Let i0 and i1 be the evident inclusions of M in M I. Lemma 2.1. Let e : N ! P be a map such that e* : H*(N) ! H*(P ) is a monomor- phism in degree s and an epimorphism in degree s - 1. Then, given maps f, g, an* *d h such that f|F s(t) = hi0 and eg = hi1 in the following diagram, there are maps * *"gand "h that make the entire diagram commute. i0 s i1 s F s(t)___________//F (t) Ioo_________F_(t) | tt | g ww | | httt | www | | ttt | ww | | yytt |e --ww | | P oo_______|_________NeeJ | | xx;; JJ | ccGGG | | f xx JJJ | GG | | xxx JJJ | GGG | fflffl|xx "h J fflffl| "g G fflffl| CF s(t)_____i____//_CF s(t)ooI________ CF s(t) 0 i0 Proof. Let i = is(t) [0] and j = is(t) [I] be the basis elements of CF s(t), * *so that d(j) = (-1)si. Then eg(i) = h(i [1]) and f(i) = h(i [0]), hence d(h(i [I]) - f(j)) = (-1)s+1eg(i): Since eg(i) bounds in P , g(i) must bound in N, say d(n0) = g(i). Then p e(n0) + (-1)s(h(i [I]) - f(j)) is a cycle. There must be a cycle n 2 N and a chain q 2 P such that d(q) = p - e(n): Define "g(j) = (-1)s(n0- n) and "h(j [I]) = q. __|_| Theorem 2.2 (HELP). Let L be a cell submodule of a cell A-module M and let e : N ! P be a quasi-isomorphism of A-modules. Then, given maps f : M ! P , g : L ! N, and h : L I ! P such that f|L = hi0 and eg = hi1 in the following OPERADS, ALGEBRAS, MODULES, AND MOTIVES 61 diagram, there are maps "gand "hthat make the entire diagram commute. i0 i1 L ___________//_L Ioo_________L | ww | g ""| | hwww | """ | | ww | "" | | --ww |e """ | | P oo_____|______N_ | | ">> ccGGG | ``AA | | f""" GGG | AAA | | "" GG | AA | fflffl|"""h G fflffl| "gA fflffl| M ______i0___//M Ioo__i1_____M Proof.By induction up the filtration {Mn} and pullback along cells not in L, th* *is quickly reduces to the case (M; L) = (CF s(t); F s(t)) of the lemma. __|_| For objects M and N of any category Cat, let Cat(M; N) denote the set of morp* *hisms in Cat from M to N. Theorem 2.3 (Whitehead). If M is a cell A-module and e : N ! P is a quasi- isomorphism of A-modules, then e* : hMA(M; N) ! hMA(M; P ) is an isomorphism. Therefore a quasi-isomorphism between cell A-modules is a homotopy equivalence. Proof.Take L = 0 in HELP to see the surjectivity. Replace (M; L) by the pair (M I; M (@I)) to see the injectivity. When N and P are cell A-modules, we may take M = P and obtain a homotopy inverse f : P ! N. __|_| Theorem 2.4 (Cellular approximation). Let L be a cell submodule of a CW A- module M, let N be a CW A-module such that Hs(N=Ns) = 0 for all s, and let f : M ! N be a map whose restriction to L is cellular. Then f is homotopic rela* *tive to L to a cellular map. Therefore any map M ! N is homotopic to a cellular map, and any two homotopic cellular maps are cellularly homotopic. Proof.By Lemma 1.2, we may change the sequential filtration of M to one for whi* *ch f is sequentially cellular. Proceeding by induction up the filtration {Mn}, we co* *nstruct compatible cellular maps gn : Mn ! Nn and a homotopy hn : Mn I ! Nn from f|Mn to gn. The result quickly reduces to the case of a single cell of M that is not* * in L and thus to the case when (M; L) = (CF s(t); F s(t)). The conclusion follows by app* *lication of Lemma 2.1 to the inclusions e : (Nn)s-1! Nn. __|_| Remark 2.5.If Hs(A) = 0 for all s > 0, then the hypothesis holds for all N, and we can work throughout with CW A-modules and cellular maps rather than with cell A-modules. Of course, if we regrade homologically, then this means that Hs(A) =* * 0 for s < 0, which matches the intuition: CW theory works topologically because t* *he homotopy groups of the zero sphere spectrum are zero in negative degrees. 62 IGOR KRIZ AND J. P. MAY Theorem 2.6 (Approximation by cell modules). For any A-module M, there is a cell A-module N and a quasi-isomorphism e : N ! M. Proof. We construct an expanding sequence Nn and compatible maps en : Nn ! M inductively. Choose a cycle 2 (q; r) in each homology class of M, let N1 be t* *he direct sum of A-modules F q(r), one for each , and let e1 : N1 ! M send the th canonical basis element to the cycle . Inductively, suppose that en : Nn ! M h* *as been constructed. Choose a pair of cycles (; 0) in each pair of unequal homolo* *gy classes on Nn that map under (en)* to the same element of H*(M). Let Nn+1 be the "homotopy coequalizer" obtained by adjoining a copy of F q(r) I to Nn alon* *g the evident map F q(r)@I ! Nn determined by each such pair (; 0) 2 (q; r). Proposit* *ion 1.5 implies that Nn+1 is a cell A-module such that Nn is a cell submodule. Any * *choice of chains 2 M such that d() = - 0 determines an extension of en : Nn ! M to en+1: Nn+1! M. Let N be the direct limit of the Nn and e : N ! M be the resulti* *ng map. Clearly, N is a cell module, e induces an epimorphism on homology since e1* * does, and e induces a monomorphism on homology by construction. __|_| Construction 2.7. For each A-module M, choose a cell A-module M and a quasi- isomorphism fl : M ! M. By the Whitehead theorem, for a map f : M ! N, there is a map f : M ! N, unique up to homotopy, such that the following diagram is homotopy commutative: f M ___//_N fl|| |fl| fflffl| fflffl| M __f__//_N Thus is a functor hMA ! hMA, and fl is natural. The derived category DA can be described as the category whose objects are the A-modules and whose morphisms a* *re specified by DA(M; N) = hMA(M; N); with the evident composition. When M is a cell A-module, DA(M; N) ~=hMA(M; N): Using the identity function on objects and on morphisms, we obtain a functor i* * : hMA ! DA that sends quasi-isomorphisms to isomorphisms and is universal with th* *is property. Let CA be the full subcategory of MA whose objects are the cell A-mod* *ules. Then the functor induces an equivalence of categories DA ! hCA with inverse the composite of i and the inclusion of hCA in hMA. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 63 Therefore the derived category and the homotopy category of cell modules can * *be used interchangeably. Homotopy-preserving functors on A-modules that do not pre* *serve quasi-isomorphisms are transported to the derived category by first applying , * *then the given functor, a point that we return to in Section 4. More emphasis is pl* *aced on this procedure in the algebraic than the topological literature: topologists* * routinely transport constructions to the stable category by passing to CW spectra, withou* *t change L of notation. For example, topologists always write ^ rather than ^ for the der* *ived smash product. In fact, while a great deal of modern work depends heavily on ha* *ving a good underlying category of spectra in which to work, early constructions of * *the stable homotopy category did not even allow spectra that were more general than* * CW spectra. For this and other reasons, topologists are accustomed to work with CW spectra and their cells in a concrete calculational way, not as something esote* *ric but rather as something much more basic and down to earth than general spectra. An analogous view of differential graded A-modules is rather intriguing. 3. Brown's representability theorem Functors of cohomological type on DA are of considerable interest, and we her* *e recall a categorical result that characterizes when they can be represented in the for* *m D(?; N). The topological analogue has long played an important role. We have said that we think of the F q(r) as analogs of sphere spectra. Just a* *s maps out of spheres calculate homotopy groups and therefore detect weak equivalences, so* * maps out of the F q(r) calculate homology groups and therefore detect quasi-isomorph* *isms. We display several versions of this fact for later use: for all A-modules N, (3.1) Hq(N)(r) ~=hMk(k; N Sq(r)) ~=hMk(S-q(-r); N) ~=hMA(F -q(-r); N) ~=DA(F -q(-r); N): The category DA has "homotopy limits and colimits". For example, the homotopy pushout of maps f : L ! M and g : L ! N is obtained from M (L I) N by identifying l [0] with f(l) and l [1] with g(l). More precisely, we first ap* *ply cell approximation and then apply the cited construction. We used a similar homotopy coequalizer in the proof of Theorem 2.6. The homotopy colimit, or telescope Tel* *Mi, of a sequence of maps fi : Mi ! Mi+1is the homotopy coequalizer of Id: Mi ! Mi and fi : Mi ! Mi; equivalently, it is the cofiber of g : Mi ! Mi, where g(m) = m - fi(m) for m 2 Mi. Homotopy pushouts, homotopy coequalizers, and homotopy colimits of sequences are weak pushouts, weak coequalizers, and weak c* *olimits in the sense that they satisfy the existence but not the uniqueness property of* * categorical pushouts, coequalizers, and colimits. 64 IGOR KRIZ AND J. P. MAY We now have enough information to quote the categorical form of Brown's repr* *e- sentability theorem given in [13], but we prefer to run through a quick concret* *e version of the proof. Theorem 3.2 (Brown). A contravariant functor J : DA ! Setsis representable in the form J(M) ~=DA(M; N) for some A-module N if and only if J converts direct s* *ums to direct products and converts homotopy pushouts to weak pullbacks. Proof. Necessity is obvious. Thus assume given a functor J that satisfies the s* *pecified direct sum and Mayer-Vietoris axioms. Since homotopy coequalizers and telescope* *s can be constructed from sums and homotopy pushouts, J converts homotopy coequalizer* *s to weak equalizers and telescopes to weak limits. Write f* = J(f) for a map f. Con* *sider pairs (M; ) where M is an A-module and 2 J(M). Starting with an arbitrary pair (N0; 0), we construct a sequence of pairs (N* *i; i) and maps fi: Ni! Ni+1such that f*i(i+1) = i. Let N1 = N0 (F q(r)), where there is a copy of F q(r) for each element OE of each set J(F q(r)). Let 1 have coordina* *tes and the elements OE, and let f0 : N0 ! N1 be the inclusion. Inductively, given (Ni;* * i), let Li be the sum of a copy of F q(r) for each (q; r) and each unequal pair (x; y) of * *elements of Hq(Ni)(r) such that, when thought of as maps F q(r) ! Niin DA, x*(i) = y*(i). Let fi: Ni! Ni+1be the coequalizer of the pair of maps Li! Nigiven by the x's a* *nd the y's. By the weak equalizer property, there is an element i+12 J(Ni+1) such * *that f*i(i+1) = i. Let N = TelNi. By the weak limit property, there is an element 2 J(N) that pulls back to i for each i. For an A-module M, define : DA(M; N) ! J(M) by (f) = f*(). Then, by construction, is a bijection for all F q(r). We claim t* *hat is a bijection for all M. Suppose given elements x; y 2 DA(M; N) such that (x) = (y). Replacing M by a cell approximation if necessary, we can assume that x and y are given by m* *aps M ! N. Let c : N ! N00be the homotopy coequalizer of x and y and choose an element 002 J(N00) such that c*(00) = . Construct a pair (N0; 0) by repeating t* *he construction above, but starting with the pair (N00; 00). Let j : N00! N0be the* * evident map such that j*(0) = 00. Then, since (jc)*(0) = and both and 0 are bijectio* *ns for all F q(r), jc : N ! N0 is an isomorphism in DA. Since cx = cy by construct* *ion, it follows that x = y. Therefore is an injection for all A-modules M. Finally, let ! 2 J(M) for any module M. Repeat the construction, starting w* *ith the zeroth pair (M N; (!; )). We obtain a new pair (N0; 0) together with a map i : M ! N0 such that i*(0) = ! and a map j : N ! N0 such that j*(0) = . Again, OPERADS, ALGEBRAS, MODULES, AND MOTIVES 65 j is an isomorphism in DA since both and 0 are bijections for all F q(r). The* *refore ! = (ij-1)*() and is a surjection for all A-modules M. __|_| Observe that we can start with N0 = 0, in which case N can be given the struc* *ture of a cell A-module. Of course, it is formal that the module N that represents J is* * unique up to isomorphism in DA and that natural transformations between representable fun* *ctors are represented by maps in DA. There is an analog due to Adams that applies when the functor J is only given* * on finite cell A-modules. The proof is a direct translation from topology to algeb* *ra of that given in [1] and will be omitted. Theorem 3.3 (Adams). A contravariant group-valued functor J defined on the ho- motopy category of finite cell A-modules is representable in the form J(M) ~=DA* *(M; N) for some cell A-module N if and only if J converts finite direct sums to direct* * products and converts homotopy pushouts to weak pullbacks of underlying sets. Here N is usually infinite and is unique only up to non-canonical isomorphism* *. More precisely, maps g; g0: N ! N0 are said to be weakly homotopic if gf is homotopi* *c to g0f for any map f : M ! N defined on a finite cell A-module M. There is a resul* *ting weak homotopy category of cell A-modules, and N is unique up to isomorphism in * *that category. 4. Derived tensor product and Hom functors: Tor and Ext We first record some elementary facts about tensor products with cell A-modul* *es. Lemma 4.1. Let N be a cell A-module. Then the functor M A N preserves exact sequences and quasi-isomorphisms in the variable M. Proof.With differential ignored, N is a free A-module, and preservation of exac* *t se- quences follows. The sequential filtration of N gives short exact sequences of* * free A-modules 0 -! Nn -! Nn+1- ! Nn+1=Nn -! 0; where the subquotients Nn+1=Nn are direct sums of sphere A-modules. The preserv* *a- tion of quasi-isomorphisms holds trivially if N is a sphere A-module, and the g* *eneral case follows by passage to direct sums, induction up the filtration, and passag* *e to colimits. __|_| L It is usual to define the derived tensor product, denoted M AN, by replacing* * the left A-module N (or the right A-module M) by a suitable resolution P and taking* * the ordinary tensor product M A P , in line with the standard rubric of derived fun* *ctors (see e.g. Verdier [60], who restricts to bounded below modules). Our procedure * *is the 66 IGOR KRIZ AND J. P. MAY same, except that we take approximation by quasi-isomorphic cell A-modules as o* *ur L version of resolution. That is, we define the derived tensor product M AN in D* *k to be M A N. The lemma shows that the definition makes sense. We leave it as an exerc* *ise to verify that this definition of the derived tensor product agrees with the us* *ual one. (For example, one might use Theorem 4.13 below.) We can also use the lemma to s* *how that the derived category DA depends only on the quasi-isomorphism type of A. Proposition 4.2. Let OE : A ! A0be a quasi-isomorphism of DGA's. Then the pull- back functor OE* : DA0! DA is an equivalence of categories with inverse given b* *y the L derived extension of scalars functor A0 A(?). Proof. For M 2 MA and M02 MA0, we have MA0(A0A M; M0) ~=MA(M; OE*M0): The functor A0A(?) preserves sphere modules and therefore cell modules. This im* *plies formally that the adjunction passes to derived categories, giving L 0 * 0 DA0(A0 AM; M ) ~=MA(M; OE M ): If M is a cell A-module, then OE Id: M ~=A A M -! OE*(A0A M) is a quasi-isomorphism of A-modules. These maps give the unit of the adjunction* *. Its counit is given by the maps of A0-modules IdOEfl : A0A M0- ! A0A0M0~=M0; where M0 is a cell A0-module, M0 is a cell A-module and fl : M0- ! M0 is a quas* *i- isomorphism of A-modules. Since the composite of this map with the quasi-isomor* *phism OEId for the A-module M0coincides with fl, this map too is a quasi-isomorphism.* * __|_| For left A-modules M and N, let Hom A(M; N)q(r) be the k-module of homomor- phisms of A-modules of bidegree (q; r) with the standard differential (df)(m) = d(f(m)) - (-1)qf(d(m)): For k-modules K, (4.3) MA(K M; N) ~=Mk(K; HomA (M; N)); where A acts on L M through its action on M (with the usual sign convention). This isomorphism clearly passes to homotopy categories. Letting L run through * *the sphere k-modules and using (3.1) and the Whitehead theorem, we see that if M is* * a cell A-module then the functor Hom A(M; N) preserves quasi-isomorphisms in N. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 67 This allows us to define the right derived Hom functor in DA by letting RHom * *A(M; N) be given by Hom A(M; N), where M is a cell approximation of M. This gives a wel* *l- defined functor such that, for k-modules K, L (4.4) DA(K M; N) ~=Dk(K; RHom A(M; N)): Remark 4.5.The argument we have just run through is a special case of a general* * one. If S and T are left and right adjoint functors between two categories of the so* *rt that we are considering, then S preserves objects of the homotopy type of cell modules * *if and only T preserves quasi-isomorphisms, and in that case the resulting induced fun* *ctors on derived categories are still adjoint. See [40, I.5.13] for a precise categorica* *l statement. We can now define differential Tor and Ext (or hyperhomology and hypercohomol* *ogy) groups as follows. We cheerfully ignore questions of justification in terms of * *standard homological terms: these are of little interest to us, and such language would * *be un- available in the precisely analogous E1 context of Part V (let alone the topol* *ogical context of [25]). Definition 4.6.Define L * * Tor*A(M; N) = H*(M AN) and ExtA(M; N) = H (RHom A(M; N)): These are Adams graded k-modules (with notation for the Adams grading sup- pressed). However Tor and Ext are defined, the essential point is to have Eilen* *berg- Moore, or hyperhomology, spectral sequences for their calculation. Theorem 4.7. There are natural spectral sequences of the form (4.8) Ep;q2= Torp;qH*A(H*M; H*N) =) Torp+qA(M; N) and (4.9) Ep;q2= Extp;qH*A(H*M; H*N) =) Extp+qA(M; N): These are both spectral sequences of cohomological type, with (4.10) dr : Ep;qr! Ep+r;q-r+1r: In (4.8), p is the negative of the usual homological degree, the spectral seque* *nce is non-zero only in the left half-plane, and it converges strongly. In (4.9), the* * spectral sequence is non-zero only in the right half plane, and it converges strongly if* *, for each fixed (p; q), only finitely many of the differentials (4.10) are non-zero. (The* * best study of the convergence of spectral sequences, unfortunately still unpublished, is g* *iven in [10].) 68 IGOR KRIZ AND J. P. MAY Our construction of the spectral sequences follows [30], which is a precurso* *r of the present approach to derived categories. Let ffl : P ! N be a quasi-isomorphism * *of left A-modules, where P is a cell A-module. Refilter P by setting F 1-nP = Pn. Thus 0 = F 1P F 0P F -1P . . .F -nP . .:. Suppressing the Adams grading, we see that this filtration gives rise to a spec* *tral sequence that starts from Ep;q0P = (F pP=F p+1P )p+q~=A (P p;*)p+q; where P p;*is k-free on the canonical basis elements of the open cells of P1-p.* * The definition of a cell module implies that d0 = d 1. Therefore Ep;*1P ~=H*(A) P p;*: Thinking of N as filtered with F 1N = 0 and F pN = N for p 0, we see that E*;** *1P gives a complex of left H*(A)-modules (4.11) . .!.Ep-1;*1P ! Ep;*1P ! . .!.E0;*1P ! H*(N) ! 0: Definition 4.12.Let P be a cell A-module. A quasi-isomorphism ffl : P ! N is sa* *id to be a distinguished resolution of N if the sequence (4.11) is exact, so that * *{Ep;*1P } is a (negatively indexed) free H*(A)-resolution of H*(N). Observe that ffl : P ! N is necessarily a homotopy equivalence if N is a cel* *l A-module, by Whitehead's theorem. The following result of Gugenheim and May [30, 2.1] sho* *uld be viewed as a greatly sharpened version of Theorem 2.6: it gives cell approxim* *ations with precisely prescribed algebraic properties. Theorem 4.13 (Gugenheim-May). For any A-module N, every free H*(A)-resolu- tion of H*(N) can be realized as {Ep;*1P } for some distinguished resolution ff* *l : P ! N. A distinguished resolution ffl : P ! N of a cell A-module A-module N induces* * a homotopy equivalence M A P ! M A N for any (right) A-module M. Filtering M A P by F p(M A P ) = M A (F pP ); p 0; we obtain the spectral sequence (4.8). Similarly, a distinguished resolution ffl : P ! M of a cell A-module A-modul* *e M in- duces a homotopy equivalence Hom A(M; N) ~=Hom A(P; N) for any (left) A-module * *N, and the filtration F pHomA(P; N) = Hom A(P=F 1-pP; N); p 0; gives rise to the spectral sequence (4.9). OPERADS, ALGEBRAS, MODULES, AND MOTIVES 69 In both cases, the identification of E2-terms is immediate from the definitio* *n of a distinguished resolution. Details and applications may be found in [30]. A di* *fferent construction of the spectral sequences can be obtained by specialization of Vx7* *. It will be immediate from the discussion in the next section that, when A is commu* *ta- tive, Tor*A(M; N) and Ext*A(M; N) are H*(A)-modules and the spectral sequences * *are spectral sequences of differential H*(A)-modules. 5. Commutative DGA's and duality Let A be commutative throughout this section. We give DA a structure of a sym* *metric monoidal category (= tensor category [21, 1.2]) with internal hom objects. We * *also discuss duality, characterizing the strongly dualizable objects or, in another * *language, L identifying the largest rigid tensored subcategory of DA. Again, in DA, M AN i* *s given by M A N. Of course, since A is commutative, this is an A-module. From our present point of view, it makes good sense to resolve both variables since we n* *ow have the canonical isomorphisms F q(r) A F s(t) ~=F q+s(r + t): As in Proposition 1.5(iii), this directly implies that tensor products of cell * *A-modules are cell A-modules. Proposition 5.1.Let M and M0 be cell A-modules. Then M A M0 is a cell A- P module with sequential filtration { p(MpA Nn-p)}. It has a (q + s; r + t)-cell* * for each pair consisting of a (q; r)-cell of Mp and an (s; t)-cell of M0n-p, 0 p n. For A-modules M and N, Hom A(M; N) is an A-module such that (5.2) MA(M A N; P ) ~=MA(M; HomA (N; P )): In DA, RHom A(M; N) is given by Hom A(M; N), and we have an isomorphism L (5.3) DA(M AN; P ) ~=DA(M; RHom A(N; P )): The standard coherence isomorphisms (= associativity and commutativity constrai* *nts) on the tensor product pass to the derived category, which is thus a symmetric m* *onoidal closed category in the sense of [43, 36]. There are general accounts of duality theory in such a context in the literat* *ure of both algebraic geometry [21, x1], [19] and algebraic topology [23]; we follow [40, I* *II xx1-2]. Observe first that, by an easy direct inspection of definitions, the functor Ho* *m A(M; N) preserves cofiber sequences in both variables. (Actually, in the variable M, th* *e functor Hom A converts an exact triangle into the negative of an exact triangle.) 70 IGOR KRIZ AND J. P. MAY The dual of an A-module M, denoted M_ or DM, is defined to be Hom A(M; A). The adjunction (5.2) specializes to give an evaluation map ffl : M_ A M ! A and* * a map j : A ! Hom A(M; M). There is a natural map (5.4) : Hom A(L; M) A N ! Hom A(L; M A N); which specializes to (5.5) : M_ A M ! Hom A(M; M): A cell A-module M is said to be "finite" or "strongly dualizable" or "rigid" if* * there is a coevaluation map j : A ! M A M_ such that the following diagram commutes in DA, where o is the commutativity isomorphism. j _ A _________//M A M (5.6) j || o|| fflffl| fflffl| Hom A(M; M) oo__M_ A M The definition has many purely formal implications. The map of (5.4) is an * *iso- morphism in DA if either L or N is finite. The map of (5.5) is an isomorphism * *in DA if and only if M is finite, and, in DA, the coevaluation map j is then the c* *omposite fl-1j in (5.6). The natural map ae : M ! M__ is an isomorphism in DA if M is finite. The natural map : Hom A(M; N) A Hom A(M0; N0) ! Hom A(M A M0; N A N0) is an isomorphism in DA if M and M0 are finite or if M is finite and N = A. Say that a cell A-module N is a direct summand up to homotopy of a cell A-mo* *dule M if there is a homotopy equivalence of A-modules between M and N N0 for some ce* *ll A-module N0. Theorem 5.7. A cell A-module is finite in the sense just defined if and only i* *f it is a direct summand up to homotopy of a finite cell A-module. Proof. Observe first that F q(r) is finite with dual F -q(-r), hence any finite* * direct sum of A-modules F q(r) is finite. Observe next that the cofiber of a map between * *finite A-modules is finite. In fact, the evaluation map ffl induces a natural map L _ L ffl# : DA(L; N AM ) ! DA(L AM; N); and M is finite if and only if ffl# is an isomorphism for all L and N [40, III.* *3.6]. Since both sides turn cofiber sequences in the variable M into long exact sequences, * *the five OPERADS, ALGEBRAS, MODULES, AND MOTIVES 71 lemma gives the observation. We conclude by induction on the number of cells t* *hat a finite cell A-module is finite. It is formal that a direct summand in DA of a* * finite A-module is finite. For the converse, let M be a cell A-module that is finite * *with coevaluation map j : A ! M A M_. Clearly j factors through N A M_ for some finite cell subcomplex N of M. By [40, III.1.2], the bottom composite in the fo* *llowing commutative diagram is the identity in DA: 1^ffl ~= N A5M_5A M ____//_N A A____//_N kkk kkkk | | | kkkk | | | kkk fflffl| fflffl|~= fflffl| M ~=A A M _j^1//_M A M_ A M _1^f//fl_M A_A__//M Therefore M is a retract up to homotopy and thus, by a comparison of exact tria* *ngles, a direct summand up to homotopy of N. (Retractions split in triangulated categori* *es.) __|_| Let F CA be the full subcategory of CA whose objects are the direct summands * *up to homotopy of finite cell A-modules. In the language of [21, 1.7], the theorem* * states that the homotopy category hF CA is the largest rigid tensored subcategory of t* *he derived category DA. Note that the sequential filtration of a finite cell A-mod* *ule can be arranged so that a single cell is attached at each stage. That is, such a m* *odule is just a finite sequence of extensions by free modules on a single generator, * *and each quotient module Mn=Mn-1has the form F q(r) for some (q; r). A direct summand up* * to homotopy of a finite cell A-module, which is the appropriate analog in DA of a * *finitely generated projective A-module, need not be an actual direct summand and need not be isomorphic in DA to a finite cell A-module. The situation demands the introd* *uction and study of the K-theory K0(F CA), but we shall desist. 6.Relative and unital cell A-modules We here revert to a general DGA A, not necessarily commutative, and we assume given a fixed A-module K. There is a theory of cell A-modules relative to K tha* *t is exactly like the absolute theory, except that we start with M0 = K rather than * *M0 = 0 in the definition of a cell module (Definition 1.1(i)). When A is augmented, so* * that k is an A-module, this theory applies with K = k to give a theory of unital cell A-m* *odules. It will be needed in Part V. The relative theory is adapted to the study of the category of A-modules unde* *r K, by which we understand A-modules with a given map of A-modules j : K ! M; a map f : M ! N of A-modules under K must satisfy f O j = j. We let MAK denote the category of A-modules under K. We observe an obvious difficulty: the sum of maps under K is not a map under K, hence MAKis certainly not an additive category. W* *e say 72 IGOR KRIZ AND J. P. MAY that two maps under K are homotopic (or homotopic rel K) if they are homotopic * *via a chain homotopy h of A-modules such that h O j = 0. That is, if we regard h as* * a map M I ! N, then h(j(x) [I]) = 0 for all x 2 K. The notion of quasi-isomorphism is unchanged: a map under K is a quasi-isomorphism if it induces an isomorphism on homology. We have the homotopy category hMAK of A-modules under K, and we construct the derived category DKAfrom the homotopy category by formally invert* *ing the quasi-isomorphisms. The theory of relative cell A-modules makes this definition rigorous. In fac* *t, if K L M, where L is a relative cell submodule of the relative cell A-module M, then HELP (Theorem 2.2) applies verbatim, by the same proof. The relative Whitehead theorem reads as follows. Theorem 6.1. If M is a relative cell A-module and e : N ! P is a quasi-isomorp* *hism under K, then e* : hMAK(M; N) ! hMAK(M; P ) is an isomorphism. Therefore a quasi-isomorphism under K of relative cell A-modules is a homotopy equivalence * *under K. Proof. We see that e* is surjective by applying HELP to the pair (M; K), taking* * g = j, h to be the evident homotopy j ' j rel K, and f : M ! P to be any given map und* *er K. Injectivity is shown as in the proof of Theorem 2.3. __|_| Approximation by relative cell A-modules works exactly as in Theorem 2.6. Theorem 6.2. For any A-module M under K, there is a relative cell A module N a* *nd a quasi-isomorphism ffl : N ! M under K. Now Construction 2.7 applies verbatim to the category of A-modules under K. Corollary 6.3. The category DKA is equivalent to the homotopy category of relat* *ive cell A-modules. The forgetful functor MAK ! MA obviously preserves quasi-isomorphisms and so induces a functor DKA! DA. However, this functor fails to take relative cell A-* *modules to A-modules of the homotopy type of cell A-modules unless K itself is of the h* *omotopy type of a cell A-module, which is generally not the case in the applications. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 73 Part IV. Rational derived categories and mixed Tate motives We shall do some rational differential homological algebra_alias rational hom* *otopy theory_and use it to prove that two proposed definitions of rational mixed Tate* * motives agree [19, 18, 6]. One of them has been proved to admit Hodge andetale realizat* *ions [6], but is intrinsically restricted to the rational world. The other can be li* *nked up to a proposed definition of integral mixed Tate motives (or modules). We describe* * our results in Section 1, which recapitulates much of our announcement [38], and pr* *ove them in the remaining sections. We refer the reader to [38] for a number of re* *lated conjectures and speculations. 1.Statements of results Let A be a commutative, differential graded, and Adams graded k-algebra, abbr* *e- viated DGA, where k is a field of characteristic zero. Thus A is bigraded via * *k- modules Aq(r), where q 2 Z and r 0. We assume that Aq(r) = 0 unless 2r q. The differential and product behave as follows with respect to the gradings: d : Aq(r) ! Aq+1(r) and Aq(r) As(t) ! Aq+s(r + t): We assume that A has an augmentation ffl : A ! K. Write Hq(A)(r) for the coho- mology of A in bidegree (q; r). In the following three theorems, we assume that* * A is "cohomologically connected" in the sense that Hq(A)(r) = 0 ifq < 0; H0(A)(r) = 0 ifr > 0; and ffl induces an isomorphism H0(A)(0) ! k. While the Adams grading is present and important in our motivating examples, all of our results apply verbatim to * *DGA's without Adams grading. Let DA be the derived category of cohomologically bounded below A-modules. Its objects are differential bigraded A-modules M, where Mq(r) may be non-zero for * *any pair of integers (q; r), such that Hq(M)(r) = 0 for all sufficiently small q. A* *ll of our A-modules are to satisfy this cohomological condition. We assume familiarity wi* *th the cell theory of Part III. As explained there, DA is equivalent to the homotopy c* *ategory hCA of (cohomologically bounded below) cell A-modules. For a cell A-module M, we define the k-module QM of indecomposable elements of M by setting QM = k A M. Note that QM inherits a differential from M. Recall the notion of a t-structure and its heart from [3, x1.3], and recall t* *he definition of a neutral Tannakian category from [21, 2.19]. We shall prove the following r* *esult in Section 4, after reviewing the theory of minimal DGA's in Section 2 and develop* *ing the theory of minimal modules over DGA's in Section 3. Let HA be the full subcatego* *ry 74 IGOR KRIZ AND J. P. MAY of DA consisting of the cell A-modules M such that Hq(QM) = 0 for q 6= 0. Let F* * HA be the full subcategory of HA consisting of the modules M such that H0(QM) is f* *inite dimensional and define !(M) = H0(QM). Theorem 1.1. The triangulated category DA admits a t-structure whose heart is * *HA. In particular, HA is Abelian. Moreover, F HA is a (graded) neutral Tannakian ca* *tegory over k with fiber functor !. When A is a polynomial algebra on finitely many generators of even positive * *degree, most of this is proven in [5, pp.93-101]. It follows from [21, 2.11] that F HA * *is equivalent (in possibly many ways) to the category of finite dimensional representations o* *f an affine group scheme. What amounts to the same thing [21, 2.2], F HA is equivalent to t* *he category of finite dimensional comodules over a Hopf algebra (= bialgebra). We * *next specify an explicit such Hopf algebra. The algebra A has a bar construction B(A). Let IA denote the augmentation id* *eal of A. Then Bq(A)(r) is the direct sum over p 0 of the submodules of the p-fold tensor power of IA in bigrading (q + p; r). As we recall in Section 2, we can a* *rrange without loss of generality that A is connected, so that Aq = 0 for q < 0 and A0* * ~=k. In that case, B0(A) is additively isomorphic to the tensor algebra on the (Adams g* *raded) k-module A1. Following [6], let OA = H0B (A). This is a commutative Hopf algebr* *a, and it turns out to be a polynomial algebra. Its k-module of indecomposable ele* *ments is a co-Lie algebra, which we denote by flA. (The notation MA was used in [6],* * but this conflicts with our notation for the category of A-modules.) We think of OA* * as a kind of universal enveloping Hopf algebra of flA. We shall prove the following * *theorem in Section 5, where its undefined terms are specified. It gives a concrete and * *explicit description of the categories HA and F HA. Theorem 1.2. Let A be a connected DGA. Then the following categories are equiv* *a- lent. (i) The heart HA of DA. (ii) The category of generalized nilpotent representations of the co-Lie algebr* *a flA. (iii) The category of comodules over the Hopf algebra OA. (iv) The category TA of generalized nilpotent twisting matrices in A. The full subcategories of finite dimensional objects in the categories (i), (ii* *), and (iii) and of finite matrices in the category (iv) are also equivalent. The hypothesis that A be connected and not just cohomologically connected is* * needed to allow use of the category TA. The other three categories are invariant under* * quasi- isomorphisms of cohomologically connected DGA's. The DGA A has a "1-minimal OPERADS, ALGEBRAS, MODULES, AND MOTIVES 75 model" : A<1> ! A. The map induces an isomorphism on H1 and a monomorphism on H2. A quick construction, explained in Section 2 and justified in Section 6,* * is to let A<1> = ^(flA[-1]), with differential induced by the cobracket on flA, where* * flA[-1] denotes a copy of flA concentrated in degree one. We say that "A is a K(ss; 1)* *" if is a quasi-isomorphism. It is apparent from the equivalence of (i) and (ii) in * *Theorem 1.2 that the Abelian category HA depends only on A<1>. We shall prove the follo* *wing result in Section 7. Theorem 1.3. The derived category of bounded below chain complexes in HA is eq* *uiv- alent to the derived category DA<1>. Let k(r) be a copy of k concentrated in bidegree (0; r) and regarded as a rep* *resentation of flA in the evident way. Corollary 1.4.If A is a K(ss; 1), then ExtqHA(k; k(r)) ~=Hq(A)(r): While the results above are statements in differential homological algebra, w* *e for- mulated them as general results that would have to be true if two seemingly dif* *ferent definitions of mixed Tate motives were to agree. We briefly explain the relevan* *ce. Let X be a (smooth, quasi-projective) variety over a field F . As we recalled in II* *x6, Bloch [7] defined an Adams graded simplicial Abelian group Z(X) whose homology groups are the Chow groups of X: (1.5) CHr(X; q) = Hq(Z(X))(r): Bloch [7, 9] (see also Levine [39]) proved that (1.6) CHr(X; q) Q ~=(Kq(X) Q)(r); where the right side is the nr-eigenspace of the Adams operation n (for any n * *> 1), and Kq(X) Q is the direct sum of these eigenspaces. The simplicial Abelian group Z(X) has a partially defined product. In IIx6, w* *e con- structed an E1 algebra A (X) quasi-isomorphic to the associated chain complex * *of Z(X). We also constructed a commutative DGA AQ(X) and a quasi-isomorphism of E1 algebras A (X) Q ! AQ(X). These objects are graded homologically. Cohomo- logical considerations dictate the regrading (1.7) N 2r-p(X)(r) = Ap(X)(r) and NQ2r-p(X)(r) = (AQ)p(X)(r): Since Ap(X) = 0 if p < 0, N q(X)(r) = 0 unless 2r q. Thinking of the eigenspac* *es on the right side of (1.6) as successive terms of the associated grading with r* *espect to 76 IGOR KRIZ AND J. P. MAY the fl-filtration, we may rewrite (1.6) in the form (1.8) Hq(NQ(X))(r) = grrfl(K2r-q(X) Q): The "Beilinson-Soule conjecture for X" asserts that these groups are zero if q * *< 0 or if q = 0 and r 6= 0, and that the group in bidegree (0; 0) is Q. That is, the Beil* *inson-Soule conjecture is that NQ(X) is cohomologically connected. When it holds, our gene* *ral results above apply to NQ(X). Specializing to X = Spec(F ), let N denote the E1 algebra N (Spec(F )) and* * let NQ denote the commutative DGA NQ(Spec(F )). Even without the Beilinson-Soule conjecture, [6] proposed the following definition. Definition 1.9.Let Omotdenote the Hopf algebra ONQ = H0B (NQ). Define the cat- egory of (rational) mixed Tate motives of the field F , denoted M T M (F ), to * *be the category of finite dimensional comodules over Omot. Such a definition had been suggested in general terms by Deligne [18]. Actu* *ally, since the equivalence between categories (ii) and (iii) in Theorem 1.2 was not * *yet un- derstood, the preprint version of [6] confused this category with the category * *of all finite dimensional representations of flNQ. Technically, [6] worked with the rationali* *zation of a cubical version of the Chow complex Z(Spec(F )). The simplicial version is k* *nown to be quasi-isomorphic to the cubical one ([6, 39]). Lack of commutativity make* *s the cubical version ill-suited to an integral theory, although it is conceivable th* *at a suitable E1 operad acts on it_we have not explored this possibility. Theorem 1.2 specia* *lizes to give the following equivalence of categories. Theorem 1.10. If the Beilinson-Soule conjecture holds for Spec(F ), then M T M* * (F ) is equivalent to the category F HNQ. Deligne [20] first suggested that, if a suitable commutative DGA NQ could in* * fact be constructed, then F HNQ should give an appropriate definition of M T M (F ) * *when the Beilinson-Soule conjecture holds for Spec(F ). Thus Theorem 1.10 is the pro* *mised equivalence of two approaches to mixed Tate motives. In view of (1.8), Corollar* *y 1.4 has the following immediate consequence. Theorem 1.11. If NQ is a K(ss; 1), then ExtpMT M(F)(Q; Q(r)) ~=grrfl(K2r-p(F ) Q): This verifies one of the key properties desired of a category of mixed Tate * *motives. The results of [6] start from Definition 1.9 and give realization functors from* * M T M (F ) to the category of mixed Tate l-adic representations inetale theory and to the * *category of mixed Tate Hodge structures in Hodge theory. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 77 The reinterpretation Definition 1.9 given by Theorem 1.10 leads to a category* * of integral mixed Tate motives that is related to M T M (F ) by extension of scala* *rs. In fact, in Part V, we shall construct the derived category DA associated to an E1* * algebra A. Just as if A were a DGA, DA is a triangulated tensor category satisfying al* *l of the usual properties. Like the derived category of modules over a DGA, DA can * *be described as a homotopy category of cell modules. The convenience and workabili* *ty of such a description will become apparent in our proofs of Theorems 1.1-1.3. Deligne [20], [17, x3] suggested that the resulting derived category DN shou* *ld be an integral "categorie triangulee motivique D(F )", and he gave speculations ab* *out its motivic role. One can define Adams graded Ext groups ExtqN(M; N) = D(F )(M; N[q]) for modules M and N. These agree with the cohomology groups of the right derived module Hom N (M; N) that we shall define in Part V, and we shall there construc* *t a spectral sequence that converges from Ext*;*H*(N()H*(M); H*(N)) to Ext*N(M; N). Here H*(N ) is the integral Chow ring of Spec(F ), regraded as * *dic- tated by (1.7) and (1.8). Little is known about the integral Chow groups and there is only speculation * *as to their relationship to the higher algebraic K-groups of F . However, our results* * on derived categories work equally well if we reduce mod n, and Suslin [59] has recently p* *roven that if F is an algebraically closed field of characteristic prime to n and X i* *s a smooth affine variety over F , then, for r dim(X), CHr(X; q; Z=n) ~=H2r-qet(X; Z=n (r)): 2. Minimal algebras, 1-minimal models, and co-Lie algebras In the interests of intelligibility, we first review some basic rational homo* *topy theory, working over our given field k of characteristic zero. We assume once and for a* *ll that all DGA's in this Part are commutative. Definition 2.1.A connected DGA A is said to be minimal if it is a free commutat* *ive algebra with decomposable differential: d(A) (IA)2. Definition 2.2.Let A be a connected DGA and define sub DGA's A and A as follows. (i)For n 0, let A be the subalgebra generated by the elements of degree * * n and their differentials; note that A<0> = k. 78 IGOR KRIZ AND J. P. MAY (ii)For n 1, let A = A and let A; q 0 be the subalg* *ebra generated by A [ {a|a 2 An andd(a) 2 A}: Say that A is generalized nilpotent if it is free commutative as an algebra and* * if A = [A for each n 1. This means that every element of An is in some A.* * Say that A is nilpotent if, for each n 1, there is a qn such that A = A. Proposition 2.3. A connected DGA (with Adams grading) is minimal if and only if it is generalized nilpotent. Proof. If A is generalized nilpotent, then d(A) (IA)2 by an easy double induct* *ion on n and q (e.g. [4, 7.3]). Assume that A is minimal. Suppose for a contradi* *ction that A is not generalized nilpotent and let n be minimal such that there is an * *element of An not in any A. Let a be such an element of minimal Adams degree and consider a typical summand a0a00of the decomposable element d(a). We may assume that 0 < deg(a0) deg(a00), and a0a002 A unless deg(a0) = 1. Since Aq(r)* * = 0 unless 2r q, a0and a00have strictly lower Adams grading than a. By the assumed minimality, both a0and a00are in some A. Therefore d(a) is in some A, hence so is a. __|_| Except when A is simply connected, the "only if" part would be false without* * the Adams grading, and we shall not use this implication. Without the Adams grading, the useful notion is that of a generalized nilpotent DGA (hence [4] redefined "* *minimal" to mean generalized nilpotent). The following result is standard: see [58, x5]* *, or [4, 7.7 and 7.8]. Its proof is just like that of Theorem 3.7 below, except that one* * adjoins generators of algebras rather than generators of free modules. Theorem 2.4. If B is a cohomologically connected DGA, then there is a quasi-is* *omor- phism OE : A ! B, where A is generalized nilpotent. If OE0: A0! B is another su* *ch quasi-isomorphism, then there is an isomorphism : A ! A0such that OE0 is homot* *opic to OE. Definition 2.5.An n-minimal model of B is a composite map of DGA's A A ! B; where A is generalized nilpotent and A ! B is a quasi-isomorphism. The 1-minimal model admits a canonical description in terms of co-Lie algebr* *as, as we recall next. Here and later, we write X_ = Hom (X; k) and we regard the dual* * of a OPERADS, ALGEBRAS, MODULES, AND MOTIVES 79 map X ! Y Z of k-modules to be the evident composite Y _ Z_ ! (Y Z)_ ! X_ . Definition 2.6.A co-Lie algebra is a k-module fl together with a cobracket map * *fl ! fl fl such that the dual fl_ is a Lie algebra via the dual homomorphism. Here * *fl is concentrated in ordinary grading zero; its Adams grading (if it has one), is co* *ncentrated in positive degrees. It is natural to think of the bracket of a Lie algebra L as defined on the su* *bspace of invariants with respect to the involution xy ! -yx in LL. The sign suggests that one should think of elements of L as having degree 1. Dually, it is natural to * *think of the cobracket operation of a co-Lie algebra fl as a k-linear map d : fl[-1] ! ^* *2(fl[-1]), where fl[-1] is a copy of fl concentrated in degree 1 and ^2(fl[-1]) is the sec* *ond exterior power. Sullivan observed the following fact, [58, p. 279]. Lemma 2.7. A co-Lie algebra fl determines and is determined by a structure of * *DGA on ^(fl[-1]): That is, the (dual) Jacobi identity is equivalent to the assertion that d ind* *uces a differential on the exterior algebra ^(fl[-1]). Explicitly, if {ar} is an order* *ed basis for fl[-1] and if X (2.8) d(ar) = krp;qap^ aq; p of A is isomorphic to ^(flA[-1]). (ii) The Hopf algebras OA<1>and OA are isomorphic, hence the co-Lie algebras fl* *A<1>and flA are isomorphic. 3.Minimal A-modules By III.3.4, we have the following invariance statement; an E1 generalization* * will be proven in Vx4. Proposition 3.1. If OE : A ! A0is a quasi-isomorphism of cohomologically connec* *ted DGA's, then OE induces an equivalence OE* : DA0! DA of triangulated tensor cate* *gories. In particular, by Theorem 2.4, we can and will assume that our given DGA A is connected. Remember that we require all modules to be cohomologically bounded below. Let M be a cell A-module and recall that QM = k A M. Ignoring the differenti* *al, M is A-free on the canonical basis elements of its open cells, and this bas* *is projects to a canonical basis of QM. We write X d = ai;j; where runs through the basis elements of the open cells. Define Mn M to b* *e the sum of those open cells with basis elements in (ordinary) degree n. Note that * *Mn is not necessarily closed under the differential. Definition 3.2.A bounded below cell A-module M is minimal if it is A-free and h* *as decomposable differential: d(M) (IA)M. Proposition 3.3. The following conditions on a bounded below cell A-module M are equivalent. (i) M is minimal. (ii) QM = H0(QM); that is, d = 0 on QM. (iii) All coefficients ai;jhave positive degree. (iv) Each Mn is closed under d and is thus a cell submodule of M. If f : M ! N is a quasi-isomorphism between minimal A-modules, then f is an isomorphism. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 81 Proof.Since Aq = 0 for q < 0 and ffl : A0 ! k is an isomorphism, the equivalenc* *e of (i)-(iv) is immediate by inspection of definitions. A quasi-isomorphism f : M * *! N induces a quasi-isomorphism Qf : QM ! QN and, if M and N are minimal, Qf itself is then an isomorphism. Thus the last statement follows by Nakayama's le* *mma: a map f of bounded below free A-modules is an isomorphism if and only if Qf is * *an isomorphism. __|_| There is an equivalent condition in terms of generalized nilpotency. Definition 3.4.Let M be a bounded below A-module (not a priori a cell module) a* *nd define sub A-modules M and M as follows. (i)Let M be the sub A-module generated by the elements of degree n and their differentials; note that M = 0 for n sufficiently small. (ii)Let M = M and let M, q 0, be the sub A-module ge* *n- erated by M [ {m|m 2 Mn and d(m) 2 M}: (iii)Define the "nilpotent filtration" {FtM} by letting F0M = 0 and, inductiv* *ely, letting FtM be the sub A-module generated by Ft-1M [ {m|d(m) 2 Ft-1M}: Say that M is generalized nilpotent if it is free as an A-module and if M = [M for each n. This means that every element of Mn is in some M. Say that M * *is nilpotent if, for each n, there is a qn such that M = M. In marked contrast with the case of algebras, the following result for module* *s is true regardless of whether or not there is an Adams grading. Proposition 3.5.A bounded below A-module M is generalized nilpotent if and only* * if it is a minimal cell A-module, and then {FtM} specifies a canonical choice of s* *equential filtration for the cell structure on M. Proof.Suppose that M is generalized nilpotent. Then d(M) (IA)M since an A-basis element in degree n must have differential in the sub A-module generated by the* * Mj for j n. We claim that M is a cell A-module with {FtM} as sequential filtratio* *n. Certainly M is the union of the FtM since, if not, there would be a minimal pai* *r (n; q) in the lexicographic ordering such that M was not contained in the cited * *union and this would contradict the generalized nilpotency. Assuming inductively that* * Ft-1M is A-free, we easily check that FtM is A-free with basis obtained by extending * *a basis 82 IGOR KRIZ AND J. P. MAY for Ft-1M. Conversely, assume that M is a minimal cell A-module. Suppose for a contradiction that M is not generalized nilpotent and let n be minimal such tha* *t there is an element of Mn that is not in any M. Let m be such an element of min* *imal sequential filtration. By the definition of a cell A-module, d(m) has lower seq* *uential filtration than m. But then d(m) is in some M and m is in M. Th* *is proves the result. __|_| Remark 3.6. A minimal A-module M need not have bounded below Adams grading, as we see by considering infinite direct sums. However, if M has bounded below Ada* *ms grading, then it admits a second canonical sequential filtration {FtAdM}. Preci* *sely, let {rt|t 1} be the ordered set of integers for which the free A-module M has a ba* *sis element of Adams grading rt. Then F0AdM = 0 and FtAdis the sub A-module spanned by the basis elements of Adams grading at most rt. Clearly FtAdM FtM, and the inclusion can be proper. Theorem 3.7. Let N be an A-module. Then there is a quasi-isomorphism e : M ! N, where M is a minimal A-module. If e0: M0 ! N is another such quasi-isomorphism, then there is an isomorphism f : M ! M0 such that e0f is homotopic to e. Proof. Let n0 be sufficiently small that Hq(N) = 0 for q < n0 and let M[n0; 0] * *= 0. Assume inductively that an A-map e : M[n; 0] ! N has been constructed such that* * e* is an isomorphism on Hi for i < n and a monomorphism on Hn. Then, proceeding by induction on q, construct A-maps e : M[n; q] ! N for q 0 as follows. If q = 0,* * choose a set {ns} of representative cycles in N for a basis of Coker(Hn(M[n; q]) ! Hn(N)): If q 0, choose a set {mr} of representative cycles in Hn+1(M[n; q]) for a basi* *s of Ker(Hn(M[n; q]) ! Hn(N)) and choose elements nr in N such that d(nr) = (-1)n+1e(mr). Construct M[n; q + * *1] from M[n; q] by attaching n-cells js (if q = 0) and ir via attaching cycles 0 a* *nd mr; thus the basis elements of the adjoined open n-cells satisfy d = 0 and d = (-1)n+1mr: Extend e to M[n; q + 1] by setting e = ns and e = nr. An easy colimit a* *rgument shows that if we define M[n + 1; 0] = [M[n; q] and let e : M[n + 1; 0] ! M be the induced map, then e* is an isomorphism on Hi* * for i n and a monomorphism on Hn+1. Define M = [M[n; 0]. Then the induced map OPERADS, ALGEBRAS, MODULES, AND MOTIVES 83 e : M ! N is a quasi-isomorphism, and M is minimal since it is generalized nilp* *otent with M = M[n; q]. For the last statement, the Whitehead theorem (III.2.3)* * gives a map f : M ! M0such that e0f is homotopic to f. Obviously f is a quasi-isomorphi* *sm, and it is therefore an isomorphism by Proposition 2.3. __|_| 4. The t-structure on DA We here prove Theorem 1.1. Let A be a cohomologically connected DGA. We agree* * to abbreviate notation by writing D = DA, and similarly for other categories that * *depend on A. Definition 4.1.Define full subcategories Dn and Dn of D by Dn = D0 [-n] = {M|Hq(QM) = 0 forq > n} and Dn = D0 [-n] = {M|Hq(QM) = 0 forq < n}: Observe that D0 D1 and D0 D1 . Define H = D0 \ D0 = {M|Hq(QM) = 0 forq 6= 0}: The following result is a more explicit statement of the first part of Theore* *m 1.1. Theorem 4.2. Definition 4.1 specifies a t-structure on D. Proposition 3.1 implies that the result will be true for A if it is true for * *a DGA quasi-isomorphic to A. Therefore, by Theorem 2.4, we may as well assume that A * *is connected. This allows us to use the theory of minimal A-modules. Taken togethe* *r, the following two lemmas constitute a restatement of Theorem 4.2. Lemma 4.3. For M 2 D, there is an exact triangle M0 ! M ! M=M0 in D with M0 in D0 and M=M0 in D1 . Proof.It suffices to assume that M is minimal, in which case the conclusion is * *immediate from Proposition 3.3(iv). __|_| Lemma 4.4. If M is in D0 and N is in D1 , then D(M; N) = 0. Proof.It suffices to assume that M and N are minimal. In that case, (QM)q = 0 f* *or q > 0 and Nq = 0 for q 0, hence there are no non-zero maps of A-modules M ! N. __|_| 84 IGOR KRIZ AND J. P. MAY Remark 4.5. Theorem 1.1 would be false without the restriction to cohomological* *ly bounded below A-modules. An unbounded A-module M can have non-zero cohomology and yet satisfy H*Q(M) = 0. For example, if ff 2 Hn(A) is represented by a cycl* *e a and M is the telescope of the sequence of A-maps a : A[-qn] ! A[-(q + 1)n]; then H*(M) is the localization H*(A)[ff-1] and H*(QM) = 0. Clearly Lemma 4.4 will usually fail in this situation since Hq(M)(r) ~=D(F; M), where F is free o* *n one generator of bidegree (-q; -r). The following lemma implies that F H is a rigid Abelian tensor category. Lemma 4.6. The subcategory F H is closed under passage to tensor products and duals in D. Proof. By III.5.1, if M and N are (finite) cell A-modules, then M A N is a (fin* *ite) cell A-module such that (*) Q(M A N) ~=QM QN: If M and N are minimal, then, by Proposition 3.3(ii), so is M A N. If the indec* *om- posable elements of minimal A-modules M and N are concentrated in degree zero, * *then so are the indecomposable elements of M A N, and this proves closure under tens* *or products. For duals, it is easy to check that the k-modules Q(M_) and (QM) _ a* *re isomorphic when M is a finite cell A-module. __|_| The following lemma completes the proof of the last statement of Theorem 1.1. Lemma 4.7. ! = H0Q : F H ! QM is a faithful exact tensor functor. Proof. An easy formal elaboration of (*) shows that ! is a tensor functor. The * *functor Q is exact since cell A-modulesare A-free. Therefore H0Q is exact on H by virtu* *e of the long exact sequences associated to short exact sequences obtained by applyi* *ng Q to short exact sequences 0 ! M0! M ! M00! 0 of cell A-modules. Alternatively, we can check that it suffices to restrict at* *tention to short exact sequences of minimal A-modules. Finally, ! is faithful since tw* *o A- maps between minimal A-modules in H are equal if they are equal on indecomposab* *le elements. Note that there is no room for homotopies since there are no element* *s of degree -1: a map in H between minimal A-modules is just a map of A-modules. __* *|_| OPERADS, ALGEBRAS, MODULES, AND MOTIVES 85 5.Twisting matrices and representations of co-Lie algebras We here prove Theorem 1.2. We begin by describing HA in terms of matrices. We then show that representations of co-Lie algebras admit a precisely similar des* *cription. We tie in comodules at the end. In view of Theorem 2.4, Proposition 3.1, and t* *he quasi-isomorphism invariance of the homology of the bar construction, we may as* * well assume that A is generalized nilpotent. Let M be a minimal A-module in HA. Then M is A-free on basis elements of degree zero and Adams degree r(j). Here the nilpotent filtration of Definition * *3.4 is given by FtM = M<0; t>. Each lies in FtM - Ft-1M for some positive integer * *t, which we denote by t(j) and think of as the order of nilpotency. The different* *ial is given by X d = ai;j; where ai;jhas degree one and Adams degree r(j) - r(i); in particular, ai;j= 0 if r(j) r(i). For each , only finitely many of the ai;jare non-zero, and ai;j=* * 0 if t(i) t(j). Order the basis and write a = (ai;j) and da = (d(ai;j)). Then the c* *ondition dd = 0 is easily seen to take the form of the matrix equation da = -aa, and thi* *s makes sense even when M is infinite dimensional. Note in particular that each d(ai;j)* * must be a decomposable element of the algebra A. Now consider a map f : M ! N of minimal A-modules, where the differentials on P M and N are given by the matrices a and b in A1. Let f = kj;i, where runs through the canonical basis of N0 and the kj;iare elements of the ground field.* * Here kj;i= 0 unless and have the same Adams degree. Moreover, since f preser* *ves the nilpotent filtration, kj;i= 0 if t(j) > t(i). Write k = (kj;i). Then the condit* *ion df = fd is easily seen to take the form of the matrix equation bk = ka. These observati* *ons lead to the following definition (compare Sullivan [58, x1]) and proposition. By an * *"initial segment of the positive integers", we understand either the set of all positive* * integers or the set {1; 2; : :;:n} for some finite n. Definition 5.1.A "twisting matrix" in A is an ordered set I, a function r : I !* * Z, and a row finite (I x I)-matrix a = (ai;j) with entries in A1 such that ai;jhas* * Adams degree r(j) - r(i) and da = -aa. We say that a is indexed on r. A twisting matr* *ix a is generalized nilpotent if there is a surjection t from I to an initial segment o* *f the positive integers such that ai;j= 0 if t(i) t(j). A morphism from a twisting matrix a i* *ndexed on r : I ! Z to a twisting matrix b indexed on s : J ! Z is a row finite (J x I* *)-matrix k = (kj;i) with entries in the ground field such that kj;i= 0 if r(i) 6= s(j) a* *nd bk = ka. If a and b are generalized nilpotent (with nilpotency functions both denoted t)* *, then we require morphisms to satisfy kj;i= 0 if t(i) > t(j). With composition specified* * by the 86 IGOR KRIZ AND J. P. MAY usual product of matrices, there results a category TA of generalized nilpotent* * twisting matrices in A. Proposition 5.2. The category HA is equivalent to the category TA. Proof. The category HA is equivalent to its full subcategory of minimal A-modul* *es, maps in HA between minimal A-modules are just maps of modules, and the discussi* *on above gives the conclusion. __|_| We next recall the notion of a representation of a co-Lie algebra fl. Recall* * Definition 2.6 and Lemma 2.7. Definition 5.3.A representation of a co-Lie algebra fl is a k-module V together* * with a coaction map : V ! fl V such that the dual V _is a module over the Lie alge* *bra fl_ via the dual homomorphism. Here V is concentrated in ordinary grading zero;* * its Adams grading (if it has one) is unrestricted. Dualizing and reinterpreting, we see that a representation on V can equally * *well be viewed as a k-linear map : V ! fl[-1] V such that (d 1) coincides with the m* *ap obtained by passage to coinvariants from the composite (1 ); that is (5.4) (1 ) = (d 1) : V ! ^2fl[-1] V: However, we do not want to allow all such representations. Definition 5.5.Let V be a representation of a co-Lie algebra fl. Define the nil* *potent filtration {FtV } by letting F0V = 0 and letting FtV be the subspace generated * *by the union of Ft-1V and {v|(v) 2 fl[-1] Ft-1V }. Say that V is generalized nilpoten* *t if it is the union of the FtV . Say that V is nilpotent if V = FtV for some finite t. Remark 5.6. A generalized nilpotent representation V need not have bounded below Adams grading. If a representation V has bounded below Adams grading, then it is generalized nilpotent and has the Adams filtration {FtAdV } specified by lettin* *g F0AdV = 0 and letting FtAdV be the subspace of elements with Adams grading at most rt, * *where {rt|t 1} is the ordered set of integers for which V has an element of Adams gr* *ading rt. As in Remark 3.6, FtAdV FtV , and the inclusion can be proper. Let V be a generalized nilpotent representation of fl. Fix a basis {vi} for * *V indexed on an ordered set I. Define r : I ! Z by letting r(i) be the Adams degree of vi* * and define a surjection from I to an initial segment of the positive integers by le* *tting t(i) P be minimal such that vi2 Ft(i)V . Let (vj) = ai;j vi. Then ai;jhas Adams degr* *ee r(j) - r(i) and ai;j= 0 if t(i) t(j). We again write a = (ai;j) and da = (d(ai* *;j)). Then (5.4) takes the form of the matrix identity da = -aa. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 87 Similarly, let f : V ! W be a map of generalized nilpotent representations of* * fl and P write f(vi) = kj;i(wj), where wj is the chosen basis of W . Then kj;i= 0 if r* *(i) 6= r(j) or if t(j) > t(i). Write k = (kj;i). Then the identity (1 f) = !f takes the fo* *rm of the matrix identity bk = ka, where ! : W ! fl[-1] W is specified by the matrix* * b. These observations imply the following result. Proposition 5.7.The category of generalized nilpotent representations of a co-L* *ie algebra fl is equivalent to the category TA, where A = ^(fl[-1]). Corollary 5.8.The categories (i), (ii), and (iv) of Theorem 1.2 are equivalent. Proof.This is immediate from Proposition 5.2 and Proposition 5.7, applied to th* *e co-Lie algebra flA of Theorem 2.10. Note that TA is equivalent to TA<1>since flA is is* *omorphic to flA<1>. __|_| To complete the proof of Theorem 1.2, we must connect up the category of como* *dules over OA. First recall exactly how a module V over a Lie algebra L determines a * *module over its universal enveloping algebra U(L): the given action map L V ! V induc* *es an action T (L) V ! V of the tensor algebra T (L), by iteration, and this map factors through the quotient map T (L) V ! U(L) V to induce the required acti* *on U(L) V ! V . We shall dualize this description. Definition 5.9.Define the universal enveloping Hopf algebra O(fl) of a co-Lie a* *lgebra fl to be OA, where A = ^(fl[-1]). Let T (fl) be the tensor coalgebra of fl. Additively, it is the same as the t* *ensor algebra, and it has the coproduct given by X (c1 . . .cn) = (c1 . . .ci) (ci+1 . . .cn): i+j=n We shall prove the following result in the next section. Recall that fl is sai* *d to be generalized nilpotent if ^(fl[-1]) is generalized nilpotent and that this alway* *s holds when ^(fl[-1]) is Adams graded. Proposition 5.10.Let fl be a generalized nilpotent co-Lie algebra. Then there i* *s a canonical commutative diagram of algebras T (fl_)___//_T_(fl) | | | | fflffl| fflffl| U(fl_)____//O(fl)_: 88 IGOR KRIZ AND J. P. MAY _ Here T (fl_) ! U(fl_) is the obvious quotient map, U(fl_) ! O(fl) is the map* * of algebras induced by the inclusion of Lie algebras dual to the quotient map of c* *o-Lie algebras O(fl) ! fl, T (fl_) ! T (fl)_is the map of algebras induced by the dua* *l of the evident quotient map of k-modules T (fl) ! fl, and T (fl)_! O(fl)_is dual t* *o a canonical embedding of O(fl) as a subcoalgebra of T (fl) that will be explained* * in the next section. The dual of a O(fl)-comodule V is a O(fl)_-module and therefore a U(fl_)-mod* *ule. Equivalently, it is a fl_-module, and of course the action of fl_ is the restri* *ction of the action of U(fl_). If the coaction of O(fl) is given by : V ! O(fl)V , then we * *obtain an induced coaction of fl by composing with the projection O(fl)V ! fl V . Convers* *ely, let V be a representation of fl with coaction : V ! fl V . Then the dual V _i* *s a fl_-module under the dual of . Equivalently, V _is a U(fl_)-module. We ask when this action results from dualization of a coaction by O(fl). By * *iteration, induces a map n : V ! fln V for each n 0, where fln denotes the n-fold tensor power of fl and 0 is understood to be the identity map of V . Under the proviso* * that, for each v 2 V , n(v) = 0 for all sufficiently large n, the sum : V ! T (fl) * *V of the maps n makes sense. It must take values in O(fl) V and specify a structure* * of O(fl)-comodule on V , by consideration of the dual situation. A moment's refle* *ction on Definition 5.5 will convince the reader that the proviso holds if and only i* *f V is generalized nilpotent. Again, reflection on the dual situation shows that if w* *e start with a coaction of O(fl) on V , project to obtain a coaction of fl on V , and* * then take the sum of the iterates n, we must get back . Since is defined by finite * *sums, this means that V is generalized nilpotent. These arguments, which can be car* *ried out less intuitively and more precisely without use of dualization, lead to the* * following conclusion, which completes the proof of Theorem 1.2. Proposition 5.11. The category of generalized nilpotent representations of a ge* *neral- ized nilpotent co-Lie algebra fl is equivalent to the category of comodules ove* *r O(fl). Remark 5.12. When fl is not generalized nilpotent, the co-Lie algebra of indeco* *mposable elements of O(fl) specifies the "generalized nilpotent completion" of fl. Equiv* *alently, for a minimal DGA A with degree one indecomposable elements, the DGA ^(flA[-1]) specifies the "generalized nilpotent completion" of A. This corresponds topolog* *ically to generalized nilpotent completion of rational K(ss; 1)'s. 6. The bar construction and the Hopf algebra OA We prove Theorem 2.10 and Proposition 5.10 here, and we also develop prelimi* *naries that will be needed in the proof of Theorem 1.3. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 89 We first recall the basic facts about the bar construction (e.g. from [30, Ap* *pendix], or [6]). We shall use the sign conventions of [30]. The two-sided bar constru* *ction B(M; A; N) is defined for A-modules M and N. Even though A is commutative, we think of M as a right and N as a left A-module to keep track of signs. As a ch* *ain complex, B(M; A; N) is obtained by totalization (as in IIx5) of the usual simpl* *icial chain complex B*(M; A; N) with Bp(M; A; N) = M Ap N: Since our totalization includes normalization, B(M; A; N) is additively the dir* *ect sum of the vector spaces M (IA)p N. (Logically, the cokernel of the unit k ! A sho* *uld appear in place of the isomorphic k-module IA.) We grade B(M; A; N) so that the homological degree is negative. Thus elements of M (IA)p N have degree their internal degree minus p; the (total) differential on such elements is given by * *the map X (-1)pd + (-1)idi; where d is the internal differential on the tensor product M (IA)p N. With the evident right action by A, B(M; A; N) is a differential A-module and B(M; A; N) = B(M; A; A) A N: We may think of B(M; A; N) as an explicit model for the derived tensor product * *of M and N, and, as in IIIx4, we have an Eilenberg-Moore spectral sequence TorH*(A)(H*(M); H*(N)) =) H*(B(M; A; N)): Therefore quasi-isomorphisms of its variables induce quasi-isomorphisms of the * *bar construction. The following minor technical point will become relevant in the * *next section. Remark 6.1.While B(M; A; A) is a right differential A-module with the evident r* *ight action, there is no choice of signs in B(M; A; N) for which both this and its a* *nalog for B(A; A; N) are true. To make B(A; A; N) a left differential A-module, one must * *modify the obvious action by a sign, defining a new action of A by a . x = (-1)pdeg(a)* *ax, where x has homological degree p. The required formula d(a . x) = d(a) . x + (-1)deg(* *a)a . d(x) is easily checked. As usual we abbreviate B (A) = B(k; A; k). The product OE on B A is the shuf* *fle product X OE([a1| : :|:ar] [ar+1| : :|:as]) = (-1)oe()[a(1)| : :|:a(r+s)]; 90 IGOR KRIZ AND J. P. MAY where the sum runs over the (r; s)-shuffles in the symmetric group r+s; oe() i* *s the sum over (i; j) such that 1 i r, r < j r+s, and (j) < (i) of deg(a(i))deg(a(* *j)). The coproduct on B(A) is X ([a1| : :|:ap]) = (-1)o(i)[a1| : :|:ai] [ai+1| : :|:ap]; where the sum runs over 0 i p and o(i) = (p - i)(deg(a1) + . .+.deg(ai)): Remark 6.2. This coalgebra structure on the tensor algebra T (IA) is isomorphic* * to the usual one. In fact, the isomorphism specified by [a1| : :|:ap] ! (-1)(p)[a1| : :|:ap]; where (p) = pdeg(a1) + (p - 1)deg(a2) + . .+.deg(ap) throws the coproduct defin* *ed with signs onto the coproduct defined without signs. To prove Theorem 2.10, we may assume without loss of generality that A is ge* *neral- ized nilpotent. Since A is connected, there are no non-zero elements of negativ* *e degree in B(A). Thus there are no degree zero boundaries and OA = H0B (A) embeds in B(* *A) as its k-module of cycles of degree zero. Since OA inherits its Hopf algebra st* *ructure from the Hopf algebra structure on B(A), this embedding must be a map of Hopf a* *lge- bras. Note however that, even in simple cases, it is not obvious how to identif* *y cycles explicitly. The elements of degree zero in B(A) are the elements of the (A1)p, * *so that OA depends only on the elements of A1 and their differentials. When A is genera* *lized nilpotent, this means that OA = OA<1>, and this already implies Theorem 2.10(ii* *). The last part of the following calculational description of OA is Theorem 2.10(i). Theorem 6.3. Let A = A<1>. Then the following conclusions hold. (i) The embedding OA ! B(A) is a quasi-isomorphism. (ii) OA is isomorphic to the polynomial algebra generated by a copy of A1, tran* *slated to lie in degree zero. (iii) There is a degree 1 k-map q : OA ! A1 which is the composite of the quoti* *ent homomorphism OA ! flA and an isomorphism flA ! A1 and which makes the following diagram commute, where OE is the multiplication of A and is the comultiplicat* *ion of OA: OA ___________//_OA OA q|| qq|| fflffl| fflffl| A1 _d_//_A1ooOA1EA1_ (iv) A can be identified with the DGA ^(flA[-1]). OPERADS, ALGEBRAS, MODULES, AND MOTIVES 91 Proof.Let A# be the underlying algebra of A, with differential zero. Filtering * *B(A) by homological degree, we obtain a spectral sequence that converges from the homol* *ogy of B (A#), which is TorA#(k; k), to the homology of B (A). Here the convergenc* *e of the spectral sequence follows by induction and passage to colimits from the gen* *eralized nilpotency of A. Since A# is the exterior algebra generated by A1, TorA#(k; k)* * is the divided polynomial algebra generated by a copy of A1 concentrated in bidegr* *ee (-1; 1). Since char(k) = 0, a divided polynomial algebra is isomorphic to a pol* *ynomial algebra. The generators are permanent cycles, by obvious degree considerations,* * hence E2 = E1 . Thus the homology of B(A) is a polynomial algebra concentrated in deg* *ree zero since its associated graded algebra is a polynomial algebra with generator* *s of bidegree (-1; 1). This proves (i) and (ii). We see from this argument that the * *elements of A1, thought of as elements [a] in BA, extend to cycles by addition of summan* *ds of lower homological degree. The map q sends a generating cycle "[a]+ lower terms"* * to a. That is, q is induced from the homomorphism B A ! A1 that is the identity on A1 and is zero on all elements other than those of degree 1 and homological deg* *ree -1. To compute the coproduct on generators of OA, we must compute the coproduct* * on generating cycles. Observe that qq annihilates all summands not of the form [a0* *][a00] with a0; a002 A1. With the notation of (2.8), the definition of the differentia* *l on B(A) forces our basic cycles to have the form X [ar] - krp;q[ap|aq] + terms of lower homological degree. p. Let us first observe t* *hat Corollary 1.4 is an immediate consequence. 92 IGOR KRIZ AND J. P. MAY Proof of Corollary 1.4.By III.3.1, Hq(A)(r) ~= D(A; F q(r)), where F q(r) is th* *e free A-module on one generator of bidegree (q; r). By Theorem 1.3, this is isomorphi* *c to DH (k; k(r)[-q]), where k and k(r) are regarded as chain complexes concentrated* * in degree zero, and this is ExtqH(k; k(r)). __|_| To begin the proof of Theorem 1.3, we construct a functor S : DH ! D. For t* *his, we need only assume that A is connected. Consider a bounded below chain complex M* = {Mn; ffi : Mn ! Mn+1} in H . Since H D, each Mn is an A-module (with differential d) and each ffi i* *s a map of A-modules. Any such chain complex M* is quasi-isomorphic to a chain comp* *lex of minimal A-modules in H , by Theorem 3.7 and the Whitehead theorem (III.2.3), hence we may assume without loss of generality that each Mn is minimal. Then the differential on Mn is specified by a generalized nilpotent twisting matrix an a* *nd ffi is specified by matrices kn such that an+1kn = knan. We define a cell A-module SM*, called the summation of M*, with one n-cell for each 0-cell of Mn. We specify * *the differential on the canonical basis element of an open n-cell by X X (7.1) d = kni;j + ani;j |i|=n+1 |i|=n where |i| denotes the degree of a canonical basis element . If N* is a chain* * complex specified by matrices bn and ln and f* : M* ! N* is a chain map, then f* is giv* *en by matrices OEnwith entries in k such that OEn+1kn = lnOEn. We define Sf* : SM* ! * *SN* by letting Sf* be prescribed by the matrix OEnon the canonical basis for the op* *en n- cells. If f* is a quasi-isomorphism of chain complexes, then Sf* is a quasi-iso* *morphism of A-modules by a little spectral sequence argument. Now consider a general cell A-module M. If |j| = n, we can write X X X (7.2) d = kni;j + ani;j + bni;j: |i|=n+1 |i|=n |i| and , then M is isomorphic to SM*, where Mn is the A-module in H specifi* *ed by the twisting matrix an and where the differential ffin : Mn ! Mn+1 is specif* *ied by the matrix kn. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 93 A map of A-modules g : SM* ! SN* is given on a canonical basis element of* * M of degree n by X X (7.4) g = nj;i + ffnj;i: |j|=n |j|, any cell A-module, wi* *th differential of the form (7.2), is quasi-isomorphic to some cell A-module with * *differential of the form (7.1). We shall exploit the following conceptual procedure for cons* *tructing A-modules of the form SM* out of general A-modules. Construction 7.5.Let O be a coalgebra (Adams graded, but concentrated in degree zero with respect to the ordinary grading) and suppose given a map q : O ! A1 A of Adams graded k-modules such that the following diagram commutes, where OE is* * the multiplication of A and is the comultiplication of O: O __________//_O O q || |qq| fflffl| fflffl| A __d_//AooOEA_ A Let M be an A-module. Define a new A-module fi(A; O; M) by letting fi(A; O; M) * *be A O M as an A-module, with differential d 1 1 + (OE(1 q) 1 1)(1 1) - (1 1 (q 1))(1 1) + 1 1 d; where : A M ! M is the action of A on M. A lengthy but purely formal diagram chase shows that d2 = 0. The standard sign convention on tensor products of morphisms, (f g)(x y) = (-1)deg(g)deg(x)f(x) g(y); is used; observe that this implies, for example, that (1 d)(d 1) = -d d. The following special case will lead to the proof of Theorem 1.3. We assume * *that A = A<1> in the rest of this section. Definition 7.6.As in Theorem 6.2(iii), let q : OA ! A be the composite of the q* *uotient map from OA to flA and the evident identification of flA with flA[-1]. Define a* * functor R from A-modules to A-modules by R(M) = fi(A; OA; M): Another little spectral sequence argument shows that the functor R preserves * *quasi- isomorphisms, and this will also follow from Proposition 7.8 below. 94 IGOR KRIZ AND J. P. MAY Proposition 7.7. Let A = A<1> and let M be a cell A-module. Then R(M) is a cell A-module whose differential is given by formula (7.1). If f : M ! N is a map of* * cell A-modules, then g = R(f) is a map of cell A-modules such that the coefficients * *ffnj;iin (7.4) are zero. Therefore R induces a functor R0: D ! DH such that R = SR0. Proof. We must specify a sequential filtration {Ftfi(A; OA; M)}. We are given a* * sequen- tial filtration {FtM} of M. Let J be the subspace of M spanned by the basis ele* *ments of its open cells, so that M = AJ as an A-module. We have an induced filtration {F* *tJ} such that FtM = A FtJ. We also have the nilpotent filtration {FtA} of A, namely FtA = A<1; t> in the notation of Definition 2.2. Via the tensor product filtrat* *ion of the summands (IA)p, there results a filtration of the bar construction B(A) and thu* *s a fil- tration of its subspace OA; here F0OA = k. The filtration of A has the property* * that, for P 0 00 0 00 any element a, d(a) = a a with each a and a of strictly lower filtration th* *an a. The P 0 00 filtration of OA has the property that, for any element x, (x) = x1+1x+ x x with each x0and x00of strictly lower filtration than x. We define Ftfi(A; OA; M) = A Ft(OA A J); where the filtration {Ft(OA A J)} must still be specified. Note first that th* *e tensor product of the three filtrations just specified does not work because, in the d* *ifferential (7.2) on M, we have no control on the filtrations of the ani;jand bni;j. Rather* *, thinking of the filtration as given by a partial ordering of basis elements, we define a le* *xicographic filtration by first taking the filtration on J, next the filtration on OA, and * *last the filtration on A. Formally, this involves an arbitrary choice of total ordering* * of the lexicographically ordered set of triples (q; r; s) of non-negative integers. Th* *e elements of filtration t are linear combinations of elements x a such that if t co* *rresponds to (q; r; s), then 2 FqJ, x 2 FrOA, and a 2 FsA. We obtain a basis for the * *open cells of filtration t by extending a basis of Ft-1(OA A J) to a basis of Ft(OA A * *J). In Construction 7.5, of the four summands of the differential, the first is just t* *he differential on A in the free A-module structure, the second gives the decomposable summands ani;j in (7.1), and the third and fourth both give indecomposable summands k* *ni;j. The statement about maps is clear and the last statement follows from Lemma 7.3* *. __|_| The following two results complete the proof of Theorem 1.3 by showing that * *the functors R : D ! DH and S : DH ! D are inverse equivalences of categories. Proposition 7.8. For cell A-modules M, there is a natural quasi-isomorphism RM = SR0M ! M. Proof. With the signs given in Remark 6.1, we have an A-module B(A; A; M). As noted in II.4.2, there is a natural map of A-modules ffl : B(A; A; M) ! M that * *is a OPERADS, ALGEBRAS, MODULES, AND MOTIVES 95 chain homotopy equivalence and thus a quasi-isomorphism. It suffices to constru* *ct a quasi-isomorphism : R(M) = fi(A; OA; M) ! B(A; A; M): Additively, B(A; A; M) = A B(A) M, and OA is contained in B(A) as its sub Hopf algebra of cycles of total degree zero. The resulting inclusion OA M ! B(A) M extends to the desired map of A-modules (but the extension involves insertion * *of the sign dictated by Remark 6.1). We must show that commutes with the differentials. In homological degree p, * *the differential on the subspace B(A) M of B(A; A; M) can be written as the sum of* * the following four terms: (i)The zeroth face operator d0. (ii)The last face operator (-1)pdp. (iii)(-1)p(1 d), where d is the differential on M. (iv)d 1, where d is the differential on the chain complex B(A). Observe that, in B (A) itself, the zeroth and last face operators are zero. Wh* *en we restrict to O(A) M, the term (iv) is zero. An inspection of definitions shows * *that the remaining three terms sum to the differential on the subspace OA M of fi(A; OA* *; M), the essential point being that the zeroth and last faces in the bar constructio* *n can be written in terms of its coproduct in the fashion given in Construction 7.5. The* * rest of the verification that commutes with differentials is just a check of signs. Finally, we must prove that is a quasi-isomorphism. Filter the source and ta* *rget of by the sum of the degrees of the first coordinate A and last coordinate M; * *that is, a x m is in F tif deg(a) + deg(m) t. Then the differential on the E1-term of the resulting spectral sequence for fi(A; OA; M) is zero, while the differen* *tial on the E1-term for B(A; A; M) is induced by term (iv) above. Therefore the induced map* * of E2-terms is an isomorphism by Theorem 6.3(i). __|_| Proposition 7.9.Let M* be a chain complex of minimal A-modules in H . Then there is a natural quasi-isomorphism M* ! R0SM* of chain complexes in H . Proof.We change our point of view. Let V *be the chain complex of OA-modules th* *at corresponds to M* under the equivalence of categories given in Theorem 1.2 and * *let : V n! OA V nbe the coaction. Observe that, as an A-module, Mn = A V n. Let !* be the composite ____ * *___1j1___// * !* : V * //OA V = OA k V OA A V ; 96 IGOR KRIZ AND J. P. MAY where j is the unit of A. We claim that !* is a quasi-isomorphism from V *to th* *e chain complex of OA-comodules that corresponds to R0SM*. On translation back to H , t* *his will imply the result. We must first show that !* is a map of OA-comodules. Clearly OA A V nmay be identified with the k-module of indecomposable elements of R0SMn. The coaction * *of OA arises in the manner described above Proposition 5.11 from the coaction of f* *lA, and this arises from the decomposable portion of the differential (7.1). This porti* *on comes from the second term of the differential in Construction 7.5, which reduces on * *OA M to (q 1 1)( 1). This implies that the coaction on OA A V nis the obvious * *one induced by the diagonal map on OA. It is now clear from the relation ( 1) = (1) that !* is a map of OA-comodules. We must next show that !* is a map of chain complexes. The differential on t* *he chain complex of OA-comodules that corresponds to R0SM* is given by the indecomposable portion of the differential (7.1), applied to RSM*. This portion comes from the* * last two terms of the differential in Construction 7.5, which reduce on OA M to the* * sum of the following two terms: (i)-(1 (q 1))( 1) (i0)1 d. With M replaced by A V n, regarded as part of SM*, the factor d : A V n! A V* * n in the second summand 1 d is itself the sum of the following three terms: (ii)d 1, where d is the differential on A. (iii)The decomposable part of the differential (7.1) on V n Mn, which is giv* *en by the coaction of flA on V n. (iv)The indecomposable part of the differential (7.1) on Mn, which is given * *by the differential V n! V n+1. On the image of !*, the term (ii) obviously vanishes, the term (i) reduces to -(1 q 1)( 1); and the sum of the terms (i) and (iii) is zero by a little diagram chase based * *on the identity ( 1) = (1 ): Thus the differential on the image of !* is given by (iv), and it follows that * *!* is a map of chain complexes. It remains to prove that !* is a quasi-isomorphism. To see this, assume firs* *t that the coaction of flA on each V nis zero, so that term (iii) vanishes. The inclusion * *of OA in OPERADS, ALGEBRAS, MODULES, AND MOTIVES 97 B (A) induces an inclusion n : OA A V n! B(A) A V n= B(k; A; A V n): The differential on the target is the sum of three terms: the differential on B* *(A), the part of the differential coming from the last face operator, and the differenti* *al on the factor A of A V n. The first of these is zero on OA, and the second and third * *agree under the inclusion with the terms (i) and (ii). Thus is a map of chain comple* *xes. Filtering by degrees in A V n, we see by a little spectral sequence argument t* *hat is a quasi-isomorphism because OA ! B (A) is a quasi-isomorphism. Since V nis just* * a k-module, we have the standard quasi-isomorphism ffln : B(k; A; A V n) ! V n: The composite fflnn!n : V n! V nis the identity map. So far we have ignored the differential V n! V n+1, but if we filter OA A V *and B(A) A V *by degrees in V *, then the differential on the resulting E1-terms is that obtained by ign* *oring the differential in V *and, on E2-terms, we obtain copies of the chain complex V *.* * Thus ffl*, *, and !* are quasi-isomorphisms when the V nare trivial representations * *of flA. Finally, we must take account of the coaction of flA. The V nare generalized * *nilpotent representations of flA. Since the nilpotent filtration of Definition 5.5 is nat* *ural, V *is the union of its subcomplexes FtV *, and the quotients FtV *=Ft-1V *are complex* *es of trivial representations. Therefore !* is a quasi-isomorphism in general. __|_| 98 IGOR KRIZ AND J. P. MAY Part V. Derived categories of modules over E1 algebras Let k be a commutative ring and let C be an E1 operad. We defined C -algebra* *s and modules over C -algebras in Part I, and we showed how to convert partial C -alg* *ebras and modules into genuine C -algebras and modules in Part II. Interesting exampl* *es arise in both topology and algebraic geometry. In this part, we will demonstrate that the derived category of modules over * *a C - algebra has the same kind of structure as the derived category of modules over * *a com- mutative DGA. The essential point is that there is a derived tensor product tha* *t satisfies all of the usual properties, but even the rigorous construction of the derived * *category will require a little work. The standard tools of projective resolutions and de* *rived func- tors are not present here, and our theory is based on the non-standard approach* * to the classical derived categories of DGA's that we presented in Part III. As discussed in Part IV, our original motivation came from Deligne's suggest* *ion [20] that the derived category of modules over the E1 algebra N (Spec(F )) that* * we associated to Bloch's higher Chow complex Z(Spec(F )) in IIx5 is an appropriate* * derived category of integral mixed Tate motives of F . We shall say nothing more about * *that here. We are confident that the present theory will have other applications. It* * has been developed in parallel with a precisely analogous, but more difficult, theory of* * derived categories of modules over E1 ring spectra in algebraic topology [25], and that* * theory has already had very substantial applications. As in Part III, k-modules will mean Z-bigraded k-complexes, with an ordinary* * grading and an Adams grading, and the grading will be cohomological. We let Mk denote t* *he category of such k-modules. Each of the k-modules C (j) of an operad C is conce* *ntrated in Adams grading zero. In view of our cohomological grading, the ordinary gradi* *ng of the C (j) is concentrated in negative degrees. When C is an E1 operad, C (j) is* * a free k[j]-resolution of k. Although our interest is in modules over general E1 algebras, we shall first* * concen- trate on the study of "E1 modules" over the ground ring k. In fact, the theory* * of this Part is based on the idea of changing underlying ground categories from the cat* *egory of ordinary k-modules to that of E1 k-modules. For a given operad C , we agree to write C = C (1) for brevity. Clearly C * *is a differential graded k-algebra via fl : C C ! C; it is usually not commutative,* * but it is homotopy commutative when C is an E1 operad. For a unital operad C , it is e* *asy to see that the category of C-modules can be identified with the category of op* *eradic k-modules of I.4.1, where we regard k as a C -algebra via the augmentation C ! * *N of I.2.2(iii). In fact, what is equivalent, C coincides with the universal envelop* *ing algebra OPERADS, ALGEBRAS, MODULES, AND MOTIVES 99 of k as defined in I.4.9. Recall that the derived category Dk of k-modules is obtained from the homotop* *y cat- egory of k-modules by adjoining formal inverses to the quasi-isomorphisms. Simi* *larly, we have the derived category DC of C-modules. If C ! k is a quasi-isomorphism, * *then, by III.4.2, the categories Dk and DC are equivalent. When C is an E1 operad, we* * think of C-modules as E1 k-modules. As we explain in Section 1, there is a particularly convenient choice of an E* *1 operad C , and there is no loss of generality if we restrict attention to that choice.* * The proofs of these claims are deferred until Section 9. We agree to work with this parti* *cular E1 operad C throughout the rest of this Part. With this choice, we find that * *the category of C-modules admits an associative and commutative "tensor product", w* *hich we denote by to distinguish it from = k. Since Cis not commutative, the exist* *ence of the operation is a remarkable phenomenon. Under the equivalence between DC L * * L and Dk, the new derived tensor product agrees with the derived tensor product* * . Similarly, there is an internal Hom functor Hom on the category of C-modules * *whose derived functor RHom agrees with the derived functor RHom under the equivalence between DC and Dk. In Section 2, we study phenomena connected with the fact that k is not a unit* * for , although there is a natural unit map : k M ! M. We define certain variants of the new tensor product of C-modules that apply when one or both of the given* * C- modules M is unital, in the sense that it comes with a prescribed map k ! M. We write M C N and N B M for the new tensor products of a unital C-module M and a non-unital C-module N, and we write M N for the new tensor product defined when both M and N are unital. The product is associative, commutative, and unital u* *p to coherent natural isomorphism. Thus we have a symmetric monoidal category of uni* *tal C -modules, which we denote by MCu. In Section 3, we prove that A1 and E1 algebras, defined with respect to the p* *articu- lar E1 operad C , are exactly the monoids and commutative monoids in the symmet* *ric monoidal category MCu. This drastically simplifies the study of these algebraic* * struc- tures. We also give an appropriate analog for modules over such algebras. With these preliminaries, we can proceed in precise analogy with the theory o* *f Part III. In fact, we find in Section 4 that the theory of cell modules over a DGA g* *eneralizes verbatim to give a theory of cell modules over an A1 algebra A. The only change* * is that the free functor from k-modules to A-modules has a slightly different desc* *ription. We define and study the tensor product of modules over A in Section 5. We defi* *ne and study the concomitant Hom functor in Section 6. We also describe the varian* *ts of 100 IGOR KRIZ AND J. P. MAY the tensor product for unital A-modules and prove that quasi-isomorphic A1 alge* *bras have equivalent derived categories. In Section 7, we define generalized Tor an* *d Ext groups as the homology groups of derived tensor product and Hom modules, and we construct Eilenberg-Moore spectral sequences for their calculation in terms of * *ordinary Tor and Ext groups. The conclusions are precisely the same as if A were a DGA. * *In Section 8, we specialize to E1 algebras. Here our tensor product of A-modules i* *s again an A-module, and similarly for Hom. The discussion of duality in Part III carri* *es over directly to the E1 context. 1.The category of C-modules and the product For the moment, let C be any operad. Since C = C (1) is a DGA, the theory of Part III applies to it. The free functor F from k-modules to C-modules is give* *n by F M = C M, and the free C-modules generated by suspensions of k play the role * *of sphere C-modules. The derived category DC is equivalent to the homotopy catego* *ry of cell C-modules. When C is unital and the augmentation ffl : C ! k is a quas* *i- isomorphism, the derived categories Dk and DC are equivalent. A key point is th* *at the action C M ! M is then a quasi-isomorphism for any cell C-module M. Via instances of the structural maps fl, we have a left action of C and a ri* *ght action of C C on C (2), and these actions commute with each other. Thus we have a bimod* *ule structure on C (2). Let M and N be left C-modules. Clearly M N is a left C C- module via the given actions. This makes sense of the following definition. Definition 1.1.For C-modules M and N, define M N to be the C-module M N = C (2) CC M N: We have a Hom functor on C-modules that is related to the tensor product by* * an adjunction of the usual form. In fact, the desired adjunction dictates the defi* *nition. Definition 1.2.Let M and N be (left) C-modules. Define Hom (M; N) = Hom C(C (2) C M; N): Here, when forming C (2) C M, C acts on C (2) through j Id: C = k C ! C C; when forming Hom C, C acts on C (2) C M via its left action on C (2). The right* * action of C on C (2) through Id j : C = C k ! C C induces a left action of C on Hom (M; N). Lemma 1.3. There is a natural adjunction isomorphism MC(M N; P ) ~=MC(M; Hom (N; P )): We must consider the commutativity, associativity, and unity properties of . OPERADS, ALGEBRAS, MODULES, AND MOTIVES 101 Lemma 1.4. There is a canonical commutativity isomorphism of C-modules o : M N -! N M: Proof.Use the action of the transposition oe 2 2 on C (2) together with the tra* *nspo- sition isomorphisms C C ! C C and M N ! N M. __|_| The following result is fundamental to our work. It comes from our parallel t* *opo- logical work with Elmendorf [25]. We defer the proof to Section 9. Note that k * *is a C -module via the augmentation C ! k. Theorem 1.5. There is an E1 operad C , called the "linear isometries operad",* * for which there is a canonical associativity isomorphism of C-modules (L M) N ~=L (M N): In fact, for any j-tuple M1; : :;:Mj of C-modules, there is a canonical isomorp* *hism M1 . . .Mj~= C (j) Cj(M1 . .M.j); where the iterated product on the left is associated in any fashion. For j 2, * *the j-fold -power Cj is isomorphic to C (j) as a (C ; Cj)-bimodule, and C (j) is isomorph* *ic to C as a left C-module. Lemma 1.6. There is a natural map of C-modules : k N ! N. The symmetrically defined map M k ! M coincides with the composite o. Moreover, under the associativity isomorphism, o Id= Id : M k N -! M N: Proof.The degeneracy map oe1 : C (2) ! C of I.3.5, fflId : CC ! kC ~=C , and the isomorphism k N ~=N together give the required map : k N ! CCN ~=N. The symmetry is clear. Under the isomorphisms of their domains with C (3)MkN, both o Idand Id agree with the tensor product over Idffl Idof oe2 : C (3) -! C (2) and the isomorphism M k N ~=M N. __|_| In our motivating examples from algebraic geometry, we started with partial a* *lgebras and converted them to C -algebras, where C was an arbitrarily chosen E1 operad. Clearly, we may as well choose C to be the linear isometries operad. However, w* *e have the following result. Its proof is a bar construction argument similar to those* * used in Part II; we defer it to Section 9. Theorem 1.7. Let C and C 0be any two E1 operads. There is a functor V that ass* *igns a quasi-isomorphic C 0-algebra V A to a C -algebra A. There is also a functor V* * that assigns a quasi-isomorphic V A-module V M to an A-module M. 102 IGOR KRIZ AND J. P. MAY We construct the derived category of A-modules from the homotopy category of A-modules by adjoining formal inverses to the quasi-isomorphisms, where a map of A-modules is a quasi-isomorophism if it induces an isomorphism on homology, tha* *t is, if it is a quasi-isomorphism when regarded as a map of k-modules. The theorem c* *an be elaborated to give an equivalence of the derived category of A-modules with the* * derived category of V A-modules. Thus there is no loss of generality if we restrict attention to the linear i* *sometries E1 operad C , and we do so throughout the rest of this Part. We repeat that C* * is an abbreviated notation for C (1). By use of cell approximations of C -modules, L and Hom induce a derived tensor product and a derived Hom functor R Hom on DC. We must relate these fuctors to the derived tensor product and Hom functors* * on k-modules. This depends on another special property of the operad C that will * *be proven in Section 9. Lemma 1.8. Define : C ! C C by (c) = c 1 and regard the right C C-module C (2) as a right C-module by pullback along . Choose a degree zero cycle x 2 C * *(2) that augments to 1 2 C, observe that x cannot be a boundary, and define o : C !* * C (2) by o(c) = fl(x c 1). Then o is a homotopy equivalence of right C-modules with homotopy inverse the degeneracy map oe1 : C (2) ! C. Proposition 1.9. The product satisfies the following properties. (i)If 0 -! N0- ! N -! N00-! 0 is an exact sequence of C-modules, where N00 is a cell C-module, then 0 -! M N0- ! M N -! M N00-! 0 is an exact sequence of C-modules for any C-module M. (ii)If f : M ! M0 is a quasi-isomorphism of C-modules and N is a cell C-modu* *le, then f Id: M N -! M0 N is a quasi-isomorphism of C-modules. (iii)If N is a cell C-module, then M N is naturally quasi-isomorphic as a k-* *module to M N. Therefore the equivalence of derived categories DC ! Dk that is induced by t* *he L L forgetful functor from C-modules to k-modules carries to . Proof. With differential ignored, N00is a free C-module, and the exact sequence* * given in (i) is therefore split exact. Upon tensoring with M we obtain an algebraical* *ly split exact sequence of C2-modules, and (i) follows. In view of (i), induction up the* * sequential filtration and passage to colimits show that (ii) will hold for general cell C-* *modules N if it holds when N is the free C-module F K generated by a free k-module K. By * *the OPERADS, ALGEBRAS, MODULES, AND MOTIVES 103 definition of , M (F K) is naturally isomorphic to C (2)C (M K). By the lemma, this is homotopy equivalent to M K, and (ii) follows. For (iii), choose x 2 C * *(2) as in the lemma. Then x determines a homotopy equivalence k ! C (2). This equivale* *nce and the definition of give us natural maps of k-modules M N -! C (2) M N -! M N; the first of which is a homotopy equivalence. By Lemma 1.8 and the proof of (ii* *), the composite is a homotopy equivalence when N = F K. Therefore, by induction up the sequential filtration of N and passage to colimits, the composite is a quasi-is* *omorphism for any cell C-module N. __|_| Although the unit map : k N ! N is not an isomorphism in general, it induces a natural isomorphism on the level of derived categories. Corollary 1.10.If N is a cell C-module, then the unit map : k N ! N is a L quasi-isomorphism. Therefore induces a natural isomorphism k N ! N of functors on the derived category DC. Proof.Consider the following commutative diagram: C (2) k N ____//k N | oe1Id|| || fflffl| fflffl| C (1) N_______//_N: Here is the action of C (1) on N. By Proposition 1.9 and its proof, all arrows* * except are quasi-isomorphisms, hence so is . __|_| We need a lemma to obtain the analog of Proposition 1.9 for Hom . Lemma 1.11. For k-modules K and L, there are isomorphisms of C-modules F K F L ~=F (K L) and Hom (K; L) ~=Hom (F K; L): For cell C-modules M and N, M N is a cell C-module. Proof.The first isomorphism is immediate from the isomorphism C (2) ~=C (1) giv* *en by the last statement of Theorem 1.5. The second follows in view of the chain of n* *atural isomorphisms Mk(K0; Hom (F K; L)) ~=MC(F K0; Hom (F K; L)) ~=MC(F K0 F K; L) ~=MC((F (K0 K); L) ~=Mk(K0 K; L) ~=Mk(K0; Hom(K; L)): As in III.1.5(iii) or III.5.1, the last statement follows from the first isomor* *phism. __|_| 104 IGOR KRIZ AND J. P. MAY Proposition 1.12. Let N be an arbitrary C-module. (i)If 0 -! M0- ! M -! M00-! 0 is an exact sequence of C-modules, where M00 is a cell C-module, then 0 -! Hom (M00; N) -! Hom (M; N) -! Hom (M0; N) -! 0 is an exact sequence of C-modules. (ii)If M is a cell C-module and f : N ! N0 is a quasi-isomorphism of C-modul* *es, then Hom (Id; f) : Hom (M; N) -! Hom (M; N0) is a quasi-isomorphism of C-modules. (iii)There is an induced adjunction isomorphism DC(M L N; P ) ~=DC(M; R Hom (N; P )): (iv)If M is a cell C-module, then Hom (M; N) is quasi-isomorphic as a k-mod* *ule to Hom (M; N). Therefore the equivalence of derived categories DC ! Dk that is induced by t* *he forgetful functor from C-modules to k-modules carries R Hom to R Hom . Proof. Part (i) is clear since the given exact sequence splits as a sequence of* * C-modules with differential ignored. Parts (ii) and (iii) follow formally from the lemma;* * see III.4.5. If M is a cell C-module and K is a k-module, the quasi-isomorphism F K M -! F K M of Proposition 1.9 and the natural quasi-isomorphism K -! F K give rise * *to the composite Mk(K; Hom (M; N)) ~=MC(F K; Hom (M; N)) ~=MC(F K M; N) ! Mk(F K M; N) ! Mk(F K M; N) ~=Mk(F K; Hom(M; N)) ! Mk(K; Hom(M; N)): Since the last two arrows are induced by quasi-isomorphisms, the composite indu* *ces a natural isomorphism on passage to derived categories, and the image of the id* *entity map is a natural quasi-isomorphism of k-modules Hom (M; N) -! Hom (M; N): __|_| Corollary 1.13. There is a natural isomorphism N -! R Hom (k; N) in the derived category DC. Proof. This is immediate from the natural isomorphisms L __ DC(M; N) ~=DC(M k; N) ~=DC(M; R Hom (k; N)): |_| OPERADS, ALGEBRAS, MODULES, AND MOTIVES 105 Remark 1.14.It would be of interest to construct an E1 operad with the properti* *es of Theorem 1.5 by purely algebraic methods. There are intrinsic limitations to the* * present construction. For example, in contrast to the analogous topological situation o* *f [25], we do not have that : k k ! k is a quasi-isomorphism, and we could only arrange * *this by having C (2) ~=C2, which would sacrifice freeness under the 2-action. It wou* *ld be desirable to have an operad with the additional property that C (2) is chain ho* *motopy equivalent to C2 as a right C2-module (of course, not 2-equivariantly). This pr* *operty would ensure that M N is quasi-isomorphic to M N for all C-modules M and N. 2.Unital C-modules and the products C, B, and The fact that is not an isomorphism before passage to the derived category l* *eads us to introduce some further products. By a unital C-module M, we understand a C -module M together with a map of C-modules j : k ! M. We regard k itself as a unital C-module via the identity map k ! k. An augmentation of a unital C-module M is a map ffl : M ! k of unital k-modules, so that fflj = Id. For a non-unital* * k-module M, we let M+ denote the unital C-module M k. Clearly an augmented C-module M is isomorphic to (Ker ffl)+ as a unital C-module. Our formal arguments will a* *pply to arbitrary unital C-modules, but some of our arguments about quasi-isomorphisms * *will apply only to augmented C-modules. It is possible to generalize these arguments* *, but the extra verbiage does not seem to be warranted since the applications we envi* *sage are to augmented C -algebras and since Theorem 2.9 will give a way around such diff* *iculties. A cell theory adapted to unital C-modules is given in IIIx6 and is used to cons* *truct a derived category of unital C-modules. Given this, our results on quasi-isomorph* *isms lead to conclusions about derived categories. We shall leave the formulation of* * these interpretations to the reader. Definition 2.1.Let M be a unital C-module and let N be any C-module. Define M C N to be the pushout displayed in the following diagram of C-modules: jId k N ____//_M N || || fflffl| fflffl| N ______//M C N: Define N B M by symmetry. Proposition 2.2.Let M and N be C-modules. Then M+ C N ~=N (M N): 106 IGOR KRIZ AND J. P. MAY If N is a cell C-module, then the canonical map M+ N -! M C N is a quasi-isomorphism. Proof. The first statement is clear, and the cited canonical map reduces to Id : (M N) (k N) -! (M N) N: Thus Corollary 1.10 gives the second statement. __|_| The commutativity and associativity of imply the following commutativity and associativity isomorphisms relating and C; these isomorphisms imply various ot* *hers. Lemma 2.3. Let L and M be unital C-modules and let N and P be any C-modules. Then there are natural isomorphisms M C N ~=N B M; M C (N P ) ~=(M C N) P; and L C (N B M) ~=(L C N) B M: We have a Hom functor and a suitable adjunction. Definition 2.4.Let M be a unital C-module and let N be any C-module. Define Hom C(M; N) to be the C-module displayed in the following pullback diagram: Hom C(M; N)____//Hom (M; N) | |j* | | fflffl| fflffl| N ________//_Hom(k; N); here the bottom arrow is adjoint to the unit map o : N k ~=k N ! N. Lemma 2.5. For a unital C-module M and any C-modules L and N, there is a natu* *ral adjunction isomorphism MC(L B M; N) ~=MC(L; HomC (M; N)): Definition 2.6.Let M and N be unital C-modules. The coproduct of M and N in the category of unital C-modules is the pushout M [k N. There is an analogous pusho* *ut (M k) [kk (k N), and the unit maps determine a natural map of C-modules : (M k) [kk (k N) ! M [k N: The restrictions to k k of the maps Id j : M k ! M N and j Id: k N ! M N OPERADS, ALGEBRAS, MODULES, AND MOTIVES 107 coincide, hence these maps determine a map : (M k) [kk (k N) -! M N: Define M N to be the pushout displayed in the following diagram of C-modules: (M k) [kk (k N) ____//M [k N || || fflffl| fflffl| M N __________//_M N: Then M N is a unital C-module with unit the composite of the unit k ! M [k N and the displayed canonical map M [k N ! M N. Lemma 2.7. Let M and N be C-modules. Then (M+) (N+) ~=(M N) M N k: Remark 2.8.We have a compatible decomposition (M+) (N+) ~=(M N) (M k) (k N) (k k): If we knew that : k k ! k were a quasi-isomorphism, it would follow from Corol* *lary 1.10 that the canonical map M N -! M N is a quasi-isomorphism when M and N are cell C-modules. However, such a result * *would be of limited utility since the "augmentation ideals" of augmented A1 or E1 alg* *ebras are unlikely to be of the homotopy types of cell C-modules. In the applications of the analogous topological theory, it is vital to overc* *ome the problem pointed out in the previous remark. One way to do this is to to approxi* *mate a given A1 or E1 algebra A by its monadic bar construction BA of II.4.2, which* * is quasi-isomorphic to A and therefore has an equivalent derived category. We shal* *l be more explicit about the definitions and shall prove the following result in Sec* *tion 9. Theorem 2.9. For an A1 or E1 algebra A, there is an A1 or E1 algebra BA and a natural quasi-isomorphism ffl : BA ! A. For an A-module M, there is a BA-module BM and a natural quasi-isomorphism of BA-modules ffl : BM ! M. If A and A0are augmented A1 or E1 algebras, there are natural quasi-isomorphisms of k-modules BA BA0-! BA BA0and BA BM -! BA C BM: The purpose of introducing the products C and is to obtain good algebraic pr* *op- erties on the domains of definition of the multiplications on A1 and E1 algebra* *s and of their actions on modules. The theorem shows that, by use of bar construction* * ap- proximations, we can obtain such algebraic control without changing the underly* *ing 108 IGOR KRIZ AND J. P. MAY quasi-isomorphism type. The following algebraic properties of are easily deri* *ved from the associativity and commutativity of together with formal arguments from the definition. Lemma 2.10. The following associativity relation holds, where M and N are uni* *tal C -modules and P is any C-module: (M N) C P ~=M C (N C P ): Proposition 2.11. The category of unital C-modules is symmetric monoidal under * *the product ; that is, is associative, commutative, and unital up to coherent natu* *ral isomorphism. 3.A new description of A1 and E1 algebras and modules Let C be the linear isometries operad. Recall from I.2.1 that a C -algebra * *A is a k-module together with an associative, unital, and equivariant system of action* * maps : C (j) Aj! A: Recall from I.4.1 that an A-module M is a k-module together with an associative, unital, and equivariant system of action maps : C (j) Aj-1 M ! M: By Theorem 1.7, up to quasi-isomorphism, all E1 algebras and modules are C -alg* *ebras and modules. Similarly, if we drop the equivariance conditions, then, up to qu* *asi- isomorphism, all A1 algebras and modules are of this form. We agree to refer * *to C -algebras and modules, with and without equivariance, as E1 and A1 algebras a* *nd modules in the rest of this Part. Restricting the action to j = 0 and j = 1, we see that an A1 algebra is a un* *ital C- module with additional structure. The category MCuof unital C-modules is symmet* *ric monoidal under the product . As with any symmetric monoidal category, we define* * a monoid in MCuto be an object A with an associative and unital product OE : AA !* * A; A is commutative if OEo = OE. The following result is the precise analog of a t* *heorem first discovered in the deeper topological context of [25]. Theorem 3.1. An A1 algebra A determines and is determined by a monoid structure on its underlying unital C-module; A is an E1 algebra if and only if it is a co* *mmutative monoid in MCu. While this is the most elegant form of the theorem, it admits an equivalent * *form expressed in terms of the product. In fact, the following result is immediate * *from the description of in terms of and . OPERADS, ALGEBRAS, MODULES, AND MOTIVES 109 Lemma 3.2. A monoid structure on a unital C-module A determines and is determi* *ned by a product OE : A A ! A such that the following diagrams commute: jId Idj IdOE k A J____//A Aoo__A k and A A A ____//A A JJ ttt JJJ OE| ttt OEId| OE| JJJ | ttt | | JJ%%fflffl|yytt fflffl|OE fflffl| A A A _______//_A; A is commutative if the following diagram commutes: ______o____ A AG //A A GGG wwww GGG www OE G##G--wOEw A: The analog of Theorem 3.1 for modules reads as follows; we incorporate the an* *alog of Lemma 3.2 in the statement. Theorem 3.3. Let A be an A1 or E1 algebra with product OE : A A ! A. An A-module is a C-module M together with a map : A C M ! M such that the following diagrams commute, where the second diagram implicitly uses the isomor* *phism (A A) C M ~=A C (A C M): jCId IdC k C ML____//LA C M and A A C M ____//A C M LLL | OECId| | ~=LLLLL| | | L&&fflffl| fflffl| fflffl| M A C M _______//_M: Equivalently, an A-module is a C-module M together with a map : AM ! M such that the following diagrams commute: jId Id k M L____//A M and A A M ____//A M LL LLL | OEId| | LLL | | | LL%%fflffl| fflffl| fflffl| M A M _______//_M: We illustrate the force of these results by giving some formal consequences. * *Recall that the tensor product of commutative DGA's is their coproduct in the category* * of commutative DGA's. The proof consists of categorical diagram chases that now ca* *rry over to our more general context. Corollary 3.4.Let A and B be A1 algebras. Then A B is an A1 algebra. If M is an A-module and N is a B-module, then M N is an A B-module. If A and B are E1 algebras, then A B is an E1 algebra and is the coproduct of A and B in * *the category of E1 algebras. 110 IGOR KRIZ AND J. P. MAY The following corollary will become important in Section 5. We first recall * *a standard categorical definition [43, VI.6]. Definition 3.5.Working in an arbitrary category, suppose given a diagram __e_ g A ____////B__//C f in which ge = gf. The diagram is called a split coequalizer if there are maps h : C ! B and k : B ! A such that gh = IdC, fk = IdB, and ek = hg. It follows that g is the coequalizer* * of e and f. Observe that, while covariant functors need not preserve coequalizers in gen* *eral, they clearly do preserve split coequalizers. Corollary 3.6. Let A be an A1 algebra. Then the following diagram of unital k- modules is a split coequalizer: __OEId___ OE A A A _________////A A__//_A: IdOE If M is a left A-module, then the following diagram of k-modules is also a spli* *t coequal- izer: __OEId___ (A A) C M ~=A C (A C M) _________////A C_M__//M: IdC Proof. The first statement is true for monoids in any symmetric monoidal catego* *ry. The required maps h and k are j Idand j Id Id. The second statement is equally trivial. __|_| Remark 3.7. In MCu, as in any symmetric monoidal category, we have operads M and N such that an M -algebra is a monoid and an N -algebra is a commutative monoi* *d; compare I.2.2. Thus these operads define A1 and E1 algebras. There result monads M and N in MCuwhich define the free A1 and E1 algebras. We can start with a k-module K and form the free unital C-module (C K)+ = (C K) k. The free A1 algebra it generates must be the free A1 algebra generated by K. That is, X M((C K)+) ~=C(K) = C (j) Kj: Similarly, reinterpreting C in the E1 sense, X N((C K)+) ~=C(K) = C (j) j Kj: OPERADS, ALGEBRAS, MODULES, AND MOTIVES 111 In fact, verification of the relations just given provides one way of proving* * Theorem 3.1. As a matter of category theory, to prove the theorem in the E1 case, it su* *ffices to show that the monad C in Mk that defines E1 algebras is the "composite" of the * *monad N in MCuthat defines commutative monoids in that category and the monad in Mk that defines unital C-modules. Rather than run through the relevant category th* *eory, we sketch a more direct proof in the following two lemmas. The proof of Theorem* * 3.3 is precisely analogous and will be left to the reader. Lemma 3.8. Let A be an A1 algebra. Then : C (2)AA -! A induces a product OE : A A C (2) CC A A -! A such that the first two diagrams of Lemma 3.2 commute. If A is an E1 algebra, t* *hen the third diagram also commutes. Proof.That factors through the tensor product is immediate from the associati* *vity diagram in the definition, I.1.1, of an operad action. Since j : k ! A is taken* * to be : C (0) ! A, it is easy to check the commutativity of the unit diagrams from L* *emma 1.6 and I.1.1. The associativity diagram is more interesting and depends on the* * proof of Theorem 1.5. In fact, the two squares in the following diagram commute by I.* *1.1: Id 2 C (2) C (1) C (2) A A2____//_C (2) A flId|| || fflffl| fflffl| C (3)OOA3________________//AOO flId|| || | Id | 2 C (2) C (2) C (1) A2 A ____//C (2) A : The horizontal arrows factor through tensor products over C3 in the terms in th* *e left column and through tensor products over C2 in the terms at the top and bottom r* *ight corners, and the diagram then becomes IdOE A (A A) ____//_A A | ~=|| OE| |fflffl fflffl|| C (3) C3A3O______//_AOOO | ~=|| OE|| | OEId | (A A) A ____//A A: The two arrows labelled ~=are isomorphisms by the proof of Theorem 1.5 in Secti* *on 9, and they give the associativity isomorphism that is implicit in the claim th* *at the associativity diagram of Lemma 3.2 commutes. If A is an E1 algebra, then the map 112 IGOR KRIZ AND J. P. MAY : C (2) A A ! A is 2-equivariant, and the commutativity of the last diagram * *of Lemma 3.2 follows. __|_| Lemma 3.9. Let A be a monoid in the category of unital C -modules. Its monoid structure is uniquely determined by an A1 algebra structure, and A is commutati* *ve if and only if the A1 structure is an E1 structure. Proof. The unit and C-action of A give : C (0) ! A and : C (1) A ! A. The product OE : A A ! A induces : C (2) A2 ! A. The associativity of OE shows t* *hat it defines an unambigous map Aj ! A, where Aj denotes the j-fold -power of A. Since Aj ~=C (j) CjAj; OE induces a map : C (j)Aj! A for each j 2. The verification that the associa* *tivity and unity diagrams of I.1.1 commute are laborious diagram chases from the defin* *ition and the arguments of Section 9. It is not hard to see that the equivariance dia* *grams commute if A is commutative. It is clear that the resulting A1 algebra struct* *ure determines the given monoid structure. Conversely, by the associativity diagram* *s, the higher maps of an A1 algebra structure are determined by the second map, so th* *at the A1 structure is uniquely determined by the monoid structure. __|_| 4. Cell A-modules and the derived category of A-modules Fix an A1 algebra A. We first observe that the category of A-modules is cl* *osed under various constructions in the underlying categories of k-modules and C-mod* *ules. Modules mean left modules unless otherwise specified; right modules are defined* * by symmetry in terms of action maps M A ! M, or M B A ! M. Proposition 4.1. Let M and N be A-modules, let L be a C-module, and let K be a k-module. (i)Any categorical colimit or limit (in the category of k-modules) of a dia* *gram of A-modules is an A-module. (ii)M K and Hom (K; M) are A-modules and MA(M K; N) ~=MA(M; Hom(K; N)): (iii)M L and Hom (L; M) are A-modules and MA(M L; N) ~=MA(M; Hom (L; N)): (iv)Hom (M; L) is a right A-module. (v) The cofiber of a map of A-modules is an A-module. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 113 Proof.(i) Note first that colimits and limits of C-modules are calculated as co* *limits and limits of underlying k-modules. The functor commutes with colimits in each* * of its variables. Thus, for a direct system Miof C-modules, A (ColimMi) ~=Colim(A Mi): If the Mi are A-modules, their structure maps induce a structure map for ColimM* *i. For an inverse system Mi, canonical projections give a map A (Lim Mi) -! Lim(A Mi): This implies the analogous conclusion for limits, and this conclusion also foll* *ows from the fact, recalled below, that the forgetful functor from A-modules to k-module* *s is a right adjoint. (ii) Certainly M K is a C-module, and A (M K) ~=(A M) K: By applying A (?) to the evaluation map Hom (K; M) K ! M and taking the adjoint of the resulting map, we obtain a map of C-modules A Hom (K; M) -! Hom (K; A M): Therefore, by composition, the structure map for M induces structure maps for M* * K and Hom (K; M). The adjunction is a formal verification. (iii) The argument is just like the proof of (ii). (iv) The action of A on Hom (M; L) is the adjoint of the composite Id ffl Hom (M; L) A M ____//Hom (M; L) M ____//L; where is the action of A on M and ffl is the evaluation map. The last statemen* *t of Lemma 1.6 is needed for the verification of the unit property. (v) This follows easily from (i) and (ii). __|_| To develop the cell theory of A-modules, we need a free functor from k-module* *s to A-modules, and we already have a free functor from k-modules to C-modules, name* *ly K ! C K. The following observation shows that A C L is the free functor from C -modules to A-modules. Lemma 4.2. For C-modules L and A-modules M, MA(A C L; M) ~=MC(L; M): 114 IGOR KRIZ AND J. P. MAY Proof. The unit k ! A induces a C-map j : L ~=k C L ! A C L, the product on A induces a structure of A-module on A C L, and an A-module structure on M is giv* *en by an A-map : A C M ! M. Therefore an A-map g : A C L ! M induces the C-map g O j : L ! M and a C-map f : L ! M induces the A-map O (Id C f) : A C L ! M. These are inverse correspondences. __|_| Definition 4.3.For a k-module K, define an A-module F K by F K = A C (C K): Lemma 4.4. For k-modules K and A-modules M, MA(F K; M) ~=Mk(K; M): At this point, we recall that we have already constructed the free A-module * *functor in I.4.9 and I.4.10: since the category of A-modules is isomorphic to the categ* *ory of U(A)-modules, the free A-module generated by a k-module K must be U(A) K. We are entitled to the following consequence, which is special to the linear is* *ometries operad. Note that the unit of U(A) determines a natural k-map K ! U(A) K. Proposition 4.5. For k-modules K, the natural map F K ! U(A) K is an isomor- phism of A-modules. In particular, F k is isomorphic to U(A); we can read off the resulting prod* *uct on F k by comparison with I.4.9. The following basic result is intuitively obvious* *, but we assume that A is augmented in order to obtain a quick proof. Proposition 4.6. Assume that A is augmented. If K is a cell k-module, then the A-map ff : F K ! A K induced by the canonical k-map K ! A K is a quasi- isomorphism. If K is a free k-module with zero differential, then H*(F K) is th* *e free H*(A)-module generated by K. Proof. By inspection of definitions, F K ~=F k K. Thus the result will hold in* * general if it holds when K = k. We have the following commutative diagram of maps of k-modules: O A COO____//C (2) A C____//A C || |fi| || | fflffl| fflffl| A _____Id____//Aoo__ff__ A C C: Here is the canonical inclusion, O is determined by a chosen degree zero cycle* * x 2 C (2) that augments to 1 2 k, and fi is given by fi(d a c) = (d a (c 1)) OPERADS, ALGEBRAS, MODULES, AND MOTIVES 115 for d 2 C (2), a 2 A, and c 2 C . The composite fiO is multiplication by the u* *nit 1 2 A under the product determined by x, and the very definition of an E1 operad action implies that fiO ' Id. Clearly and O are quasi-isomorphisms, hence so i* *s fi. The unlabelled arrows in the right-hand square are quasi-isomorphisms by the pr* *oof of Proposition 1.9 and by Proposition 2.2, hence ff is also a quasi-isomorphism* *. The second statement follows. __|_| At this point, we can simply parrot the theory of Part III in our more genera* *l context, replacing the free functor A(?) used there with the free functor F (?) = A C (C* * (?)). To begin with, we define "sphere A-modules" F s(t) by F s(t) = F (Ss(t)); and we observe that the cones on spheres satisfy CF s(t) ~=F (CSs(t)): Part III has been written with this generalization in mind, and we reach the fo* *llowing conclusion. Theorem 4.7. Without exception, every statement and proof in Sections 1, 2, 3,* * and 6 in Part III applies verbatim to modules over A1 algebras. Of course, for an actual DGA A, we now have two categories of A-modules in si* *ght, namely ordinary ones and A1 ones. The latter are the same as U(A)-modules, and * *we have the following expected consistency statement. Proposition 4.8.If A is a DGA, then the map ff : U(A) ~=F k ! A is a map of DGA's. It induces an equivalence of categories between the ordinary derived cat* *egory DA and the E1 derived category DU(A). Proof.The first statement is an immediate verification since C acts on A throug* *h the augmentation C ! N . Since ff is a quasi-isomorphism, the second statement foll* *ows from III.4.2. __|_| 5.The tensor product of A-modules We have not yet defined tensor products of modules over A1 algebras. We can m* *imic classical algebra. Definition 5.1.Let A be an A1 algebra and let M be a right and N be a left A- module. Define M A N to be the coequalizer (or difference cokernel) displayed i* *n the 116 IGOR KRIZ AND J. P. MAY following diagram of C-modules: ___Id____ (M B A) N ~=M (A C N) _________////M N___//_M A N; Id where and are the given actions of A on M and N; the canonical isomorphism of the terms on the left is implied by Lemma 2.3. Remark 5.2. We have given the definition in the form most convenient for our la* *ter proofs. However, it is equivalent to define M A N more intuitively as the coequ* *alizer in the following diagram: _Id______ M A N ________////_M N__//_M A N: Id In fact, by the definitions of our products, there is a natural epimorphism ss : (M A N) (M N) -! (M B A) N ~=M (A C N): The composites ( Id) O ss and (Id) O ss restrict to Idand Id on M A N, and both composites restrict to the identity on M N. L As we shall justify shortly, we construct A as a functor rDA x `DA ! Dk by approximating one of the variables by a cell A-module; here "r" and "`" indi* *cate right and left A-modules. We have used the notation A to avoid confusion with A in the case of a DGA A regarded as an A1 algbra. We have the following consistency statement. Remarks 5.3.When A = k, M B k = M, k C N = N, and we are coequalizing two identity maps. Therefore our new M k N coincides with M N. When A is a DGA and M and N are A-modules regarded as E1 A-algebras, the quasi-isomorphisms constructed in Proposition 1.9 can be elaborated to obtain comparisons of coequ* *alizer L diagrams that show that the derived tensor product M AN of M and N regarded as E1 modules is isomorphic in the derived category Dk to the classical derived t* *ensor L product M AN. An A1 algebra A with product OE : A A ! A has an opposite algebra Aopwith product OE O o, and a left A-module with action is a right Aop-algebra with ac* *tion O o. A simple comparison of coequalizer diagrams gives the following commutati* *vity isomorphism. Lemma 5.4. For a right A-module M and left A-module N, M A N ~=N AopM: OPERADS, ALGEBRAS, MODULES, AND MOTIVES 117 Lemma 5.5. For a C-module L, L (M A N) ~=(L M) A N and (M A N) L ~=M A (N L): For A1 algebras A and B, we define an (A; B)-bimodule to be a left A and rig* *ht B-module M such that the following diagram commutes: A M B ____//M B | | | | fflffl| fflffl| A M _______//_M: The previous lemma and comparisons of coequalizer diagrams give the following associativity isomorphism and unit map. Lemma 5.6. Let L be an (A; B)-bimodule, M be a (B; C)-bimodule, and N be a (C; D)-bimodule. Then L B M is an (A; C)-bimodule and (L B M) C N ~=L B (M C N) as (A; D)-bimodules. Lemma 5.7. The action : A N -! N of a left A-module N factors through a map of A-modules : A A N -! N: Observe that, for a C-module L, L B A ~=A C L is an (A; A)-bimodule. In parti* *cu- lar, this applies to the free left A-module F K = A C (C K) generated by a k-m* *odule K, which may be identified with the free right A-module generated by K. The fol* *low- ing result and its corollary will be used in conjunction with the quasi-isomorp* *hism of Proposition 1.9 relating the -product of C-modules with their ordinary tensor p* *roduct as k-modules. Recall from Theorem 1.5 that C C ~=C as a left C-module. Proposition 5.8.Let L and L0be C-modules and let N be an A-module. There is a natural isomorphism of A-modules (L B A) A N ~=L N: There is also a natural isomorphism of (A; A)-bimodules (L B A) A (A C L0) ~=A C (L L0): Proof.Applying the functor L (?) to the representation of N as a split coequal* *izer in Lemma 3.6 and using isomorphisms from Lemmas 2.3 and 2.10, we find that the resulting split coequalizer diagram is isomorphic to the diagram that defines F* * L A N. Similarly, we obtain the second isomorphism by applying the functor (?) C (L L* *0) to the representation of A as a split coequalizer in Lemma 3.6. __|_| 118 IGOR KRIZ AND J. P. MAY Corollary 5.9. Let K and K0 be k-modules and let N be an A-module. There is a natural isomorphism of A-modules F K A N ~=(C K) N: There is also a natural isomorphism of (A; A)-bimodules F K A F K0~=F (K K0): L To justify the definition of A, we must consider the behavior of A on cell * *A-modules N. The sequential filtration of N gives short exact sequences 0 -! Nn -! Nn+1- ! Nn+1=Nn -! 0; where the quotient is a direct sum of sphere A-modules F Sq(r). Just as for DGA* *'s, the sequence is algebraically split when we ignore the differentials, and this * *implies that the sequence is still exact when we apply the functor M A (?) for any M. T* *his allows us to reduce proofs for general N to the case N = F k, using commutation* * with suspension to handle sphere modules, commutation with direct sums to handle fil* *tration subquotients, induction and five lemma arguments to handle the Nn, and passage * *to colimits to complete the proof. For example, we have the following result, whic* *h is just III.4.1 restated in our new context. Lemma 5.10. Let N be a cell A-module. Then the functor M A N preserves exact sequences and quasi-isomorphisms in the variable M. Proof. Both statements are clear from Corollary 5.9 and Proposition 1.9 if N is* * a sphere A-module. The general case follows by passage to direct sums, induction * *up the filtration, and passage to colimits. For the exactness, one uses a 3 x 3 lemma * *to prove the inductive step. __|_| L This justifies our definition of the derived tensor product A, and it is un* *ital by the following result. Corollary 5.11. If A is augmented and N is a cell A-module, then the unit map : A A N ! N is a quasi-isomorphism. Proof. It suffices to prove this for the sphere N = F k = A C C , and A A F k ~= A C. Comparing the coequalizer diagram that defines A A N with the coequalizer representation of N in Corollary 3.6 and using Definition 2.1, we see that coi* *ncides with the canonical map A C -! A C C. This is a quasi-isomorphism by Proposition 2.2. __|_| OPERADS, ALGEBRAS, MODULES, AND MOTIVES 119 The following result will be the starting point for the construction of a spe* *ctral sequence for the computation of H*(M A N). Corollary 5.12.Let K be a free k-module with zero differential and let N be a c* *ell A-module. Then there is an isomorphism H*(F K A N) ~=(H*(A) K) H*(A)H*(N) that is natural in the A-modules F K and N. Proof.The subtle point is that naturality in F K and not just K will be essenti* *al in Section 7. Recall that A is defined by a coequalizer diagram like that used to * *define A. Recall too that Proposition 4.6 gives a quasi-isomorphism of A-modules F K ! A * * K and that the functor (?)A N preserves quasi-isomorphisms. We obtain a commutati* *ve diagram H*(F K) H*(A)H*(N) ______//_H*(F K A N) | | | | fflffl| fflffl| H*(A K) H*(A)H*(N) ____//H*((A K) A N) in which the vertical arrows are isomorphisms. We see by Corollary 5.9 that the* * top arrow is an isomorphism when N is a sphere A-module, and it follows by our usual induction and passage to colimits that it is an isomorphism for any N. __|_| 6.The Hom functor on A-modules; unital A-modules We have a Hom functor to go with our new tensor product. Its definition is di* *ctated by the desired adjunction. Let A be an A1 algebra. Definition 6.1.Let M and N be left A-modules. Define Hom A(M; N) to be the equalizer displayed in the following diagram of C-modules: _*__ Hom A(M; N) ___//_Hom (M; N)_!__////Hom(A C M; N): Here * = Hom (; Id) and ! is the adjoint of the composite IdCffl A C (M Hom A(M; N)) ____//A C N___//_N; where ffl is the evaluation map of the adjunction in Lemma 1.3. As we shall justify shortly, we define the derived Hom functor by letting R H* *om Abe Hom A(M; N), where M is a cell A-module quasi-isomorphic to M. Remark 6.2.If A = k, and M and N are E1 k-modules, then our new Hom k(M; N) is identical to Hom (M; N). 120 IGOR KRIZ AND J. P. MAY Lemma 6.3. For C-modules L and left A-modules M and N, there is a natural ad- junction isomorphism MA(L M; N) ~=MC(L; HomA (M; N)): Just as in ordinary module theory, we have the following complementary adjun* *ction. Lemma 6.4. For C-modules L, right A-modules M, and left A-modules N, there is* * a natural adjunction isomorphism MC(M A N; L) ~=MA(M; Hom (N; L)): Proposition 5.8 and Corollary 5.9 imply the following results. Proposition 6.5. Let L be a C-module and N be an A-module. There is a natural isomorphism of A-modules Hom A (A C L; N) ~=Hom (L; N): Proof. This is immediate from the following composite of isomorphisms of repres* *ented functors, in which L0is a C-module: MC(L0; HomA (A C L; N)) ~=MA(L0 (A C L); N) ~=MA((L0B A) A (A C L); N) ~=MA(A C (L0 L); N) ~=MC(L0 L; N) ~=MC(L0; Hom (L; N)): __|_| Corollary 6.6. Let K be a k-module and N be an A-module. There is a natural isomorphism of A-modules HomA (F K; N) ~=Hom (C K; N): Arguing as in IIIx5, we obtain the following analog of Lemma 5.10. Lemma 6.7. Let M be a cell A-module. Then the functor Hom A(M; N) preserves exact sequences and quasi-isomorphisms in the variable N. It also preserves ex* *act sequences of cell A-modules in the variable M. This justifies our definition of R Hom A(M; N). As in IIIx5, we conclude that L (6.8) DA(L M; N) ~=DC(L; R Hom A(M; N)): Now Corollary 5.11 has the following formal consequence; compare Corollary 1.12. Corollary 6.9. The adjoint N ! Hom A(A; N) of o : N A A ~=AA N ! N induces a natural isomorphism of functors on the derived category DA. Again, as in Corollary 5.12, we can use Proposition 4.6 to deduce the follow* *ing calculational consequence of Corollary 6.6. It will be needed in the next secti* *on. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 121 Corollary 6.10.Let K be a free k-module with zero differential and let N be a c* *ell A-module. Then there is an isomorphism H*(Hom A(F K; N)) ~=HomH*(A)(H*(A) K; H*(N)) that is natural in the A-modules F K and N. We briefly indicate some further developments of the theory, including the im* *portant invariance result parallel to III.4.2. By a unital A-module, we understand an A* *-module M together with a map j : A -! M of A-modules. Definition 6.11.Define the mixed tensor product M CA N of a unital right A-modu* *le M and a left A-module N by replacing k by A and by A in Definition 2.1. Define BA by symmetry. Define the unital tensor product M A N of a unital right A-modu* *le M and a unital left A-module N by replacing k by A and by A in Definition 2.6. When A is an E1 algebra, it will follow from the discussion in Section 8 that* * these products all take values in A-modules. The properties of C, B, and listed in S* *ection 2 generalize in the expected fashion. Moreover, the new products admit alterna* *tive descriptions in terms of coequalizer diagrams like that which defines A. Lemma 6.12. For a unital right A-module M and a left A-module N, M CA N can be identified with the coequalizer displayed in the diagram __CId____ (M C A) C N ~=M C (A C N) _________////M C_N__//M CA N: IdC For a unital right A-module M and a unital left A-module N, M A N can be identi* *fied with the coequalizer displayed in the diagram __Id_____ (M A) N ~=M (A N) _________////M N___//M A N: Id Proof.It is easy to check this on augmented A-modules M = M0 A, and the general case follows by a formal argument; compare [25, IIIx3]. __|_| Proposition 6.13.Let OE : A ! A0 be a quasi-isomorphism of augmented DGA's. Then the pullback functor OE* : DA0! DA is an equivalence of categories with in* *verse L given by the derived extension of scalars functor A0CA(?). Proof.We regard OE as a map of A-modules in forming A0CA M. With this modificat* *ion of A0A (?), we have an adjunction isomorphism MA0(A0CA M; M0) ~=MA(M; OE*M0) 122 IGOR KRIZ AND J. P. MAY for M 2 MA and M0 2 MA0. For a C-module L, a formal argument (compare [25, IIIx3]) gives a natural isomorphism A0CA (A C L) ~=A0C L: Thus the functor A0CA (?) preserves sphere modules and therefore cell modules. * *This implies that the adjuction passes to derived categories. The essential point is* * that OE CA Id: M ~=A CA M -! A0CA M is a quasi-isomorphism when M is a cell A-module. This will hold in general if * *it holds when M is a sphere A-module. However, when M = A C L for a C-module L, OE CA Id reduces to OE C Id: A C L -! A0C L: For a cell C-module L, OE C Idis a quasi-isomorphism by Propositions 1.9 and 2.* *2; the latter result applies in view of our assumption that A and A0are augmented. The* * rest of the argument is the same as in III.4.2. __|_| Definition 6.14.Let A be an E1 algebra. Define an A-algebra B and a B-module M by replacing , C and by the corresponding products over A in the diagrammatic descriptions of A1 k-algebras and their modules given in Theorem 3.1, Lemma 3.2* *, and Theorem 3.3. We can carry out homological algebra in this more general context, as sugges* *ted by the results of the next section. For example, we can construct the Hochschild h* *omology of A1 algebras by mimicking the definition of the standard complex for its comp* *utation. We refer the interested reader to our topological paper [25]. It carries the pa* *rallel theory considerably further, and its arguments can be adapted to the present algebraic* * context. 7. Generalized Eilenberg-Moore spectral sequences Fix an A1 algebra A. Since our derived tensor product and Hom functors gener* *alize those of DGA's, the following definition generalizes III.4.4. Definition 7.1.Define L Tor*A(M; N) = H*(M AN) and Ext*A(M; N) = H*(RHom A(M; N)): These functors enjoy the same general properties as in the case of DGA's: e* *xact triangles in either variable induce long exact sequences on passage to Tor or E* *xt, Tor preserves direct sums in either variable, and Ext converts direct sums in M to * *direct products and preserves direct products in N. The behavior on free modules is (7.2) Tor*A(M; F K) ~=H*(M K) and Ext*A(F K; N) ~=H*(Hom (K; N)): OPERADS, ALGEBRAS, MODULES, AND MOTIVES 123 The crucial point of our general definition of Tor and Ext is that we still h* *ave Eilenberg-Moore spectral sequences for their calculation. Write M* = H*(M) for brevity of notation. Theorem 7.3. There are natural spectral sequences of the form (7.4) Ep;q2= Torp;qA*(M*; N*) =) Torp+qA(M; N) and (7.5) Ep;q2= Extp;qA*(M*; N*) =) Extp+qA(M; N): These are both spectral sequences of cohomological type, with (7.6) dr : Ep;qr! Ep+r;q-r+1r: In (7.4), p is the negative of the usual homological degree, the spectral seque* *nce is non-zero only in the left half-plane, and it converges strongly. In (7.5), the* * spectral sequence is non-zero only in the right half plane, and it converges strongly if* *, for each fixed (p; q), only finitely many of the differentials (7.6) are non-zero. (See* * [10] for a general discussion of convergence.) The rest of this section will be devoted to* * the proof of Theorem 7.3. The starting point is the following construction. Construction 7.7.Let M be an A-module and let Q be a submodule of M* with generating set {yi}. If yi2 (qi; ri), we may think of yias a map of k-modules S* *qi(ri) ! M. Let K = Sqi(ri), let f : K ! M be the sum of the yi, and let "f: F K ! M be the induced map of A-modules. Then (F K)* is the free A*-module on generators yi2 (qi; ri), and the induced homomorphism f* : (F K)* ! M* is a map of A*-modu* *les that sends xito yi. Clearly Imf"*= Q. For a right A-module M, we choose an A*-free resolution dp ffl* (7.8) . .F.p-!Fp-1-! . .-.! F0-! M -! 0 and regrade it cohomologically, setting F p= F-p. Each F pis bigraded, via deg* *ree and Adams degree. We shall pay little attention to the Adams degree since the o* *nly complications that it introduces are notational. Let Q0 = Kerffl and Qp = Kerdp for p -1, so that dp defines an epimorphism F* * p! Qp+1. For p 0, let Kp be the sum of a copy of the sphere k-module -pks(t) = ks* *-p(t) for each basis element of F pof bidegree (s; t) and let M0 = M. Using Construct* *ion 7.7 inductively, we can construct cofiber sequences of right A-modules p p ip p-1jp-1 p (7.9) F Kp-k!M -! M -! F K for p 0 that satisfy the following properties: 124 IGOR KRIZ AND J. P. MAY (i)k0 realizes ffl on H*. (ii)H*(Mp) = -pQp+1for p -1. (iii)kp realizes -pdp : -pF p-! -pQp+1on H* for p - 1. (iv)ip induces the zero homomorphism on H* for p 0. (v) jp-1realizes the inclusion 1-pQp -! 1-pF pon H* for p 0. Observe that (iii) implies the case p - 1 of (ii) together with (iv) and (v)* *. We are actually constructing a cell A-module relative to M, in the sense of IIIx6. To obtain the spectral sequence (7.4), we assume that N is a cell A-module a* *nd we define (7.10) Dp;q1= Hp+q-1(Mp-1A N) and Ep;q1= Hp+q(F Kp A N); where we have ignored the Adams grading. The maps displayed in (6.9) give maps i (ip-1)* : Dp;q1-! Dp-1;q+11 j (jp-1)* : Dp;q1-! Ep;q1 k (kp)* : Ep;q1-! Dp+1;q1: By Lemma 5.10, these display an exact couple in standard cohomological form. We* * see from Corollary 5.12 that Ep;q1~=(F pA*N*)qand that d1 agrees under the isomorph* *ism with d 1. This proves that Ep;q2= Torp;qA*(M*; N*): Observe that k : E0;q1-! D1;q1can and must be interpreted as Hq(F K0 A N) -! Hq(M A N): On passage to E2, it induces the edge homomorphism (7.11) E0;q2= M* A*N* -! H*(M A N): The convergence is standard, although it appears a bit differently than in m* *ost spec- tral sequences in current use. Write i0;pfor the evident composite map M ! Mp a* *nd also for its tensor product with N. Filter H*(M A N) by letting F pH*(M A N) be the kernel of (i0;p-1)* : H*(M A N) -! H*(Mp-1A N): By (iv) above, we see that the telescope (= homotopy colimit) TelMp has zero ho* *mology and is thus quasi-isomorphic to the zero module. Since the functor (?) A N comm* *utes with telescopes, it follows from Lemma 5.10 that Tel(MpA N) also has zero homol* *ogy. OPERADS, ALGEBRAS, MODULES, AND MOTIVES 125 This implies that the filtration is exhaustive. Consider the (p; q)th term of t* *he associated bigraded group of the filtration. It is defined as usual by Ep;q0H*(M A N) = F pHp+q(M A N)=F p+1Hp+q(M A N); and the definition of the filtration immediately implies that this group is iso* *morphic to the image of (i0;p)* : Hp+q(M A N) -! Hp+q(Mp A N): The target of (i0;p)* is Dp+1;q1, and of course Ep;q1= Hp+q(F Kp A N) also maps* * into Dp+1;q1, via k. It is a routine exercise in the definition of a spectral sequen* *ce to check that k induces an isomorphism Ep;q1-! Im(i0;p)*: (We know of no published source, but this verification is given in [10, x6].) To see the functoriality of the spectral sequence, suppose given a map f : M * *! M0 of A-modules and apply the constructions above to M0 (writing F 0p, etc). Const* *ruct a sequence of maps of A*-modules fp : F p! F 0pthat give a map of resolutions. * *We can realize the maps fp on homology groups by A-module maps "fp: F Kp ! F K0p. Starting with f = f0 and proceeding inductively, a standard exact triangle argu* *ment allows us to construct a map fp : Mp-1 ! M0p-1 such that the following diagram * *of A-modules commutes up to homotopy: F Kp ____//_Mp___//_Mp-1___//_F Kp | | | | | | | | fflffl| fflffl| fflffl| fflffl| F K0p____//Mp____//M0p-1___//F K0p: There results a map of spectral sequences that realizes the induced map Tor*;*A*(M*; N*) -! Tor*;*A*(M0 *; N0*) on E2 and converges to (f A Id)*. Functoriality in N is obvious. To obtain the analogous Ext spectral sequence, we switch from right to left m* *odules in our resolution (7.8) of M* and its realization by A-modules, and we now assu* *me that M is a cell A-module. It is convenient to work with the homological grading dis* *played in (7.8) and to regrade (7.9) correspondingly. We obtain a cofiber sequence kp ip jp+1 (7.12) F Kp-! Mp-! Mp+1-! F Kp: With this grading, we define (7.13) Dp;q1= Hp+q(Hom A(Mp; N)) and Ep;q1= Hp+q(Hom A(F Kp; N)): 126 IGOR KRIZ AND J. P. MAY The maps displayed in (6.12) give maps i (ip-1)* : Dp;q1-! Dp-1;q+11 j (kp)* : Dp;q1-! Ep;q1 k (jp+1)* : Ep;q1-! Dp+1;q1: These display an exact couple in standard cohomological form. We see by Corolla* *ry 6.10 that Ep;q1~=HomqA*(Fp; N*) and that d1 agrees with Hom (d; 1) under this isomor* *phism. This proves that Ep;q2= Extp;qA*(M*; N*): Observe that j : D0;q1! E0;q1can and must be interpreted as Hq(Hom A(M; N)) -! Hq(Hom A(F K0; N)): On passage to E2, it induces the edge homomorphism (7.14) Hq(Hom A(M; N)) -! Hom qA*(M*; N*) = E0;q2: To see the convergence, let 0;p: Hom A(Mp; N) -! Hom A(M; N) be the map induced by the evident interate M ! Mp and filter H*(Hom A(M; N)) by letting F pH*(Hom A(M; N)) be the image of (0;p)* : H*(Hom A(Mp; N)) -! H*(Hom A(M; N)): The (p; q)th term of the associated bigraded group of the filtration is Ep;q0H*(Hom A(M; N)) = F pHp+q(Hom A(M; N))=F p+1Hp+q(Hom A(M; N)): The group Ep;q1is defined as the subquotient Zp;q1=Bp;q1of Ep;q1, where Bp;q1= j(Ker(0;p)*); and a routine exercise in the definition of a spectral sequence shows that the * *additive relation (0;p)*O j-1 induces an isomorphism Ep;q1~=Ep;q0H*(Hom A(M; N)): Since TelMp has zero homology, so does the homotopy limit, or "Microscope", MicHom A (Mp; N) ~=Hom A(TelMp; N): As usual for a countable inverse system, there is a Lim1exact sequence for the * *compu- tation of H*(Mic Hom A(Mp; N)), and we conclude that LimH*(Hom A(Mp; N)) = 0 and Lim1H*(Hom A(Mp; N)) = 0: OPERADS, ALGEBRAS, MODULES, AND MOTIVES 127 In the language of [10], this means that the spectral sequence {Ep;qr} is condi* *tionally convergent, and [10] shows that strong convergence follows if, for each pair (p* *; q), only finitely many of the differentials with source Ep;qrare non-zero. The functoria* *lity of the spectral sequence is clear from the argument for torsion products given above. 8. E1 algebras and duality We assume that A is an E1 algebra in this section, and we show that the study* * of E1 modules works exactly the same way as the study of modules over commutative DGA's. In particular, we discuss composition and Yoneda products and duality. O* *b- serve that, although it is not at all obvious from the original definitions of * *Ix4, their reinterpretation in Section 3 implies that we obtain the same A-modules whether* * we regard A as an E1 algebra or, by neglect of structure, as an A1 module. If : A M ! M gives M a left A-module structure, then O o : M A ! M gives M a right A-module structure such that M is an (A; A)-bimodule. Just as i* *n the study of modules over commutative DGA's (where the argument is too trivial to n* *eed such a pedantic formalization), this leads to the following important conclusio* *n. Theorem 8.1. If M and N are A-modules, then M A N and Hom A(M; N) have canonical A-module structures deduced from the A-module structure of M or, equi* *v- alently, N. The tensor product over A is associative and commutative, and the u* *nit maps A A M ! M and A ! Hom A(A; N) are maps of A-modules. There is a natural adjunction isomorphism (8.2) MA(L A M; N) ~=MA(L; HomA(M; N)): L The derived category DA is symmetric monoidal under A, and the adjunction pass* *es to the derived category. The analog of III.5.1 is immediate from Corollary 5.9. Proposition 8.3.Let M and M0 be cell A-modules. Then M A M0 is a cell A- P module with sequential filtration { p(MpA Nn-p)}. As in the previous section, write A* = H*(A); it is an associative and (grade* *d) commutative algebra. Corollary 8.4.Tor*A(M; N) and Ext*A(M; N) are A*-modules, and there are natural commutativity and associativity isomorphisms of A*-modules (8.5) Tor*A(M; N) ~=Tor*A(N; M) 128 IGOR KRIZ AND J. P. MAY and (8.6) Tor*A(L A M; N) ~=Tor*A(L; M A N): The spectral sequences of the previous section are spectral sequences of differ* *ential A*-modules. The formal properties of Theorem 8.1 imply many others. For example, (8.7) Hom A(M A L; N) ~=Hom A(M; HomA (L; N)) because the two sides represent isomorphic functors on modules. Using this, we* * see that the evaluation map ffl : Hom A(L; M) A L -! M induces a map MA(Hom A(M; N); HomA (M; N)) ! MA(Hom A(M; N); HomA (Hom A(L; M) A L; N)) ~= MA(Hom A(M; N); HomA (Hom A(L; M); HomA (L; N))) ~=MA(Hom A(M; N) A Hom A(L; M); HomA (L; N)): The image of the identity map of Hom A(M; N) gives a composition pairing (8.8) ss : Hom A(M; N) A Hom A(L; M) -! Hom A(L; N): This pairing is associative and commutative in the sense that the following dia* *grams commute; note that the unit of the adjunction (8.2) specializes to give a map j* * : A ! Hom A(M; M): Hom A(M; N) A AU UUUUU Idj || UUoUUUUUU fflffl| UU**U Hom A(M; N) A Hom A(M; M) _ss//_HomA(M; N); A A Hom A(L; M)U UUUUU jId|| UUUUUUUU fflffl| UU**U HomA (M; M) A Hom A(L; M) _ss//_HomA(L; M); and Idss Hom A(N; P ) A Hom A(M; N) A Hom A(L; M)____//HomA(N; P ) A Hom A(L; N) ssId|| |ss| fflffl| fflffl| Hom A(M; P ) A Hom A(L; M)_________ss_______//HomA(L; P ): OPERADS, ALGEBRAS, MODULES, AND MOTIVES 129 On passage to homology, the pairing (8.8) induces a Yoneda product on Ext. Proposition 8.9.There is a natural, associative, and unital system of pairings ss* : Ext*A(M; N) A*Ext*A(L; M) -! Ext*A(L; N): Proof.We have an associative and unital system of isomorphisms in DA L F Sq(r) AF Ss(t) ~=F Sq+s(r + t): Since Hq(M) ~=DA(F S-q(-r); M) for an A-module M, the result follows directly f* *rom the pairings. __|_| These pairings also imply pairings of spectral sequences. We content ourselve* *s with a brief indication of the proof. Proposition 8.10.The pairing Hom A(M; N)AHom A(L; M) ! Hom A(L; N) induces a pairing of spectral sequences that coincides with the algebraic Yoneda pairin* *g on the E2-level and converges to the induced pairing of Ext groups. Proof.Assume that L and M are cell A-modules and construct a sequence {Lp} as in (7.12). Then the maps M ! Mp induce a compatible system of pairings Hom A(Mp; N) A Hom A(Lp0; M) | | fflffl| Hom A(M; N) A Hom A(Lp0; M) | | fflffl| Hom A(Lp0; N): These induce the required pairing of spectral sequences. The convergence is cle* *ar, and the behavior on E2 terms is correct by comparison with the axioms or by compari* *son with the usual construction of Yoneda products. __|_| Modulo the obvious changes of notation, the formal duality theory that we ex- plained in IIIx5 applies verbatim to the present more general context. We defi* *ne M_ = Hom A(M; A), and we say that a cell A-module M is "finite" if it has a co- evaluation map j : A -! M M_ such that the analog of diagram III.5.6 commutes in DA. When M is a finite A-module, various natural maps such as ae : M -! M__ and : M_ A N -! Hom A(M; N) 130 IGOR KRIZ AND J. P. MAY induce isomorphisms in DA, exactly as if A were a classical k-algebra, without * *differen- tial, and M were a finitely generated projective A-module. The last isomorphism* * has the following implication. Proposition 8.11. For a finite A-module M and any A-module N, TornA(M_; N) ~=ExtnA(M; N): Although the relation may be obscured by the grading, this is an algebraic c* *ounterpart of Spanier-Whitehead duality in algebraic topology. We call particular attenti* *on to III.5.7, which we repeat for emphasis. Theorem 8.12. A cell A-module is finite in the sense just defined if and only * *if it is a direct summand up to homotopy of a finite cell A-module. 9.The linear isometries operad; change of operads We here prove 1.5, 1.8, 1.7, and 2.9, in that order. We first define an E1 * *operad L of topological spaces. The algebraic E1 operad C of Theorem 1.5 is obtained by applying the normalized singular chain complex functor C# to L , as discussed a* *t the start of Ix5. Let U ~=R1 be a countably infinite dimensional real inner product space, top* *ologized as the union of its finite dimensional subspaces. Let Uj be the direct sum of j* * copies of U. Define L (j) to be the set of linear isometries Uj ! U with the function * *space topology, that is, the compact-open topology made compactly generated. Note tha* *t a linear isometry is an injection but not necessarily an isomorphism. The space L* * (0) is the point i, i : 0 ! U, and L (1) contains the identity 1 : U ! U. The left act* *ion of j on Uj by permutations induces a free right action of j on L (j). The structure * *maps fl : L (k) x L (j1) x . .x.L (jk) -! L (j1+ . .+.jk) are defined by fl(g; f1; : :;:fk) = g O (f1 . . .fk): The associativity property of Theorem 1.5 stems from a special associativity* * property of L that was first observed by Hopkins. Observe that L (1) acts from the left * *on any L (i), via fl, hence L (1) x L (1) acts from the left on L (i) x L (j). Note to* *o that L (1) x L (1) acts from the right on L (2). Let us denote these actions by and* * , respectively. Lemma 9.1 (Hopkins). For i 1 and j 1, the diagram xId_ fl L (2) x L (1) x L (1) x L (i) x L_(j)////_L (2) x L (i)_x_L/(j)/_L (i + j) Idx OPERADS, ALGEBRAS, MODULES, AND MOTIVES 131 is a split coequalizer of spaces. Proof.Choose isomorphisms s : Ui ! U and t : Uj ! U and define h(f) = (f O (s t)-1; s; t) and k(f; g; g0) = (f; g O s-1; g0O t-1; s; t): It is trivial to check the identities of Definition 3.5. __|_| Proposition 9.2.Let i 1 and j 1. Then the structural map fl of the operad C = C#(L ) induces an isomorphism C (2) CC C (i) C (j) -! C (i + j): Proof.As in Ix5, let g denote the shuffle map C#(X) C#(Y ) -! C#(X x Y ) and recall that it is a monomorphism naturally split by the Alexander-Whitney m* *ap f; g is associative and we continue to write g for maps obtained from it by iterat* *ion. The covariant functor C# preserves split coequalizers, and the map of the statement* * factors as the composite g fl# C (2) CC C (i) C (j)-! C#(L (2) xL(1)xL(1)L (i) x L (j))-! C (i + j); where fl# is an isomorphism. We see that g is a split monomorphism by a compari* *son of coequalizer diagrams, and we must check that g is an epimorphism. Think of i* *so- morphisms s : Ui ! U and t : Uj ! U as singular zero simplices of the spaces L * *(i) and L (j). A singular n-simplex x : n ! L (i + j) determines a singular n-simpl* *ex y of L (2) by precomposition with s-1 t-1. When all but one variable is a zero simplex, the shuffle map takes an obvious form from which it is trivial to chec* *k that (fl# O g)(y s t) = x. __|_| Proof of Theorem 1.5.We must construct a natural isomorphism (L M) N ~=L (M N); and we claim that both sides are naturally isomorphic to C (3) C3L M N: Note that N ~=C C N. We have the isomorphisms (L M) N ~=C (2) C2(C (2) C2L M) (C C N) ~=(C (2) C2C (2) C (1)) C3(L M N) ~=C (3) C3(L M N): 132 IGOR KRIZ AND J. P. MAY The symmetric argument shows that this is also isomorphic to L (M N). In view* * of the generality of Proposition 9.2, the argument iterates to prove that all j-fo* *ld iterated products are canonically isomorphic to C (j) CjM1 . . .Mj: When all Mi = C, this gives an isomorphism Cj ~=C (j) of (C ; Cj)-bimodules. * *Fi- nally, if t : Uj ! U is an isomorphism, then composition with t and t-1 give in* *verse homeomorphisms of left L (1)-spaces between L (j) and L (1). On passage to chai* *ns, these give rise to an isomorphism of left C-modules between C (j) and C. __|_| The following observation about L implies Lemma 1.8 on passage to singular c* *hains. Lemma 9.3. Let L (1) act from the right on L (2) by gf = g O (f 1) for f 2 L* * (1) and g 2 L (2). Choose an isomorphism t : U2 ! U and define o : L (1) ! L (2) by o(f) = t O (f 1). Define oe1 : L (2) ! L (1) by oe1(g) = g O i1, where i1 : U * *! U2 is the inclusion of the first summand. Then o and oe1 are inverse L (1)-equivariant ho* *motopy equivalences. Proof. Choose a path h : I ! L (1) connecting 1 to t O i1. Since (o O oe1)(f) = t O (f 1) O i1 = (t O i1) O f; "h(f; s) = h(s) O f specifies a homotopy "h: Id' o O oe1. Similarly, ^h(g; s) =* * h(s) O g specifies a homotopy Id' ae, where ae(g) = t O i1O g. We have (oe1O o)(g) = t O* * (gi1 1). Clearly aeOi1 = oe1Oo Oi1, whereas aeOi2and oe1Oo Oi2map U onto orthogonal subs* *paces of U. A homotopy between the restrictions along i2is obtained by applying Gram-Sch* *midt orthogonalization to the obvious linear homotopy. With the constant homotopy al* *ong the restriction to i1, this gives a an L (1)-equivariant homotopy between ae an* *d oe1Oo. __|_| Proof of Theorem 1.7.The argument is the exact algebraic analog of one first us* *ed in topology in [46]. It is similar to, but simpler than, the arguments of IIxx4,5.* * We assume given any two E1 operads C and C 0, and we must construct a C 0-algebra from a * *C - algebra. The argument works equally well for A1 and E1 algebras. There is an ev* *ident notion of the tensor product of operads, with (C C 0)(j) = C (j) C (j0): We abbreviate C 00= C C 0. The augmentations of C and C 0induce maps of operads C 00! C and C 00! C 0, and these in turn induce maps of monads C00! C and C00! C0. The maps C00K ! CK and C00K ! C0K are homotopy equivalences for all k-modules K since all three operads are E1 operads. Moreover, the composite* * of CC00! CC and the product of C is a right action of the monad C00on the functor * *C, OPERADS, ALGEBRAS, MODULES, AND MOTIVES 133 and C is a (C; C00)-bifunctor in the sense of II.4.1. Similarly, if A is a C-al* *gebra, then A is a C00-algebra by pullback along C00! C. Now recall the two-sided bar constru* *ction B(F; C; A) = C#B*(F; C; A) from II.4.1. Here C# is the totalization functor from simplicial k-modules to k* *-modules discussed in IIx5. By II.4.2 and the naturality properties of this constructio* *n, for a C-algebra A we have evident natural maps of C00-algebras A - B(C; C; A) - B(C00; C00; A) -! B(C0; C00; A); all of which are quasi-isomorphisms. We let V A be the C0-algebra B(C0; C00; A* *) and have the conclusion of Theorem 1.7 on the algebra level. The argument on the mo* *dule level is the same, using the monads of Ix4. __|_| Finally, we return to the linear isometries operad and prove Theorem 2.9. Proof of Theorem 2.9.For definiteness, we work with E1 algebras. The proof for A1 algebras is similar but simpler. We abbreviate BA = B(C; C; A), and we have* * a natural map of E1 algebras ffl : BA ! A that is a homotopy equivalence of k-alg* *ebras. We also have the monad C[1] of I.4.3 such that a C[1]-algebra is a C-algebra A * *together with an A-module M. We write (BA; BM) = B(C[1]; C[1]; (A; M)): More explicitly, we apply the totalization functor C# to both coordinates of th* *e pair of simplicial k-modules B*(C[1]; C[1]; (A; M)); the first coordinate is BA and * *we call the second coordinate BM. Then BM is a BA-module, and we have a map of BA- modules ffl : BM ! M that is a homotopy equivalence of k-modules. We must const* *ruct quasi-isomorphisms of k-modules BA BA0-! BA BA0and BA BM -! BA C BM: We give the argument for the first of these; the argument for the second is pre* *cisely similar. Clearly BA BA0is constructed from constituent k-modules (C (i) (CpA)i) (C (j) (CqA0)j) ~=(C (i) C (j)) ((CpA)i) (CqA0)j) by passing to orbits over the action of ix j, passing to direct sums over i 0 * *and j 0, and then totalizing over p, q, and the internal degree; BA BA0is constru* *cted similarly from constituent k-modules (C (2) C2(C (i) (CpA)i) (C (j) (CqA0)j) ~=C (i + j) ((CpA)i) (CqA0)j): Here we have used Proposition 9.2 when i 1 and j 1; Lemma 2.7 and the convent* *ion C (0) X0 = k give the summands with i = 0 or j = 0. By choosing a degree cycle 134 IGOR KRIZ AND J. P. MAY x 2 C (2) such that ffl(x) = 1 and using the operad structural maps fl, we obta* *in a composite (ix j)-map C (i) C (j) -! C (2) C (i) C (j) -! C (i + j) for each i and j. Since C is an E1 operad, this is a map between free (ix j)- resolutions of k and is thus a (ixj)-equivariant homotopy equivalence. 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Verdier. Categories derivees, in Springer Lecture Notes in Mathematic* *s Vol 569. Springer, 1971. 61. R. J. Wellington. The unstable Adams spectral sequence for free iterated lo* *op spaces. Memoirs of the Amer. Math. Soc. Vol. 258, 1982. Igor Kriz The University of Michigan Ann Arbor, Michigan 48109, USA kriz@lsa.umich.edu J. P. May The University of Chicago Chicago, Illinois 60637, USA may@math.uchicago.edu OPERADS, ALGEBRAS, MODULES, AND MOTIVES 137 Abstract.With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work i* *s divided into five largely independent Parts: I Definitions and examples of operads and their actions IIPartial algebraic structures and conversion theorems IIIDerived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E1 algebras In differential algebra, operads are systems of parameter chain complexes for* * mul- tiplication on various types of differential graded algebras "up to homotopy", * *for example commutative algebras, n-Lie algebras, n-braid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA's and derive* *d cat- egories of modules up to homotopy over DGA's up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Ta* *te motives in algebraic geometry, both rational and integral.