MORITA THEORY IN ABELIAN, DERIVED AND STABLE MODEL CATEGORIES STEFAN SCHWEDE These notes are based on lectures given at the Workshop on Structured ring s* *pectra and their applications. This workshop took place January 21-25, 2002, at the Unive* *rsity of Glasgow and was organized by Andy Baker and Birgit Richter. Contents 1. Introduction * * 1 2. Morita theory in abelian categories * * 2 3. Morita theory in derived categories * * 6 3.1. The derived category * * 6 3.2. Derived equivalences after Rickard and Keller * * 14 3.3. Examples * *19 4. Morita theory in stable model categories * * 21 4.1. Stable model categories * * 22 4.2. Symmetric ring and module spectra * * 25 4.3. Characterizing module categories over ring spectra * * 32 4.4. Morita context for ring spectra * * 35 4.5. Examples * *38 References * *42 1.Introduction The paper [Mo58 ] by Kiiti Morita seems to be the first systematic study of * *equivalences between module categories. Morita treats both contravariant equivalences (which* * he calls dualities of module categories) and covariant equivalences (which he calls isom* *orphisms of module categories) and shows that they always arise from suitable bimodules,* * either via contravariant hom functors (for `dualities') or via covariant hom functors * *and tensor products (for `isomorphisms'). The term `Morita theory' is now used for results* * concerning equivalences of various kinds of module categories. The authors of the obituar* *y article [AGH ] consider Morita's theorem "probably one of the most frequently used sin* *gle results in modern algebra". In this survey article, we focus on the covariant form of Morita theory, so * *our basic question is: When do two `rings' have `equivalent' module categories ? We discuss this question in different contexts: o (Classical) When are the module categories of two rings equivalent as ca* *tegories ? ___________ Date: October 2, 2003. 1 2 STEFAN SCHWEDE o (Derived) When are the derived categories of two rings equivalent as tri* *angulated categories ? o (Homotopical) When are the module categories of two ring spectra Quillen* * equivalent as model categories ? There is always a related question, which is in a sense more general: What characterizes the category of modules over a `ring' ? The answer is, mutatis mutandis, always the same: modules over a `ring' are * *characterized by the existence of a `small generator', which plays the role of the free modul* *e of rank one. The precise meaning of `small generator' depends on the context, be it an abeli* *an category, a derived category or a stable model category. We restrict our attention to cat* *egories which have a single small generator; this keeps things simple, while showing the main* * ideas. Almost everything can be generalized to categories (abelian, derived or stable model c* *ategories) with a set of small generators. One would have to talk about ringoids (also called r* *ings with many objects) and their differential graded and spectral analogues. Background: for a historical perspective on Morita's work we suggest a look* * at the obituary article [AGH ] by Arhangel'skii, Goodearl, and Huisgen-Zimmermann. Th* *e history of Morita theory for derived categories and `tilting theory' is summarized in S* *ection 3.1 of the book by König and Zimmermann [KZ ]. Both sources contain lots of further re* *ferences. For general background material on derived and triangulated categories, see * *[SGA 41_2] (Appendix by Verdier), [GM ], [Ver96], or [Wei94 ]. We freely use the languag* *e of model categories, alongside with the concepts of Quillen adjoint pair and Quillen equ* *ivalence. For general background on model categories see Quillen's original article [Qui67], * *a modern introduction [DwSp95 ], or [Hov99 ] for a more complete overview. Acknowledgments: The Morita theory in stable model categories which I descri* *be in Section 4 is based on joint work with Brooke Shipley spread over many years and* * several papers; I would like to take this opportunity to thank her for the pleasant and* * fruitful collaboration. I would also like to thank Andy Baker and Birgit Richter for org* *anizing the wonderful workshop Structured ring spectra and their applications in Glasgow. 2.Morita theory in abelian categories To start, we review the covariant Morita theory for modules; this is essenti* *ally the con- tent of Section 3 of [Mo58 ]. This and related material is treated in more deta* *il in [Ba , II x3], [AF92 , x22] or [Lam , x18]. Definition 2.1. Let A be an abelian category with infinite sums. An object M o* *f A is small if the hom functor A(M, -) preserves sums; M is a generator if every obje* *ct of A is an epimorphic image of a sum of (possibly infinitely many) copies of M. The emphasis in the smallness definition is on infinite sums; finite sums ar* *e isomorphic to finite products, so they are automatically preserved by the hom functor. Fo* *r modules over a ring, smallness is closely related to finite generation: every finitely * *generated module is small and for projective modules, `small' and `finitely generated' are equiv* *alent concepts. Rentschler [Ren69 ] gives an example of a small module which is not finitely ge* *nerated. A generator can equivalently be defined by the property that the functor A(M* *, -) is faithful, compare [Ba , II Prop. 1.1]. A small projective generator is called a* * progenerator. The main example is when A = Mod-R is the category of right modules over a ring* * R. Then the free module of rank one is a small projective generator. In this case, a ge* *neral R-module MORITA THEORY 3 M is a generator for Mod-R if and only if the free R-module of rank one is an e* *pimorphic image of a sum of copies of M. Here is one formulation of the classical Morita theorem for rings: Theorem 2.2. For two rings R and S, the following conditions are equivalent. (1)The categories of right R-modules and right S-modules are equivalent. (2)The category of right S-modules has a small projective generator whose e* *ndomor- phism ring is isomorphic to R. (3)There exists an R-S-bimodule M such that the functor - R M : Mod-R -! Mod-S is an equivalence of categories. If these conditions hold, then R and S are said to be Morita equivalent. Here are some elementary remarks on Morita equivalence. Condition (1) above * *is sym- metric in R and S. So if an R-S-bimodule M realizes an equivalence of module ca* *tegories, then the inverse equivalence is also realized by an S-R-bimodule N. Since the e* *quivalences are inverse to each other, M S N is then isomorphic to R as an R-bimodule and * *N R M is isomorphic to S as an S-bimodule. Moreover, M and N are then projective as * *right modules, and N is isomorphic to Hom S(M, S) as a bimodule. If R is Morita equivalent to S, then the opposite ring Rop is Morita equival* *ent to Sop. Indeed, suppose the R-S-bimodule M and the S-R-bimodule N satisfy M S N ~=R and N R M ~=S as bimodules. Since the category of right Rop-modules is isomorphic to the cate* *gory of left R-modules, we can view M as an Sop-Rop-bimodule and N as an Rop-Sop-bimodule, a* *nd then they provide the equivalence of categories between Mod-Ropand Mod-Sop. Sim* *ilarly, if R is Morita equivalent to S and R0is Morita equivalent to S0, then R S is Morit* *a equivalent to R0 S0. Here, and in the rest of the paper, undecoratd tensor products are t* *aken over Z. Invariants which are preserved under Morita equivalence include all concepts* * which can be defined from the category of modules without reference to the ring. Examples ar* *e the number of isomorphism classes of projective modules, of simple modules or of indecompo* *sable mod- ules, or the algebraic K-theory of the ring. The center Z(R) = {r 2 R | rs = sr* * for alls 2 R} is also Morita invariant, since the center of R is isomorphic to the endomorphi* *sm ring of the identity functor of Mod-R. A ring isomorphism Z(R) - ! End (IdMod-R) is obtained as follows: if r 2 Z(R) is a central element, then for every R-modu* *le M, multi- plication by r is R-linear. So the collection of R-homomorphisms {xr : M -! M}M* *2Mod-R is a natural transformation from the identity functor to itself. For more deta* *ils, see [Ba , II Prop 2.1] or [Lam , Remark 18.43]. In particular, if two commutative rings * *are Morita equivalent, then they are already isomorphic. There is a variation of the Morita theorem 2.2 relative to a commutative rin* *g k, with essentially the same proof. In this version R and S are k-algebras, condition (* *1) refers to a k-linear equivalence of module categories, condition (2) requires an isomorph* *ism of k- algebras and in part (3), M has to be a k-symmetric bimodule, i.e., the scalars* * from the ground ring k act in the same way from the left (through R) and from the right * *(through S). 4 STEFAN SCHWEDE We sketch the proof of Theorem 2.2 because it serves as the blueprint for an* *alogous results in the contexts of differential graded rings and ring spectra. Suppose (1) holds and let F : Mod- R - ! Mod- S be an equivalence of categories. The free R-module of rank one is a small proje* *ctive gen- erator of the category of R-modules. Being projective, small or a generator are* * categorical conditions, so they are preserved by an equivalence of categories. So the S-mod* *ule F R is a small projective generator of the category of S-modules. Since F is an equivale* *nce of cate- gories, it is in particular an additive fully faithful functor. So F restricts * *to an isomorphism of rings ~= F : R ~= End R(R) - ---! End S(F R) . Now assume condition (2) and let P be a small projective S-module which gene* *rates the category Mod-S. After choosing an isomorphism f : R ~=End S(P ), we can view P * *as an R-S-bimodule by setting r . x = f(r)(x) for r 2 R and x 2 P . We show that P sa* *tisfies the conditions of (3) by showing that the adjoint functors - R P and Hom S(P, -) a* *re actually inverse equivalences. The adjunction unit is the R-linear map X -! Hom S(P, X R P ) , x 7-! (y 7-! x y) . For X = R, the map adjunction unit coincides with the isomorphism f, so it is b* *ijective. Since P is small, source and target commute with sums, finite or infinite, so t* *he unit is bijective for every free R-module. Since P is projective over S, both sides of * *the adjunction unit are right exact as functors of X. Every R-module is the cokernel of a morp* *hism between free R-modules, so the adjunction unit is bijective in general. The adjunction counit is the S-linear evaluation map Hom S(P, Y ) R P -! Y , OE x 7-! OE(x) . For Y = P , the counit is an isomorphism since the right action of R on Hom S(P* *, P ) arises from the isomorphism R ~=Hom S(P, P ). Now the argument proceeds as for the adj* *unction unit: both sides are right exact and preserves sums, finite or infinite, in th* *e variable Y . Since P is a generator, every S-module is the cokernel of a morphism between di* *rect sums of copies of P , so the counit is bijective in general. Condition (1) is a special case of (3), so this finishes the proof of the Mo* *rita theorem. Example 2.3. The easiest example of a Morita equivalence involves matrix algebr* *as. Any free R-module of finite rank n 1 is a small projective generator for the cate* *gory of right R-modules. The endomorphism ring End R(Rn) ~= Mn(R) is the ring of n x n matrices with entries in R. So R and the matrix ring Mn(R)* * are Morita equivalent. The bimodules which induce the equivalences of module categories ca* *n both be taken to be Rn, but viewed as `row vectors' (or 1 x n matrices) and `column vec* *tors' (or n x 1 matrices) respectively. Matrix rings do not provide the most general kind of Morita equivalences, as* * the example below shows. However, every ring Morita equivalent to R is isomorphic to a ring* * of the form eMn(R)e where e 2 Mn(R) is a full idempotent in the n x n matrix ring, i.e., we* * have e2 = e and Mn(R)eMn(R) = Mn(R). Indeed, if P is a small projective generator for a rin* *g R, then MORITA THEORY 5 P is a summand of a free module of finite rank n, say. Thus P is isomorphic to * *the image of an idempotent n x n matrix e, and then EndR (P ) ~=eMn(R)e as rings. Example 2.4. The following example of a Morita equivalence which is not of matr* *ix algebra type was pointed out to me by M. Künzer and N. Strickland. Consider a commutati* *ve ring R and an invertible R-module Q. In other words, there exists another R-module * *Q0 and an isomorphism of R-modules Q R Q0~= R. Then tensor product with Q over R is a* * self- equivalence of the category of right R-modules (with quasi-inverse the tensor p* *roduct with Q0). This self-equivalence is not isomorphic to the identity functor unless R i* *s free of rank one. Because tensor product with an invertible module Q is an equivalence of cate* *gories, it follows that Q is a progenerator, with endomorphism ring isomorphic to R. Moreo* *ver, the `inverse' module Q0is isomorphic to the R-linear dual Q* = Hom R(Q, R). Now we * *consider the direct sum P = R Q, which is another small projective generator for Mod-R* *. Then R is Morita equivalent to the endomorphism ring of P , EndR (P ) = Hom R(R Q, R Q) . As an R-module, EndR (P ) is thus isomorphic to R Q Q* R. So if Q is not * *free, then End R(P ) is not free over its center, hence not a matrix algebra. For a specific example we consider the ring R = Z[u]=(u2 - 5u) . We set Q = (2, u) C R, the ideal generated by 2 and u. Then Q is not free as an* * R-module, but it is invertible because the evaluation map Hom R(Q, R) R Q - ! R , OE x 7-! OE(x) is an isomorphism. Note that the inclusion Q -! R becomes an isomorphism after * *inverting 2; so after inverting 2 the module P = R Q is free of rank 2 and hence the ring* * EndR (P )[1_2] is isomorphic to the ring of 2 x 2 matrices over R[1_2]. The implication (2)=)(1) in the Morita theorem 2.2 can be stated in a more g* *eneral form, and then it gives a characterization of module categories as the cocomple* *te abelian categories with a small projective generator. Theorem 2.5. Let A be an abelian category with infinite sums and a small projec* *tive gen- erator P . Then the functor A(P, -) : A -! Mod-EndA (P ) is an equivalence of categories. Proof. We give the same proof as in Bass' book [Ba , II Thm. 1.3]. Let us say t* *hat an object X of A is good if the map A(P, -) : A(X, Y ) - ! Hom EndA(P)(A(P, X), A(P, Y )) (2.6) is bijective for every object Y of A. We note that: o The generator P is good since A(P, P ) is the free EndA (P )-module of r* *ank one. o The class of good objects is closed under sums, finite or infinite: sinc* *e P is small, A(P, -) preserves sums and both sides of the map (2.6)take direct sums i* *n X to direct products. 6 STEFAN SCHWEDE o If f : X -! X0 is a morphism between good objects in A, then the cokerne* *l of f is also good. This uses that P is projective and so that A(P, -) is an e* *xact functor and both sides of the map (2.6)are right exact in X. Since P is a generator, every object can be written as the cokernel of a morphi* *sm between sums of copies of P . So every object of A is good, which precisely means that* * the hom functor A(P, -) is full and faithful. It remains to check that every EndA (P )-module is isomorphic to a module in* * the image of the functor A(P, -). The free EndA (P )-module of rank one is the image of P . * *Since A(P, -) commutes with sums, every free module is in the image, up to isomorphism. Final* *ly, every End A(P )-module X has a presentation, so it occurs in an exact sequence of End* * A(P )- modules M g M End A(P ) -! EndA (P ) - ! X -! 0 . I J SinceLA(P, -)Lis full, the homomorphism g is isomorphic to A(P, f) for some mor* *phism f : IP -! JP in A. Since the functor A(P, -) is exact, X is the image of th* *e cokernel of f. Thus A(P, -) is an equivalence of categories. Theorem 2.5 can be applied to the abelian category of right modules over a r* *ing S; then we conclude that for every small projective generator P of Mod-S the functor Hom S(P, -) : Mod-S -! Mod- EndS (P ) is an equivalence of categories. This shows again that condition (2) in the Mor* *ita theorem 2.2 implies condition (1). 3.Morita theory in derived categories Morita theory for derived categories is about the question: When are the derived categories D(R) and D(S) of two rings R and S equivale* *nt ? Here the derived category D(R) is defined from (Z-graded and unbounded) chai* *n com- plexes of right R-modules by formally inverting the quasi-isomorphisms, i.e., t* *he chain maps which induce isomorphisms of homology groups. Of course, if R and S are Morita * *equivalent, then they are also derived equivalent. But it turns out that derived equivalen* *ces happen under more general circumstances. Rickard [Ric89a, Ric91] developed a Morita theory for derived categories bas* *ed on the notion of a tilting complex. Rickard's theorem did not come out of the blue, a* *nd he had built on previous work of several other people on tilting modules. Section 3.1 * *of the book by König and Zimmermann [KZ ] gives a summary of the history in this area; this* * book also contains many more details, examples and references on the use of derived * *categories in representation theory. We follow Keller's approach from [Kel94a], based on the (differential graded* *) endomor- phism ring of a tilting complex. A similar approach to and more applications o* *f Morita theory in derived categories can be found in the paper [DG02 ] by Dwyer and Gre* *enlees. 3.1. The derived category. In this section, R is any ring. All chain complexes* * are Z- graded and homological, i.e., the differential decreases the degree by 1. Definition 3.1. A chain complex C of R-modules is cofibrant if there exists an * *exhaustive increasing filtration by subcomplexes 0 = C0 C1 . . . Cn . . . MORITA THEORY 7 such that each subquotient Cn=Cn-1 consists of projective modules and has trivi* *al differ- ential. The (unbounded) derived category D(R) of the ring R has as objects the * *cofibrant complexes of R-modules and as morphisms the chain homotopy classes of chain map* *s. Our definition of the derived category is different from the usual one. The * *more traditional way is to start with the homotopy category of all complexes, not necessarily co* *fibrant; then one uses a calculus of fractions to formally invert the class of quasi-isomorph* *isms. These two ways of constructing D(R) lead to equivalent categories. The shift functor in D(R) is given by shifting a complex, i.e., (A[1])n = An-1 with differential d : (A[1])n = An-1 -! An-2 = (A[1])n-1 the negative of the di* *fferential of the original complex A. The mapping cone C' of a chain map ' : A -! B is define* *d by (C')n = Bn An-1 , d(x, y) = (dx + '(y), -dy) . (3.2) The mapping cone comes with an inclusion i : B -! C' and a projection p : C' -!* * A[1] which induce an isomorphism (C')=B ~=A[1]; if A and B are cofibrant, then so ar* *e the shift A[1] and the mapping cone. Remark 3.3. The following remarks are meant to give a better feeling for the no* *tion of `cofibrant complex' and the unbounded derived category. (i)The concept of a `cofibrant complex' is closely related to, but stronger* * than, a com- plex of projective modules. Indeed, if C is a cofibrant complex, then in* * every dimen- sion k 2 Z, each subquotient Cnk=Cn-1kis projective. So Cnksplits as the* * sum of the subquotients, Mn Cnk ~= Cik=Ci-1k. i=1 Since Ck is the union of the submodules Cnk, the module Ck also splits a* *s the sum of the countably many subquotients Cik=Ci-1k. In particular, Ck is a sum of* * projective modules. Hence every cofibrant complex is dimensionwise projective. If* * C is a complex of projective modules which is bounded below, then it is also co* *fibrant. For example, if C is trivial in negative dimensions, then as the filtration * *we can simply take the (stupid) truncations of C, i.e., ( Cn for n < i Cin= 0 for n i. So for bounded below complexes, `cofibrant' is equivalent to `dimensionw* *ise projec- tive'. On the other hand, not every complex which is dimensionwise projective* * is also cofibrant. The standard example is the complex C in which Ck is the fre* *e Z=4- module of rank one for all k 2 Z, and where every differential d : Ck -!* * Ck-1 is multiplication by 2. (ii)Every quasi-isomorphism between cofibrant complexes is a chain homotopy * *equiv- alence. For every complex of R-modules X, there is a cofibrant complex * *Xc and a quasi-isomorphism Xc -! X; together these two facts essentially prove * *that the derived category D(R) enjoys the universal property of the localization * *of the cat- egory of chain complexes of R-modules with the class of quasi-isomorphis* *ms in- verted. These properties are very analogous to the properties that CW-co* *mplexes 8 STEFAN SCHWEDE have among all topological spaces: every weak equivalences between CW-co* *mplexes is a homotopy equivalence and every space admits a CW-approximation. Thi* *s anal- ogy is made precise in [KM , Part III]. (iii)The concept of a cofibrant chain complex is closely related to that of * *a K-projective complex as defined by Spaltenstein [Spa88, Sec. 1.1] (who attributes thi* *s notion to J. Bernstein; Keller [KZ , 8.1.1] calls this homotopically projective). * *A chain complex is K-projective if every chain map into an acyclic complex (i.e., a comp* *lex with trivial homology) is chain null-homotopic. Every cofibrant complex is K-projective. Conversely, every K-projectiv* *e complex X is chain homotopy equivalent to a cofibrant complex. Indeed, we can c* *hoose a cofibrant replacement, i.e., a cofibrant complex Xc and a quasi-isomor* *phism q : Xc -! X; the mapping cone Cq is then acyclic. We have a short exact sequ* *ence of chain homotopy classes of chain maps in Cq, [Xc[1], Cq] - ! [Cq, Cq] - ! [X, Cq] ; both X and Xc are K-projective, so the left and right groups are trivial* *. Thus the identity map of Cq is null-homotopic and so the mapping cone of q is* * chain contractible. Thus the map q : Xc -! X is a chain homotopy equivalence. (iv)We use the term `cofibrant' complex because they are the cofibrant objec* *ts in the projective model category structure on chain complexes of R-modules [Hov* *99 , 2.3.11]. In this model structure, the weak equivalences are the quasi-isomorphism* *s, the fi- brations are the surjections and the cofibrations are the injections who* *se cokernel is cofibrant in the sense of Definition 3.1. In particular, every chain com* *plex is fibrant in the projective model structure. Since every object is fibrant and because the fibrations (= surjection* *s) and weak equivalences (= quasi-isomorphisms) already have well-established names,* * it is un- necessary, and overly complicated, to use the language of model categori* *es in order to work with the derived category of a ring. (v)There is a `dual' approach to the derived category D(R) as the homotopy * *category of `fibrant' complexes (attention: these are not the fibrant objects in* * the projec- tive model structure - there every complex is fibrant). This uses the n* *otion of a `K-injective' [Spa88, Sec. 1.1] or `homotopically injective' [KZ , 8.1.1* *] complex, or the injective model structure on the category of complexes of R-modules * *[Hov99 , 2.3.13]. It is often useful to have both descriptions available. Given a* *rbitrary chain complexes C and D, we choose a cofibrant/K-projective resolution Cc -~! * *C and a fibrant/K-injective resolution D -~! Df. Then the maps induce isomorph* *isms of chain homotopy classes of chain maps ~= c f ~= f [Cc, D] - -! [C , D ] -- [C, D ] . There is an additive functor [0] : Mod- R - ! D(R) which is a fully faithful embedding onto the full subcategory of the derived ca* *tegory con- sisting of the complexes whose homology is concentrated in dimension zero. So * *we can think of the R-modules as sitting inside the derived category D(R). Had we def* *ined the derived category from the category of all complexes of R-modules by formally in* *verting the quasi-isomorphisms, then we could define the complex M[0] by putting the R-modu* *le M in MORITA THEORY 9 dimension 0, and taking trivial chain modules everywhere else. With our present* * definition of D(R) we let M[0] be a choice of resolution Po of M by projective R-modules. * * Such a resolution is unique up to chain homotopy equivalence and it is cofibrant when * *viewed as a chain complex (by Remark 3.3 (i) above), Moreover, every R-linear map M -! N is* * cov- ered by a unique chain homotopy class between the chosen projective resolutions* *. In other words, we really get a functor from R-modules to the derived category D(R), tog* *ether with a natural isomorphism H0(M[0]) ~=M. The usual definition of Ext-groups involves a choice of projective resolutio* *n Po of the source module M, and then ExtnR(M, N) can be defined as the chain homotopy clas* *ses of chain maps from the resolution Po to N, shifted up into dimension n. We get the* * same result if N is also replaced by a projective resolution; this says that Ext-groups can* * be obtained from the derived category via ExtnR(M, N) ~= D(R)(M[0], N[n]) . (3.4) The derived category of a ring has more structure. The category D(R) is addi* *tive since the homotopy relation for chain maps is additive. But D(R) is no longer an abelian * *category such as the category of chain complexes. Indeed, notions such as `monomorphism', `ep* *imorphisms, `kernels' for chain maps do not interact well with the passage to chain homotop* *y classes. The distinguished triangles in D(R) are what is left of the abelian structure o* *n the category of chain complexes, and D(R) is an example of a triangulated category. The distinguished triangles are the diagrams which are isomorphic in D(R) to* * a mapping cone triangle. More precisely, a diagram in D(R) of the form X f-!Y g-!Z h-!X[1] (3.5) is called a distinguished triangle if and only if there exists a chain map ' : * *A -! B between cofibrant complexes and isomorphisms '1 : A ~=X, '2 : B ~=Y and '3 : C' ~=Z in * *D(R) such that the diagram ' i p A _____//B____//_C'____//A[1] | | '1|| '2|| |'3| |'1[1]| fflffl|fflffl| fflffl| fflffl| X _f__//_Y_g__//_Zh__//_X[1] commutes in D(R). We do not want to reproduce the complete definition of a triangulated catego* *ry here; the data of a triangulated category consists of (i)an additive category T , (ii)a self-equivalence [1] : T - ! T called the shift functor and (iii)a class of distinguished triangles, i.e., a collection of diagrams in T* * of the form (3.5). This data is subject to several axioms which can be found for example in [Ver96* *], [Wei94 , Sec. 10.2] [Nee01] or [KZ , 2.3]. Distinguished triangles are the source of many long exact sequences that com* *e up in nature. Indeed, the axioms which we have suppressed imply in particular that f* *or every distinguished triangle of the form (3.5)and every object W of T the sequence o* *f abelian morphism groups T (W, X) -f*!T (W, Y ) -g*!T (W, Z) -h*! T (W, X[1]) 10 STEFAN SCHWEDE is exact. One of the axioms also says that one can `rotate' triangles, i.e., a * *sequence (3.5)is a distinguished triangle if and only if the sequence -f[1] Y -g-!Z -h-!X[1] - --! Y [1] is a distinguished triangle. So if we keep rotating a distinguished triangle in* * both directions and take morphisms from a fixed object W , we end up with a long exact sequence* * of abelian groups . .T.(W, X) -f*-!T (W, Y ) -g*-!T (W, Z) -h*--!T (W, X[1]) (3.6) -f[1]* -g[1]* -h[1]* ----! T (W, Y [1]) ----! T (W, Z[1]) ----! T (W, X[2]) . * *. . The axioms of a triangulated category also guarantee a similar long exact seque* *nce when taking morphism from a triangle (and its rotations) into a fixed object W . In the derived category D(R), the long exact sequence (3.6)becomes something* * more familiar when we take W = R[0], the free R-module of rank one, viewed as a com* *plex concentrated in dimension zero. For every chain complex C, cofibrant or not, t* *he chain homotopy classes of morphisms from R to C are naturally isomorphic to the homol* *ogy module H0C; so the long exact sequence (3.6)specializes to the long exact seque* *nce of homology modules H0(') ffi . .-.! H0(A) -----! H0(B) - ! H0(C') --! H-1(A) - ! . . .. Small generators. We will often require infinite direct sums in a triangulat* *ed category. The unbounded derived category D(R) has direct sums, finite and infinite. Inde* *ed, the direct sum of any number of cofibrant complexes is again cofibrant (take the di* *rect sum of the filtrations which are required in Definition 3.1), and this also represents* * the direct sum in D(R). This is one point where it is important to allow unbounded complexes. * *There are variants of the derived category which start with complexes which are bounded o* *r bounded below. One also gets triangulated categories in much the same way as for D(R),* * but for example the countable family {R[-n]}n 0 has no direct sum in the bounded or bou* *nded below derived categories. In the Morita equivalence questions, a suitably defined notion of `small gen* *erator' pops up regularly. The following concepts for triangulated categories are analogous * *to the ones for abelian categories in Definition 2.1. Definition 3.7. Let T be a triangulated category with infinite coproducts. An o* *bject M of T is small if the hom functor T (M, -) preserves sums; M is a generator if * *there is no proper full triangulated subcategory of T (with shift and triangles induced fro* *m T ) which contains M and is closed under infinite sums. As in abelian categories, the hom functor T (M, -) automatically preserves f* *inite sums. What we call `small' is sometimes called compact or finite in the literature on* * triangulated categories. A triangulated category with infinite coproducts and a set of small* * generators is often called compactly generated. The class of small objects in any triangulated category is closed under shif* *ting in either direction, taking finite sums and taking direct summands. Moreover, if two of * *the three objects in a distinguished triangle are small, then so is the third one (one ha* *s to exploit that the morphisms from a distinguished triangle into a fixed object give rise * *to a long exact sequence). MORITA THEORY 11 There is a convenient criterion for when a small object M generates a triang* *ulated category T with infinite coproducts: M generates T in the sense of Definition 3.7 if a* *nd only if it `detects objects', i.e., an object X of T is trivial if and only if there are * *no graded maps from M to X, i.e. T (M[n], X) = 0 for all n 2 Z. For the equivalence of the two* * conditions, see for example [SS03, Lemma 2.2.1]. The complex R[0] consisting of the free module of rank one concentrated in d* *imension 0 is a small generator for the derived category D(R). Indeed, morphisms in D(R) o* *ut of the complex R[0] represent homology, i.e., there is a natural isomorphism D(R)(R[n], C) ~= Hn(C) for every cofibrant complex C. Since homology commutes with infinite sums, the * *complex R[0] is small in D(R). Moreover, if all the morphism groups D(R)(R[n], C) are * *trivial as n ranges over the integers, then the complex C is acyclic, hence contractible, * *and so it is trivial in the derived category D(R). In other words, mapping out of shifted co* *pies of the complex R[0] detects whether an object in D(R) is trivial or not, so R[0] is al* *so a generator, by the previous criterion. There is a nice characterization of the small objects in the derived categor* *y of a ring. Every bounded complex of finitely generated projective modules is built from su* *mmands of the small object R[0] by shifts and extensions in triangles. Since the class of* * small objects is closed under these operations, a bounded complex of finitely generated projecti* *ve modules is small. Conversely, these are the only small objects, up to isomorphism in D(* *R): Theorem 3.8. Let R be a ring. A complex of R-modules is small in the derived ca* *tegory D(R) if and only if it is quasi-isomorphic to a bounded complex of finitely gen* *erated projective R-modules. The proof that every small object in D(R) is quasi-isomorphic to a bounded c* *omplex of finitely generated projective modules is more involved. It is a special case of* * a result about triangulated categories T with a set of small generators. Neeman [Nee92] showed* * that every small object in T is a direct summand of an iterated extension of finitely man* *y shifted generators. The proof can also be found in [Kel94a, 5.3]. There are non-trivial triangulated categories in which only the zero objects* * are small, see for example [Kel94b] or [HS99 , Cor. B.13]. If a triangulated category has a se* *t of generators, then the coproduct of all of them is a single generator. However, an infinite c* *oproduct of non-trivial small objects is not small. So the property of having a single smal* *l generator is something special. In fact we see in Theorem 4.16 below that this condition cha* *racterizes the module categories over ring spectra among the stable model categories. A tr* *iangulated category need not have a set of generators whatsoever (one could consider all o* *bjects, but in general these form a proper class), for example K(Z), the homotopy category * *of chain complexes of abelian groups, is not generated by a set [Nee01, E.3]. Equivalences of triangulated categories. A functor between triangulated cate* *gories is called exact if it commutes with shift and preserves distinguished triangles* *. More precisely, F :S -! T is exact it is is equipped with a natural isomorphism 'X : F (X[1]) * *~=F (X)[1] such that for every distinguished triangle (3.5)the sequence F(f) F(g) 'X OF(h) F (X) ---! F (Y ) ---! F (Z) - ----! F (X)[1] 12 STEFAN SCHWEDE is again a distinguished triangle. An exact functor is automatically additive. * *An equivalence of triangulated categories is an equivalence of categories which is exact and w* *hose inverse functor is also exact. Exact equivalences between derived categories preserve all concepts which ca* *n be defined from D(R) using only the triangulated structure. One such invariant is the Grot* *hendieck group K0(R), defined as the free abelian group generated by the isomorphism cla* *sses of finitely generated projective R-modules, modulo the relation [P ] + [Q] = [P Q] . For any compactly generated triangulated category T , the Grothendieck group* * K0(T ) is defined as the free abelian group generated by the isomorphism classes of small* * objects in T , modulo the relation [X] + [Z] = [Y ] for every distinguished triangle X -! Y -! Z -! X[1] involving small objects X, Y and Z (the morphisms in the triangle do not affect* * the relation). The split triangle (1,0) (01) 0 X ---! X Y --! Y --! X[1] is always distinguished, so the relation [X Y ] = [X] + [Y ] holds in K0(T ).* * So for every ring R, the assignment K0(R) - ! K0(D(R)) , [P ] 7-! [P [0]] (3.9) defines a group homomorphism. This is in fact an isomorphism, see [Gr77 , Sec.* * 7]. The inverse takes the class in K0(D(R)) of a bounded complex C of finitely generate* *d projective modules to its `Euler characteristic', X (-1)n[Cn] 2 K0(R) . n2Z It is much less obvious that constructions such as the center of a ring, Hoc* *hschild and cyclic homology and the higher Quillen K-groups are also invariants of the derived cat* *egory. In contrast to the Grothendieck group K0, there is no construction which produces * *these groups from the triangulated structure of D(R) only. The proof that two derived equiva* *lent rings share these invariants uses the `tilting theory' which we outline in the next s* *ection. More precisely, if R and S are derived equivalent flat algebras over some commutativ* *e ground ring, then there exists a two-sided tilting complex, i.e., a chain complex C of R-S-b* *imodules such that the functor - R C induces a (possibly different) derived equivalence [Ric* *91]. Tensor product with the bimodule complex C then induces an equivalence of K-theory spa* *ces by the work of Thomason-Trobaugh [TT , Thm. 1.9.8]. Without the flatness assu* *mption, the Waldhausen categories of small, cofibrant chain complexes can be related th* *rough an intermediate category of differential graded modules, to still obtain an equiva* *lence of K- theory spaces; for more details we refer to [DuSh ]. Similarly, Hochschild hom* *ology and cohomology (see [Ric91, Prop. 2.5], or, including the Gerstenhaber bracket, se* *e [Kel03]) and cyclic homology (see [Kel96, Kel98]) are isomorphic for derived equivalent * *rings which are flat algebras over some commutative ground ring. The invariance of the cen* *ter under derived equivalence is established in [Ric89a, Prop. 9.2] or [KZ , Prop. 6.3.2]. MORITA THEORY 13 There is a certain general argument which we will use several times to verif* *y that certain triangulated functors are equivalences, so we state it as a separate propositio* *n. This Propo- sition 3.10 is a version of `Beilinson's Lemma' [Bei78] and is typically applie* *d when F is the total derived functor of a suitable left adjoint. In the following proposition,* * it is crucial that the functor F be defined and exact on the entire triangulated category S. It is* * easy to find non-equivalent triangulated categories S and T with infinite sums and small gen* *erators P and Q respectively such that S(P, P )* ~= T (Q, Q)* as graded rings. For example one can take a differential graded ring A with a n* *on-trivial triple Massey product and consider derived categories S = D(A) and T = D(H*A) (* *where the cohomology ring of A is given the trivial differential). Proposition 3.10. Let F :S -! T be an exact functor between triangulated catego* *ries with infinite sums. Suppose that F preserves infinite sums and S has a small generat* *or P such that (i)F P is a small generator of T and (ii)for all integers n, the map F : S(P [n], P ) -! T (F P [n], F P ) is bijective Then F is an equivalence of categories. Proof. We consider the full subcategory of S consisting of those Y for which th* *e map F : S(P [n], Y ) - ! T (F P [n], F Y ) (3.11) is bijective for all n 2 Z. By assumption this subcategory contains P . Since* * F is exact, the subcategory is closed under extensions. Since P and F P are small and F pr* *eserves coproducts, this subcategory is also closed under coproducts. Since P generates* * S, the map (3.11)is thus bijective for arbitrary Y . Similarly for arbitrary but fixed Y the full subcategory of S consisting of* * those X for which the map F : S(X, Y ) -! T (F X, F Y ) is bijective is closed under extens* *ions and coproducts. By the first part, it also contains P , so this subcategory is all * *of S. In other words, F is full and faithful. Now we consider the full subcategory of T of objects which are isomorphic to* * an object in the image of F . This subcategory contains the generator F P and it is closed u* *nder shifts and coproducts since these are preserved by F . We claim that this subcategory is * *also closed under extensions. Since F P generates T , this shows that F is essentially sur* *jective and hence an equivalence. To prove the last claim we consider a distinguished triangle X -f-!Y -! Z -! X[1] . Since the subcategory under consideration is closed under isomorphism and shift* * in either direction we can assume that X = F (X0) and Y = F (Y 0) are objects in the imag* *e of F . Since F is full there exists a map f0 : X0 -! Y 0satisfying F (f0) = f. We can * *then choose a mapping cone for the map f0 and a compatible map from Z to F (Cone (f0)) whic* *h is necessarily an isomorphism. 14 STEFAN SCHWEDE 3.2. Derived equivalences after Rickard and Keller. In this section we state an* *d prove Rickard's öM rita theory for derived categories". Rickard shows in [Ric89a, Thm* *. 6.4] that the existence of a tilting complex is necessary and sufficient for an equivalen* *ce between the unbounded derived categories of two rings. A tilting complex is a special small* * generator of the derived category, see Definition (3.12)below. The idea to use differential * *graded algebras in the proof is due to Keller [Kel94a], and we closely follow his approach. The notion of a tilting complex comes up naturally when we examine the prope* *rties of the preferred generator R[0] of the derived category D(R). First of all, the fr* *ee R-module of rank one, considered as a complex concentrated in dimension zero, is a small* * generator of the derived category D(R). Since R is a free module, it has no self-extensio* *ns. Because Ext groups can be identified with morphisms in the derived category (see (3.4))* *, this means that the graded self-maps of the complex R[0] are concentrated in dimension zer* *o: D(R)(R[n], R) = 0 for n 6= 0. A tilting complex is any complex which also has these properties. Hence the def* *inition is made so that the image of R[0] under an equivalence of triangulated categories * *is a tilting complex. Definition 3.12. A tilting complex for a ring R is a bounded complex T of finit* *ely generated projective R-modules which generates the derived category D(R) and whose graded* * ring of self maps D(R) (T, T )* is concentrated in dimension zero. Special kinds of tilting complexes are the tilting modules; we give examples* * of tilting mod- ules and tilting complexes in Section 3.3. The following theorem is due to Rick* *ard [Ric89a, Thm. 6.4]. Theorem 3.13. For two rings R and S the following conditions are equivalent. (1) The unbounded derived categories of R and S are equivalent as triangulat* *ed categories. (2) There is a tilting complex T in D(S) whose endomorphism ring D(S)(T, T )* * is iso- morphic to R. Moreover, conditions (1) and (2) are implied by the condition (3) There exists a chain complex of R-S-bimodules M such that the derived te* *nsor product functor - LRM : D(R) -! D(S) is an equivalence of categories. If R or S is flat as an abelian group, then all three conditions are equival* *ent. Instead of using the unbounded derived category, one can replace condition (* *1) by an equivalence between the full subcategories of homologically bounded below or sm* *all objects inside the derived categories, see for example [Ric89a, Thm. 6.4]. There is a v* *ersion relative to a commutative ring k. Then R and S are k-algebras, conditions (1) and (3) th* *en refer to k- linear equivalences of derived categories, condition (2) requires an isomorphis* *m of k-algebras and in the addendum, one of R or S has to be flat as a k-module. Remark 3.14. A derived equivalence F from D(R) to D(S) which is not already a M* *orita equivalence maps the R-modules inside D(R) (i.e., complexes with homology conce* *ntrated in dimension 0) "transversely" to the S-modules inside D(S); more precisely, for a* *n R-module M, the complex F (M[0]) can have non-trivial homology in several, or even in in* *finitely many MORITA THEORY 15 dimensions; we give an example in 3.26 below. However, the homology of F (M[0])* * is always bounded below. A related point is that we can not recover the module category Mod-R from D(* *R), viewed as an abstract triangulated category. This is because we cannot make sense of ö* *c mplexes with homology concentrated in dimension 0ü nless we specify a homology functor* * like H0, or we single out the class of complexes with homology in non-negative dimension* *s. This sort of extra structure is called a t-structures [BBD , 1.3] on a triangulated* * category. Every t-structure has a heart, an abelian category which plays the role of complexes * *in D(R) with homology concentrated in dimension zero. The most involved part of the tilting theorem is the implication (2)=)(1), i* *.e., showing that a tilting complex gives rise to a derived equivalence. The proof we give i* *s due to Keller; in the original paper [Kel94a], his setup is more general (he works in differen* *tial graded categories in order to allow `many generator' versions). In the special case o* *f interest for us, the exposition simplifies somewhat [KZ , Ch. 8]. Given a tilting complex T * *in D(S), the comparison between the derived categories of R and S passes through the derived* * category of a certain differential graded ring (generalizing the derived category of a a* *n ordinary ring), namely the endomorphism DG ring End S(T ) of the tilting complex T (generalizin* *g the endomorphism ring). So we start by introducing these new characters. Definition 3.15. A differential graded ring is a Z-graded ring A together with * *a differential d of degree -1 which satisfies the Leibniz rule d(a . b) = d(a) . b + (-1)|a|a . d(b) (3.16) for all homogeneous elements a, b 2 A. A differential graded right module (or D* *G module for short) over a differential graded ring A consists of a graded right A-module to* *gether with a differential d of degree -1 which satisfies the Leibniz rule (3.16), but wher* *e now a is a homogeneous element of the module and b is a homogeneous element of A. A homomo* *rphism of DG modules is a homomorphism of graded A-modules which is also a chain map. * *A chain homotopy between homomorphisms of DG modules is a homomorphism of graded A-modu* *les of degree 1 which is also a chain homotopy. A differential graded A-module M is cofibrant if there exists an exhaustive * *increasing filtration by sub DG modules 0 = M0 M1 . . . Mn . . . such that each subquotient Mn=Mn-1 is a direct summand of a direct sum of shift* *ed copies of A. The derived category D(A) of the differential graded ring A has as object* *s the cofi- brant DG modules over A and as morphisms the chain homotopy classes of DG module homomorphisms. Up to chain homotopy equivalence, the cofibrant DG modules are the ones whic* *h have Keller's `property (P)' in [Kel94a, 3.1]. A cofibrant differential graded modul* *e is sometimes called `semi-free' or a `cell module' [KM , Part III] (up to direct summands). Remark 3.17. We need some facts about differential graded rings and modules whi* *ch are not very difficult to prove, but which we do not want to discuss in detail. (i)Several of the remarks from 3.3 carry over from rings to DG rings. A co* *fibrant DG A-module is projective as a graded A-module, ignoring the differentia* *l. Every quasi-isomorphism between cofibrant DG modules is a chain homotopy equiv* *alence, 16 STEFAN SCHWEDE and every DG module can be approximated up to quasi-isomorphism by a cof* *ibrant one. A DG module is called homotopically projective if every homomorphis* *m into an acyclic DG module is null-homotopic. Then a DG module is homotopically p* *rojective if and only if it is chain homotopy equivalent, as a DG module, to a cof* *ibrant DG module. (ii)The derived category D(A) of a differential graded ring A is naturally a* * triangulated category. The shift functor is again given by reindexing a DG module, an* *d distin- guished triangles arise from mapping cones as for the derived category o* *f a ring. The only thing to note is that for a homomorphism f : M -! N of DG modules o* *ver A, the mapping cone becomes a graded A-module as the direct sum N M[1]* *, and this A-action satisfies the Leibniz rule with respect to the mapping con* *e differential (3.2). (iii)Suppose that f : A -! B is a homomorphism of differential graded rings,* * i.e., f is a multiplicative chain homomorphism. Then extension of scalars M 7! M A* * B is exact on cofibrant differential graded modules (since the underlying gra* *ded modules over the graded ring underlying A are projective), it takes cofibrant mo* *dules to cofibrant modules, and it preserves the chain homotopy relation. So ext* *ension of scalars induces an exact functor on the level of derived categories - LAB D(A) ____//_D(B)oo_, (3.18) f* called the left derived functor. This derived functor has an exact righ* *t adjoint f* induced by restriction of scalars along f. This is not completely obviou* *s with our definition of the derived category, since a cofibrant differential grade* *d B-module is usually not cofibrant when viewed as a DG module over A via f. If f : A -! B is a quasi-isomorphism of differential graded rings, the* *n the derived functors of restriction and extension of scalars (3.18)are inverse equiv* *alences of triangulated categories. (iv)Suppose A is a differential graded ring whose homology is concentrated i* *n dimen- sion zero. Then A is quasi-isomorphic, as a differential graded ring, t* *o the zeroth homology ring H0A. Indeed, a chain of two quasi-isomorphisms is given by A --inclusion----A+ -projection------!H0(A) . Here A+ is the differential graded sub-ring of A given by 8 >< An for n > 0 (A+ )n = Ker (d : A0 -! A-1) for n = 0, and >: 0 for n < 0. Since the homology of A is trivial in negative dimensions, the inclusion* * A+ -! A is a quasi-isomorphism. Since A+ is trivial in negative dimensions, the* * projection A+ -! H0(A+ ) is a homomorphism of differential graded rings, where the * *target is concentrated in dimension zero. This projection is also a quasi-isomorp* *hism since the homology of A, and hence that of A+ , is trivial in positive dimensi* *ons. Homomorphism complexes. Let A be a DG ring and let M and N be DG modules over A, not necessarily cofibrant. We defined the homomorphism complex Hom A(M,* * N) as MORITA THEORY 17 follows. In dimension n 2 Z, the chain group Hom A(M, N)n is the group of grade* *d A-module homomorphisms of degree n, i.e., Hom A(M, N)n = Hom A(M[n], N) . The differentials of M and N do not play any role in the definition of the chai* *n groups, but they enter in the formula for the differential which makes Hom A(M, N) into* * a chain complex. This differential d: Hom A(M, N)n -! Hom A(M, N)n-1 is defined by d(f) = dN O f - (-1)nf O dM . (3.19) Here f is a graded A-module map of degree n and the composites dN O f and f O d* *M are then graded A-module maps of degree n - 1. With this definition, the 0-cycles in Hom A(M, N) are those graded A-module * *maps f which satisfy dN O f - f O dM = 0, so they are precisely the DG homomorphisms * *from M to N. Moreover, if f, g : M -! N are two graded A-module maps, then the difference* * f - g is a coboundary in the complex Hom A(M, N) if and only if f is chain homotopic to * *g. So we have established a natural isomorphism H0(Hom A (M, N)) ~= [M, N] between the zeroth homology of the complex Hom A(M, N) and the chain homotopy c* *lasses of DG A-homomorphisms from M to N. Now suppose that we have a third DG module L. Then the composition of grade* *d A- module maps gives a bilinear pairing between the homomorphism complexes O : Hom A(N, L)m x Hom A (M, N)n - ! Hom A(M, L)m+n . Moreover, composition and the differential (3.19)satisfy the Leibniz rule, i.e.* *, for graded A-module maps f : M -! N of degree n and g : N -! L of degree m we have d(g O f) = dg O f + (-1)m g O df as graded maps from M to L. The following consequences are crucial for the remaining step in the tilting* * theorem: o for every DG A-module M, the endomorphism complex End A(M) = Hom A (M, M) is a differential graded ring under composition and M is a differential graded End* * A(M)-A- bimodule; o for every DG A-module N, the homomorphism complex Hom A(M, N) is a differenti* *al graded module over End A(M) under composition. Moreover the functor Hom A (M, * *-) : Mod- A -! Mod-End A(M) is right adjoint to tensoring with the EndA (M)-A-bimodu* *le M; o if M is cofibrant, then the functor Hom A(M, -) is exact and takes quasi-isom* *orphisms of DG A-modules to quasi-isomorphisms. Moreover, its left adjoint - EndA(M)M pres* *erves cofibrant objects and chain homotopies. So there exists a derived functor on t* *he level of derived categories - LEndA(M)M : D(End A(M)) - ! D(A) , an exact functor which preserves infinite sums. The following theorem is a special case of Lemma 6.1 in [Kel94a]. Theorem 3.20. Let A be a DG ring and M a cofibrant A-module which is a small ge* *nerator for the derived category D(A). Then the derived functor - LEndA(M)M : D(End A(M)) -! D(A) (3.21) is an equivalence of triangulated categories. 18 STEFAN SCHWEDE Proof. The total left derived functor (3.21)is an exact functor between triangu* *lated cate- gories which preserves infinite sums. Moreover, it takes the free End A(M)-modu* *le of rank one _ which is a small generator for the derived category of End A(M) _ to the * *small generator M for D(A). The induced map of graded endomorphism rings - LEndA(M)M : D(End A(M))(End A(M), EndA(M))* - ! D(A)(M, M)* is an isomorphism (both sides are isomorphic to the homology ring of EndA (M)).* * So Propo- sition 3.10 shows that this derived functor is an equivalence of triangulated c* *ategories. After all these preparations we can give the Proof of the tilting theorem 3.13.Clearly, condition (3) implies condition (1).* * Now we as- sume condition (1) and we choose an exact equivalence F from the derived catego* *ry D(R) to D(S). The defining properties of a tilting complex are preserved under exact* * equivalences of triangulated categories. Since R[0], the free R-module of rank one, concentr* *ated in di- mension zero, is a tilting complex for the ring R, its image T = F (R[0]) is a * *tilting complex for S. Moreover, F restricts to a ring isomorphism ~= F : R ~= D(R)(R[0], R[0]) - ---! D(S)(T, T ) . Hence condition (2) holds. For the implication (2)=)(1) we are given a tilting complex T in D(S) and an* * isomorphism of rings D(S)(T, T ) ~= R. The complex T is naturally a differential graded En* *d S(T )-S- bimodule, and by Theorem 3.20, the derived functor - LEndS(T)T : D(End S(T )) - ! D(S) is an equivalence of triangulated categories. The isomorphism of graded rings H*(End S(T ))~= D(S)(T, T )* and the defining property of a tilting complex show that the homology of EndS(T* * ) is concen- trated in dimension zero. So there is a chain of two quasi-isomorphisms between* * End S(T ) and the ring H0(End S(T ))~=D(S)(T, T ) ~=R. Restriction and extension of scala* *rs along these quasi-isomorphisms gives a chain of equivalences between the derived cate* *gories of the differential graded ring EndS (T ) and the derived category of the ordinary rin* *g D(S)(T, T ). Putting all of this together we end up with a chain of three equivalences of tr* *iangulated categories: D(R) ~= D(End S(T )+ ) ~= D(End S(T )) ~= D(S) . It remains to prove the implication (2)=)(3), assuming that R or S is flat. * *Let T be a tilting complex in D(S) and f : D(S)(T, T ) -! R an isomorphism of rings. The h* *omology of End S(T ) is isomorphic to the graded self maps of T in D(S), so it is conce* *ntrated in dimension 0. So the inclusion of the DG sub-ring EndS(T )+ into the endomorphis* *m DG ring End S(T ) induces an isomorphism on homology, compare Remark 3.17 (iv). Since E* *ndS(T )+ is trivial in negative dimensions, there is a unique morphisms of DG rings EndS* * (T )+ -! R which realizes the isomorphism f on H0. We choose a flat resolution of EndS(T )* *+ , i.e., a DG ring E and a quasi-isomorphism of DG rings E '-!EndS(T )+ , such that the funct* *or E - preserves quasi-isomorphisms between chain complexes of abelian groups (see for* * example [Kel99, 3.2 Lemma (a)]). We end up with a chain of two quasi-isomorphisms of DG* * rings R --' E -'-!End S(T ) . MORITA THEORY 19 The complex T is naturally a DG EndS (T )-S-bimodule, and we restrict the left * *action to E and view T as a DG E-S-bimodule. We choose a cofibrant replacement T c-~! T as * *a DG E-S-bimodule. Then we obtain the desired complex of R-S-bimodules by M = R E T c. Tensoring with M over R has a total left derived functor - LRM : D(R) - ! D(S) (3.22) (although this is not obvious with our definition since M need not be cofibrant* * as a complex of right S-modules, and then - R M does not takes values in cofibrant complexe* *s). In order to show that this derived functor is an exact equivalence we use that the* * diagram of triangulated categories - LEEndS(T) D(E) _______________//_D(End S(T )) (3.23) | | - LER || -|LEndS(T)T| fflffl| fflffl| D(R) ___________________//D(S) - LRM commutes up to natural isomorphism. Indeed, two ways around the square are giv* *en by derived tensor product with the E-S-bimodules M respectively T , so it suffices* * to find a chain of quasi-isomorphisms of DG bimodules between M and T . Since E is cofibrant as a complex of abelian groups and the composite map E * *- ! End S(T )+ - ! R is a quasi-isomorphism, E Sop models the derived tensor prod* *uct of R and S. If one of R or S are flat, then R Sop also models the derived tensor* * product, so that the map E Sop -! R Sop is a quasi-isomorphism of DG rings. Since T cis cofibrant as an E Sop-module, * *the induced map T c = (E Sop)E SopT c -! (R Sop)E SopT c ~=R E T c = M is a quasi-isomorphism. So we have a chain of two quasi-isomorphisms of E-S-bim* *odules T -'- T c -'-!M . The left and upper functors in the commutative square (3.23)are derived from* * extensions of scalars along quasi-isomorphisms of DG rings; thus they are exact equivalenc* *e of trian- gulated categories. The right vertical derived functor is an exact equivalence * *by Theorem 3.20. So we conclude that the lower horizontal functor (3.22)in the square (3.2* *3)is also an exact equivalence of triangulated categories. This establishes condition (3). 3.3. Examples. Historically, tilting modules seem to have been the first exampl* *es of derived equivalences which are not Morita equivalences. I will not try to give an acco* *unt of the history of tilting module, tilting complexes, and rather refer to [KZ , Sec. 3.* *1] or [AGH ]. Definition 3.24. Let R be a finite dimensional algebra over a field. A tilting * *module is a finitely generated R-module T with the following properties. (i)T has projective dimension 0 or 1, (ii)T has no self-extensions, i.e., Ext1R(T, T ) = 0, 20 STEFAN SCHWEDE (iii)there is an exact sequence of right R-modules 0 -! R -! T1 -! T2 -! 0 for some m 0, such that T1 and T2 are direct summands of a finite sum * *of copies of T . Note that if the tilting module T is actually projective, then condition (ii* *) is automatic and the exact sequence required in (iii) splits. So then the free R-module of * *rank one is a summand of a finite sum of copies of T , and hence T is a finitely generated * *projective generator for Mod- R. So R is then Morita equivalent to the endomorphism ring * *of the tilting module T , by the Morita theorem 2.2. For a self-injective algebra, fo* *r example a group algebra over a field, the converse also holds; indeed, every module of fi* *nite projective dimension over a self-injective algebra is already projective. So for these alg* *ebras, tilting is the same as Morita equivalence. If the projective dimension of the tilting module T is 1, then we do not get* * an equivalence between the modules over R and S = EndR (T ), but we get a derived equivalence.* * Indeed, since R is noetherian, condition (i) implies that T has a 2-step resolution P1 * *-! P0 by two finitely generated projective R-modules; this resolution is a small object * *in the derived category D(R), and its graded self-maps in D(R) are concentrated in dimension 0* * by con- dition (ii). Condition (iii) implies that the complex R[0] is contained in the* * triangulated subcategory generated by the resolution, and the resolution is thus a generator* * for D(R), hence a tilting complex. Example 3.25. For an example of a non-projective tilting module we fix a field * *k and we let A be the algebra of upper triangular 3 x 3 matrices over k, 8 0 1 9 < x11 x12 x13 = A = @ 0 x22 x23A | xij2 k . : 0 0 x ; 33 Up to isomorphism, there are three indecomposable projective right A-modules, n* *amely the row vectors P 1= {(y1, y2, y3) | y1, y2, y3 2 k} and its A-submodules P 2= {(0, y2, y3) | y2, y3 2 k} and P 3= {(0, 0, y3) | y3 2 k} . These projectives are the covers of three corresponding simple modules, namely S1 = P 1=P 2, S2 = P 2=P 3, and S3 = P 3. In particular, S3 is projective and S1 and S2 have projective dimension 1. We define the tilting module T as the direct sum T = P 1 P 2 S2 . The projective resolution 0 - --! P 1 P 2 P 3 inclusion-----!P 1 P 2 P 2 ---! S2 - --! 0 can be used to calculate Ext1A(T, T ) = Ext1A(S2, T ) = 0. Since P 1 P 2 P 3* *is a free A-module of rank one, this short exact sequence verifies tilting condition (iii* *) in Definition 3.24 for the A-module T . MORITA THEORY 21 Altogether this shows that T is a tilting module for A of projective dimensi* *on one. So A `tilts' to the endomorphism algebra of T ; this endomorphism algebra can be c* *alculated directly, and it comes out to be another subalgebra of the 3 x 3 matrices over * *k, namely 8 0 1 9 < x11 x12 x13 = EndA (T ) ~= @ 0 x22 0 A | xij2 k . : 0 0 x ; 33 (hint: the modules P 1and S2 do not map to each other nor to P 2, and the rema* *ining relevant morphism spaces are 1-dimensional over k.) The algebras A and EndA (T * *) are not Morita equivalent. Indeed, both have exactly three isomorphism classes of indec* *omposable projective modules, but in one case these modules are `directed' (i.e., linearl* *y ordered un- der the existence of non-trivial homomorphisms), whereas in the other case two * *of these indecomposable projectives do not map to each other non-trivially. The preceding example, and many other ones, are often described using repres* *entations of quivers. Indeed, the upper triangular matrices A and the tilted algebra End* * A(T ) are isomorphic to the path algebras of the A3-quivers o ____//_o___//o respectively o oo___o_____//o. Example 3.26. We obtain a tilting complex whose homology is concentrated in mor* *e then one dimension by `spreading out' the free module of rank one. Let R = R1 x R2 * *be the product of two rings. Let P1 = R1 x 0 and P2 = 0 x R2 be the two "blocks", i.e* *., the projective R-bimodules corresponding to the central idempotents (1, 0) and (0, * *1) in R. Then R = P1 P2 as an R-bimodule, and there are no non-trivial R-homomorphisms betwe* *en P1 and P2. Now take T = P1[0] P2[n] for some number n 6= 0. This is a complex of R* *-modules with trivial differential whose homology is concentrated in two dimensions. Mor* *eover, the complex T is a small generator for the derived category D(R). But the only non* *-trivial self-maps of T are of degree 0 since P1 and P2 don't map to each other. Hence T* * is a tilting complex which is not (quasi-isomorphic to) a tilting module. The endomorphisms * *of T are again the ring R, so it is a non-trivial self-tilting complex of R. Under the * *equivalence D(R) ~=D(R1) x D(R2), the self-equivalence induced by T is the identity on the * *first factor and the n-fold shift on the second factor. More examples of tilting complexes can be found in Sections 4 and 5 of [Ric8* *9b] or Chapter 5 of [KZ ] 4. Morita theory in stable model categories Now we carry the Morita philosophy one step further: we sketch Morita theory* * for ring spectra and for stable model categories. As a summary one can say that essentia* *lly, every- thing which we have said for rings and differential graded rings works, suitabl* *y interpreted, for ring spectra as well. First a few words about what we mean by a ring spectrum. The stable homotopy* * category of algebraic topology has a symmetric monoidal smash product; the monoids are h* *omotopy- associative ring spectra, and they represent multiplicative cohomology theories* *. While the notion of a homotopy-associative ring spectrum is useful for many things, it do* *es not have a good enough module theory for our present purpose. One can certainly consider s* *pectra with a homotopy-associative action of a homotopy-associative ring spectrum; but the * *mapping 22 STEFAN SCHWEDE cone of a homomorphism between such modules does not inherit a natural action o* *f the ring spectrum, and the category of such modules does not form a triangulated categor* *y. So in order to carry out the Morita-theory program we need a highly structur* *ed model for the category of spectra which admits a symmetric monoidal and homotopically* * well behaved smash product _ before passing to the homotopy category ! The first exa* *mples of such categories were the S-modules [EKMM ] and the symmetric spectra [HSS * *]; by now several more such categories have been constructed [Lyd98 , MMSS ]; The approp* *riate notion of a model category equivalence is a Quillen equivalence [Hov99 , Def. 1.3.12] * *since these equivalences preserve the `homotopy theory', not just the homotopy category; al* *l known model categories of spectra are Quillen equivalent in a monoidal fashion. For definiteness, we work in one specific category of spectra with nice smas* *h product, namely the symmetric spectra based on simplicial sets, as introduced by Hovey, * *Shipley and Smith [HSS ]. The monoids are called symmetric ring spectra, and I personally t* *hink that they are the simplest kind of ring spectra; as far as their homotopy category i* *s concerned, symmetric ring spectra are equivalent to the older notion of A1 -ring spectrum,* * and com- mutative symmetric ring spectra are equivalent to E1 -ring spectra. The good th* *ing is that operads are not needed anymore. However, using symmetric spectra is not essential and the results described * *in this section could also be developed in more or less the same way in any other of the known * *model categories of spectra with compatible smash product. Alternatively, we could ha* *ve taken an axiomatic approach and use the term `spectra' for any stable, monoidal model ca* *tegory in which the unit object `looks and feels' like the sphere spectrum. Indeed, we ar* *e essentially only using the following properties of the category of symmetric spectra: (i)there is a symmetric monoidal smash product, which makes symmetric spect* *ra into a monoidal model category ([Hov99 , 4.2.6], [SS00]); (ii)the model structure is stable (Definition 4.1); (iii)the unit S of the smash product is a small generator (Definition 3.7) o* *f the homotopy category of spectra; (iv)the (derived) space of self maps of the unit object S is weakly equivale* *nt to QS0 = hocolimn nSn and in the homotopy category, there are no maps of negative* * degree from S to itself. A large part of the material in this section is taken from a joint paper wit* *h Shipley [SS03]. Two other papers devoted to Morita theory in the context of ring spectra are [D* *GI ] by Dwyer, Greenlees and Iyengar and [BL ] by Baker and Lazarev. 4.1. Stable model categories. Recall from [Qui67, I.2] or [Hov99 , 6.1] that th* *e homotopy category of a pointed model category supports a suspension and a loop functor. * *In short, for any object X the map to the zero object can be factored X --! C -'-! * as a cofibration followed by a weak equivalence. The suspension of X is then d* *efined as the quotient of the cofibration, X = C=X. Dually, the loop object X is the fi* *ber of a fibration from a weakly contractible object to X. On the level of homotopy cate* *gories, the suspension and loop constructions become functorial, and is left adjoint to . Definition 4.1. A stable model category is a pointed model category for which t* *he functors and on the homotopy category are inverse equivalences. MORITA THEORY 23 The homotopy category of a stable model category has a large amount of extra* * structure, some of which is relevant for us. First of all, it is naturally a triangulated* * category, see [Hov99 , 7.1.6] for a detailed proof. The rough outline is as follows: by defin* *ition of `stable' the suspension functor is a self-equivalence of the homotopy category and it de* *fines the shift functor. Since every object is a two-fold suspension, hence an abelian co-group* * object, the homotopy category of a stable model category is additive. Furthermore, by [Hov9* *9 , 7.1.11] the cofiber sequences and fiber sequences of [Qui67, 1.3] coincide up to sign i* *n the stable case, and they define the distinguished triangles. The model categories which w* *e consider have all limits and colimits, so the homotopy categories have infinite sums and* * products. Objects of a stable model category are called `generators' or `small' if they h* *ave this property as objects of the triangulated homotopy category, compare Definition 3.7. A Quillen adjoint functor pair between stable model categories gives rise to* * total derived functors which are exact functors with respect to the triangulated structure; i* *n other words both total derived functors commute with suspension and preserve distinguished * *triangles. Examples 4.2. (1) Chain complexes. In the previous section, we have already seen an important* * class of examples from algebra, namely the category of chain complexes over a ring R.* * This category actually has several different stable model structures: the projective* * model structure (see Remark 3.3 (iv)) and the injective model structure (see Remark 3.3 (v)) ha* *ve as weak equivalences the quasi-isomorphisms. There is a clash of terminology here: the * *homotopy category in the sense of homotopical algebra is obtained by formally inverting * *the weak equivalences; so for the projective and injective model structures, this gives * *the unbounded derived category D(R). But the category of unbounded chain complexes admits an* *other model structure in which the weak equivalences are the chain homotopy equivalen* *ces, see e.g. [CH , Ex. 3.4]. Thus for this model structure, the homotopy category is wh* *at is commonly called the homotopy category, often denoted by K(R). The derived category D(R)* * is a quotient of the homotopy category K(R); the derived category D(R) has a single * *small generator, but for example the homotopy category of chain complexes of abelian * *groups K(Z) does not have a set of generators whatsoever, compare [Nee01, E.3.2]. The* * three stable model structure on chain complexes of modules have been generalized in v* *arious directions to chain complexes in abelian categories or to other differential gr* *aded objects, see [CH ], [Bek00 ] and [Hov01a ]. (2) The stable module category of a Frobenius ring. A different kind of alge* *braic example _ not involving chain complexes _ is formed by the stable module catego* *ries of Frobenius rings. A Frobenius ring A is defined by the property that the classes* * of projective and injective A-modules coincide. Important examples are finite dimensional sel* *f-injective algebras over a field, in particular finite dimensional Hopf-algebras, such as * *group algebras of finite groups. The stable module category has as objects the A-modules (not cha* *in complexes of modules). Morphisms in the stable category or represented by module homomorp* *hisms, but two homomorphisms are identified if their difference factors through a proj* *ective (= injective) A-module. Fortunately the two different meanings of `stable' fit together nicely; the * *stable module category is the homotopy category associated to a stable model category structu* *re on the category of A-modules, see [Hov99 , Sec. 2]. The cofibrations are the monomorph* *isms, the fibrations are the epimorphisms, and the weak equivalences are the maps which b* *ecome iso- morphisms in the stable category. Every finitely generated module is small when* * considered 24 STEFAN SCHWEDE as an object of the stable module category. As in the case of chain complexes * *of mod- ules, there is usually no point in making the model structure explicit since th* *e cofibration, fibrations and weak equivalences coincide with certain well-known concepts. Quillen equivalences between stable module categories arise under the name o* *f stable equivalences of Morita type ([Bro94, Sec. 5], [KZ , Ch. 11]). For simplicity, s* *uppose that A and B are two finite-dimensional self-injective algebras over a field k; then A* * and B are in particular Frobenius rings. Consider an A-B-bimodule M (by which we mean a k-sy* *mmetric bimodule, also known as a right module over Aop kB), which is projective as lef* *t A-module and as a right B-module separately. Then the adjoint functor pair __-_AM____//_ Mod- A oo_________ Mod- B (4.3) Hom B(M,-) is Quillen adjoint pair with respect to the `stable' model structures. A stable equivalence of Morita type consists of an A-B-bimodule M and a B-A-* *bimodule N such that both M and N are projective as left and right modules separately, a* *nd such that there are direct sum decompositions N A M ~=B X and M B N ~=A Y as bimodules, where Y is a projective A-A-bimodule and B is a projective B-B-bi* *module. In this situation, the functors - A M and - B N induce inverse equivalences o* *f the stable module categories. Moreover, the Quillen adjoint pair (4.3)is a Quillen equival* *ence. Rickard observed [Ric89b] that a derived equivalence between self-injective,* * finite-dimen- sional algebras also gives rise to a stable equivalence of Morita type. In the above algebraic examples, there is no real need for the language of m* *odel categories; moreover, `Morita theory' is covered by Keller's paper [Kel94a], which uses dif* *ferential graded categories. A whole new world of stable model categories comes from homotopy th* *eory, see the following list. The associated homotopy categories yield triangulated categ* *ories which are not immediately visible to the eyes of an algebraist, since they do not ari* *se from abelian categories. (3) Spectra. The prototypical example of a stable model category (which is * *not an `algebraic'), is `the' category of spectra. We review one model, the symmetric* * spectra of Hovey, Shipley and Smith [HSS ] in more detail in Section 4.2. Many other model* * categories of spectra have been constructed, see for example [BF78 , Rob87a , Jar97, EKMM * * , Lyd98, MMSS ]. All known model categories of spectra Quillen equivalent (see e.g., [* *HSS , Thm. 4.2.5], [Sch01a] or [MMSS ]), and their common homotopy category is referred t* *o as the stable homotopy category. The sphere spectrum is a small generator for stable h* *omotopy category. (4) Modules over ring spectra. Modules over an S-algebra [EKMM , VII.1], * *over a symmetric ring spectrum [HSS , 5.4.2], or over an orthogonal ring spectrum [MMS* *S ] form stable model categories. We recall symmetric ring spectra and their module spe* *ctra in Section 4.2. In each case a module is small if and only if it is weakly equival* *ent to a retract of a finite cell module. The free module of rank one is a small generator. More* * generally there are stable model categories of modules over `symmetric ring spectra with severa* *l objects', or spectral categories, see [SS03, A.1]. (5) Equivariant stable homotopy theory. If G is a compact Lie group, there * *is a category of G-equivariant coordinate free spectra [LMS86 ] which is a stable mo* *del category. Modern versions of this model category are the G-equivariant orthogonal spectra* * of [MM02 ] MORITA THEORY 25 and G-equivariant S-modules of [EKMM ]. In this case the equivariant suspensi* *on spectra of the coset spaces G=H+ for all closed subgroups H G form a set of small gen* *erators. (6) Presheaves of spectra. For every Grothendieck site Jardine [Jar87] cons* *tructs a stable model category of presheaves of Bousfield-Friedlander type spectra; the * *weak equiva- lences are the maps which induce isomorphisms of the associated sheaves of stab* *le homotopy groups. For a general site these stable model categories do not seem to have a * *set of small generators. A similar model structure for presheaves of symmetric spectra is d* *eveloped in [Jar00a]. (7) The stabilization of a model category. Modulo technicalities, every poin* *ted model category gives rise to an associated stable model category by `inverting' the s* *uspension functor, i.e., by passage to internal spectra. This has been carried out, unde* *r different hypotheses, in [Sch97] and [Hov01b ]. (8) Bousfield localization. Following Bousfield [Bou75 ], localized model st* *ructures for modules over an S-algebra are constructed in [EKMM , VIII 1.1]. Hirschhorn [H* *ir03] shows that under quite general hypotheses the localization of a model category is aga* *in a model category. The localization of a stable model category is stable and localizati* *on preserves generators. Smallness need not be preserved. (9) Motivic stable homotopy. In [MV , Voe98] Morel and Voevodsky introduced* * the A1-local model category structure for schemes over a base. An associated stable* * homotopy category of A1-local T -spectra (where T = A1=(A1- 0) is the `Tate-sphere') is * *an important tool in Voevodsky's proof of the Milnor conjecture [Voe]. There are several sta* *ble model cate- gories underlying this motivic stable homotopy category, see for example [Jar00* *b], [Hov01b ], [Hu03 ] or [DØR ]. 4.2. Symmetric ring and module spectra. In this section we give a quick introdu* *ction to symmetric spectra and symmetric ring and module spectra. I recommend readin* *g the original, self-contained paper by Hovey, Shipley and Smith [HSS ]. At several * *points, our exposition differs from theirs, for example, we let the spheres act from the ri* *ght. Definition 4.4. [HSS ] A symmetric spectrum consists of the following data o a sequence of pointed simplicial sets Xn for n 0 o for each n 0 a base-point preserving action of the symmetric group n * *on Xn o pointed maps ffp,q: Xp ^ Sq -! Xp+q for p, q 0 which are p x q-equiv* *ariant; here S1 = 1=@ 1, Sq = (S1)^q and q permutes the factors. This data is subject to the following conditions: o under the identification Xn ~=Xn^S0, the map ffn,0: Xn^S0 -! Xn is the i* *dentity, o for p, q, r 0, the following square commutes ffp,q^ Id r Xp ^ Sq ^ Sr__________//_Xp+q^ S (4.5) ~=|| ffp+q,r|| fflffl| fflffl| Xp ^ Sq+r____ffp,q+r_//Xp+q+r . A morphism f : X -! Y of symmetric spectra consists of n-equivariant pointed * *maps fn : Xn -! Yn for n 0, which are compatible with the structure maps in the se* *nse that fp+qO ffp,q= ffp,qO (fq ^ IdSq) for all p, q 0. The definition we have just given is somewhat redundant, and Hovey, Shipley * *and Smith use a more economical definition in [HSS , Def. 1.2.2]. Indeed, the commuting s* *quare (4.5), 26 STEFAN SCHWEDE shows that all action maps ffp,qare given by composites of the maps ffp,1: Xp^ * *S1 -! Xp+1 for varying p. A first example is the symmetric sphere spectrum S given by Sn = Sn, where t* *he symmetric group permutes the factors and ffp.q: Sp ^ Sq -! Sp+q is the canonical isomorph* *ism. More generally, every pointed simplicial set K gives rise to a suspension spectrum * *1 K via ( 1 K)n = K ^ Sn ; then we have S ~= 1 S0. A symmetric spectrum is cofibrant if it has the left lifting property for le* *velwise acyclic fibrations. More precisely, A is cofibrant if the following holds: for every mo* *rphism f : X -! Y of symmetric spectra such that fn : Xn -! Yn is a weak equivalence and Kan fi* *bration for all n, and for every morphism ' : A -! X, there exists a morphism ~': A -! * *Y such that ' = f~'. Suspension spectra are examples of cofibrant symmetric spectra. A* *n equivalent definition uses the latching space LnA, a simplicial set which roughly is the `* *stuff coming from dimensions below n'; see [HSS , 5.2.1] for the precise definition. A symme* *tric spectrum A is cofibrant if and only if for all n, the map LnA -! An is injective and sym* *metric group n is freely on the complement of the image, see [HSS , Prop. 5.2.2]. An -sp* *ectrum is defined by the properties that each simplicial set Xn is a Kan complex and all * *the maps Xn -! (Xn+1) adjoint to ffn,1are weak homotopy equivalences. The stable homot* *opy category has as objects the cofibrant symmetric -spectra and as morphisms the * *homotopy classes of morphisms of symmetric spectra. Although we just gave a perfectly good definition of the stable homotopy cat* *egory, in order to work with it one needs an ambient model category structure. One such model s* *tructure is the stable model structure of [HSS , Thm. 3.4.4]. A morphism of symmetric spect* *ra is a stable equivalence if it induces isomorphisms on all cohomology theories represented b* *y (injective) -spectra, see [HSS , Def, 3.1.3] for the precise statement. There is a notion * *of cofibration such that a symmetric spectrum X is cofibrant in the above sense if and only if* * the map from the trivial symmetric spectrum to X is a cofibration. The -spectra then * *coincide with the stably fibrant symmetric spectra. There are other model structure for * *symmetric spectra with the same class of weak (=stable) equivalences, hence with the same* * homotopy category, for example the S-model structure which is hinted at in [HSS , 5.3.6]. Stable equivalences versus ß*-isomorphisms. One of the tricky points with s* *ym- metric spectra is the relationship between stable equivalences and ß*-isomorphi* *sms. The stable equivalences are defined as the morphisms which induce isomorphisms on a* *ll coho- mology theories; there is the strictly smaller class of morphisms which induce * *isomorphisms on stable homotopy groups. The k-th stable homotopy group of a symmetric spectr* *um X is defined as the colimit ßkX = colimn ßn+k|Xn| , where |Xn| denotes the geometric realization of the simplicial set Xn. The coli* *mit is taken over the maps 1 1 (ffn,1)* ßn+k |Xn| ---^S---!ßn+k+1 |Xn| ^ S ------! ßn+k+1 |Xn+1| . (4.6) While every ß*-isomorphism of symmetric spectra is a stable equivalence [HSS* * , Thm. 3.1.11], the converse is not true. The standard example is the following: con* *sider the symmetric spectrum F1S1 freely generated by the circle S1 in dimension 1. Expli* *citly, F1S1 is given by (F1S1)n = +n^ n-1 Sn-1 ^ S1 . MORITA THEORY 27 So (F1S1)n is a wedge of n copies of Sn and in the stable range, i.e., up to ro* *ughly dimensions 2n, the homotopy groups of (F1S1)n are a direct sum of n copies of the homotopy* * groups of Sn. Moreover, in the stable range, the map in the colimit system (4.6)is a dire* *ct summand inclusion into (n + 1) copies of the homotopy groups of Sn. Thus in the colimit* *, the stable homotopy groups of the symmetric spectrum F1S1 are a countably infinite direct * *sum of copies of the stable homotopy groups of spheres. Since F1S1 is freely generated* * by the circle S1 in dimension 1, it ought to be a desuspension of the suspension spectrum of * *the circle. However, the necessary symmetric group actions `blow up' such free objects with* * the effect that the stable homotopy groups are larger than they should be. This example i* *ndicates that inverting only the ß*-isomorphisms would leave too many stable homotopy ty* *pes, and the resulting category could not be equivalent to the usual stable homotopy cat* *egory. Smash product. One of the main features which distinguishes symmetric spectr* *a from the more classical spectra is the internal smash product. The smash product of * *symmetric spectra can be described via its universal property, analogous to universal pro* *perty of the tensor product over a commutative ring. Indeed, if R is a commutative ring and* * M and N are right R-modules, then a bilinear map to another left R-module W is a map * *b : M x N -! W such that for each m 2 M the map b(m, -) : N -! W and each n 2 N the map b(-, n) : M -! W are R-linear. The tensor product M R N is the universal e* *xample of a right R-module together with a bilinear map from M x N. Let us define a bilinear morphism b : (X, Y ) -! Z from two symmetric spectr* *a X and Y to a symmetric spectrum Z to consist of a collection of px q-equivariant maps* * of pointed simplicial sets bp,q: Xp ^ Yq -! Zp+q for p, q 0, such that for all p, q, r 0, the following diagram commutes Xp ^ Yq ^ SrId^twist//_Xp ^ Sr ^ Yq (4.7) Id^ffq,rmmmmmmm|| || mmmm |bp,q^Id |ffp,r^Id vvmmmm fflffl| |fflffl Xp ^ Yq+rQ Zp+q^ Sr Xp+r^ Yq QQQQ | | QQQQ |ffp+q,r |bp+r,q bp,q+rQQQ((QQQfflffl|| |fflffl| Zp+q+r oo_1xffl_____Zp+r+q r,qx1 The automorphism 1xØr,qof Zp+q+rmay look surprising at first sight. Here 1xØr,q* *2 p+r+q denotes the block permutation which fixes the first p elements, and which moves* * the next q elements past the last r elements. This can be viewed as a topological version * *of the Koszul sign rule which says that when two symbols of degree q and r are permuted past * *each other, the sign (-1)qr should appear as well. The block permutation Ør,qhas sign (-1)q* *r and it compensates the upper vertical interchange of Yq and Sr. A good way to keep tra* *ck of such permutations is to carefully distinguish between indices such as r + q and q + * *r. Of course these two numbers are equal, but the fact that one arises naturally instead of * *the other reminds us that a block permutation should be inserted. The smash product X ^ Y is the universal example of a symmetric spectrum wi* *th a bimorphism from X and Y . In other words, it comes with a bimorphism ' : (X, Y * *) -! X ^Y such that for every symmetric spectrum Z the map Sp (X ^ Y, Z) - ! Bi-Sp ((X, Y ), Z) (4.8) 28 STEFAN SCHWEDE is bijective. If we suppose that such a universal object exist, this property c* *haracterizes the smash product and the maps 'p,q: Xp ^ Yq -! (X ^ Y )p+q up to canonical isomorp* *hism. An actual construction as a certain coequalizer is given in [HSS , Def. 2.2.3];* * in this article, we will only use the universal property of the smash product. We use the universal property to derive that the smash product is functorial* * and symmetric monoidal. For example, let f : X -! Y and f0 : X0 -! Y 0be morphisms of symmet* *ric spectra. Then the collection of maps of pointed simplicial sets æ oe fq^f0q 0 'p,q 0 Xp ^ X0q- --! Yp ^ Yq --! (Y ^ Y )p+q p,q 0 form a bilinear morphism (X, X0) -! Y ^ Y 0, so it corresponds to a unique morp* *hism of symmetric spectra f ^ f0 : X ^ X0- ! Y ^ Y 0. The universal property implies fu* *nctoriality in both arguments. For the proof of the associativity of the smash product we n* *otice that the family æ oe 'p,q^Id 'p+q,r Xp ^ Yq ^ Zr ----! (X ^ Y )p+q^ Zr - --! ((X ^ Y ) ^ Z)p+q+r p,q,r 0 and the family æ oe Id^'q,r 'p,q+r Xp ^ Yq ^ Zr ----! Xp ^ (Y ^ Z)q+r - --! (X ^ (Y ^ Z))p+q+r p,q,r 0 both have the universal property of a tri-linear morphism out of X, Y and Z. Th* *e uniqueness of universal objects gives a preferred isomorphism of symmetric spectra (X ^ Y ) ^ Z ~=X ^ (Y ^ Z) . The symmetry isomorphism X ^ Y ~=Y ^ X corresponds to the bilinear morphism n twist 'q,p fflq,p o Xp ^ Yq ---! Yq ^ Xp --! (Y ^ X)q+p --! (Y ^ X)p+q . (4.9) p,q 0 The block permutation Øq,pis crucial here: without it we would not get a biline* *ar morphism is the sense of diagram (4.7). In much the same spirit, the universal propertie* *s can be used to provide unit isomorphisms S ^ X ~=X ~=X ^ S, to verify the coherence conditi* *ons of a symmetric monoidal structure, and to establish an isomorphism of suspension spe* *ctra ( 1 K) ^ ( 1 L) ~= 1 (K ^ L) . The symmetric monoidal structure given by the smash product of symmetric spectr* *a is closed in the sense that internal function objects exist as well. For each pair of sym* *metric spectra X and Y there is a symmetric function spectrum Hom (X, Y ) [HSS , 2.2.9], and * *there are natural composition morphisms O : Hom (Y, Z) ^ Hom (X, Y ) - ! Hom (X, Z) which are associative and unital with respect to a unit map S -! Hom (X, X). Mo* *reover, the usual adjunction isomorphism Sp (X ^ Y, Z) ~= Sp (X, Hom (Y, Z)) relates the smash product and function spectra. Ring and module spectra. The smash product of symmetric spectra leads to the concomitant concepts symmetric ring spectra, module spectra and algebra spectra. MORITA THEORY 29 Definition 4.10. A symmetric ring spectrum is a symmetric spectrum R together w* *ith morphisms of symmetric spectra j : S - ! R and ~ : R ^ R - ! R , called the unit and multiplication map, which satisfy certain associativity and* * unit conditions (see [McL , VII.3]). A ring spectrum R is commutative if the multiplication map* * is unchanged when composed with the twist, or the symmetry isomorphism (4.9), of R ^ R. A mo* *rphism of ring spectra is a morphism of spectra commuting with the multiplication and * *unit maps. If R is a symmetric ring spectrum, a right R-module is a spectrum N together wi* *th an action map N ^ R -! N satisfying associativity and unit conditions (see again [McL , V* *II.4]). A morphism of right R-modules is a morphism of spectra commuting with the action * *of R. We denote the category of right R-modules by Mod-R. With the universal property of smash product we can make the structure of a * *symmetric ring spectrum more explicit. The multiplication map ~ : R ^ R -! R corresponds* * to a family of pointed, p x q-equivariant maps ~p,q: Rp ^ Rq -! Rp+q for p, q 0, which are bilinear in the sense of diagram (4.7). The maps are su* *pposed to be associative and unital with respect to the maps jp : Sp -! Rp which constitute * *the unit map j : S -! R. The commutativity isomorphism of the smash product involves the block permut* *ation Øq,p, see (4.9). So the multiplication of a symmetric ring spectrum is commutat* *ive if and only if the following diagrams commute for all p, q 0 ~p,q Rp ^ Rq ________//Rp+q twist|| |fflp,q| fflffl| fflffl| Rq ^ Rp ___~q,p_//Rq+p The block permutation Øp,qhas sign (-1)pq, so this diagram is reminiscent of th* *e Koszul sign rule in a graded ring which is commutative in the graded sense. The unit S of the smash product is a ring spectrum in a unique way, and S-mo* *dules are the same as symmetric spectra. The smash product of two ring spectra is natural* *ly a ring spectrum. For a ring spectrum R the opposite ring spectrum Rop is defined by co* *mposing the multiplication with the twist map R ^ R -! R ^ R (so in terms of the biline* *ar maps ~p,q: Rp ^ Rq -! Rp+q, a block permutation appears). The definitions of left m* *odules and bimodules is hopefully clear; left R-modules and R-T -bimodule can also be * *defined as right modules over the opposite ring spectrum Rop, respectively right modules o* *ver the ring spectrum Rop ^ T . A formal consequence of having a closed symmetric monoidal smash product is * *that the category of R-modules inherits a smash product and function objects. The smash * *product M ^R N of a right R-module M and a left R-module N can be defined as the coequa* *lizer, in the category of symmetric spectra, of the two maps M ^ R ^ N ____//_//_M ^ N given by the action of R on M and N respectively. Alternatively, one can chara* *cterize M ^R N as the universal example of a symmetric spectrum which receives a biline* *ar map 30 STEFAN SCHWEDE from M and N which is R-balanced, i.e., all the diagrams Id^ffq,r Mp ^ Rq ^ Nr__________//Mp ^ Nq+r (4.11) ffp,q^Id|| |'p,q+r| fflffl| fflffl| Mp+q^ Nr ____'p+q,r//_(M ^ N)p+q+r commute. If M happens to be a T -R-bimodule and N an R-S-bimodule, then M ^R N * *is naturally a T -S-bimodule. In particular, if R is a commutative ring spectrum, * *the notions of left and right module coincide and agree with the notion of a symmetric bimo* *dule. In this case ^R is an internal symmetric monoidal smash product for R-modules. There ar* *e also internal function spectra and function modules, enjoying the `usual' adjointnes* *s properties with respect to the various smash products. The modules over a symmetric ring spectrum R inherit a model category struct* *ure from symmetric spectra, see [HSS , Cor. 5.4.2] and [SS00, Thm. 4.1 (1)]. More precis* *ely, a mor- phism of R-modules is called a weak equivalence (resp. fibration) if the underl* *ying morphism of symmetric spectra is a stable equivalence (resp. stable fibration). The cof* *ibrations are then determined by the left lifting property with respect to all acyclic fibrat* *ions in Mod-R. This model structure is stable, so the homotopy category of modules over a ring* * spectrum is a triangulated category. The free module of rank one is a small generator. For a map R -! S of ring spectra, there is a Quillen adjoint functor pair an* *alogous to restriction and extension of scalars: any S-module becomes an R-module if we le* *t R act through the map. This functor has a left adjoint taking an R-module M to the S-* *module M ^R S. If R -! S is a stable equivalence, then the functors of restriction and* * extension of scalars are a Quillen equivalence between the categories of R-modules and S-* *modules, see [HSS , Thm. 5.4.5] and [SS00, Thm. 4.3]. Example 4.12 (Monoid ring spectra). If M is a simplicial monoid, and R is a sym* *metric ring spectrum, we define a symmetric spectrum R[M] by R[M]n = Rn ^ M+ , where M+ denotes the underlying simplicial set of M, with disjoint basepoint ad* *ded. The unit map is the composite of the unit map of R and the wedge summand inclusion * *indexed by the unit of M; the multiplication map R[M] ^ R[M] -! R[M] is induced from the b* *ilinear morphism ~p,q^mult. + (Rp ^ M+ ) ^ (Rq ^ M+ ) ~=(Rp ^ Rq) ^ (M x M)+ --------! Rp+q^ M . The construction of the monoid ring over S is left adjoint to the functor which* * takes a symmetric ring spectrum R to the simplicial monoid R0. Example 4.13 (Matrix ring spectra). Let R be a symmetric ring spectrum and cons* *ider the wedge (coproduct) R x n = R___._._.R-z____" n of n copies of the free R-module of rank 1. In the usual stable model structur* *e, the free module of rank 1 is cofibrant, hence so is R ^ n+ . We choose a fibrant replace* *ment R x n ~-! (R x n)f. The ring spectrum of n x n matrices over R is defined as the endomorp* *hism ring spectrum of (R x n)f, Mn(R) = End R((R x n)f) . MORITA THEORY 31 The stable equivalence type of the matrix ring spectrum Mn(R) is independent of* * the choice of fibrant replacement, see Corollary A.2.4 of [SS03]. Moreover, the underlying* * spectrum of Mn(R) is isomorphic, in the stable homotopy category, to a sum of n2 copies of * *R. Example 4.14 (Eilenberg-Mac Lane spectra). For an abelian group A, the Eilenber* *g-Mac Lane spectrum HA is defined by (HA)n = A Z[Sn] , i.e., the underlying simplicial set of the dimensionwise tensor product of A wi* *th the reduced free simplicial abelian generated by the simplicial n-sphere. The symmetric gro* *ups acts by permuting the smash factors of Sn. The homotopy groups of the symmetric spectru* *m HA are concentrated in dimension zero, where we have a natural isomorphism ß0HA ~=* *A. For two abelian groups A and B, a natural morphism of symmetric spectra HA ^ HB -! H(A B) is obtained, by the universal property (4.8), from the bilinear morphism (HA)n ^ (HB)m = (A Z[Sn]) ^ (B Z[Sm ]) -! (A B) Z[Sn+m ] = (H(A B))n+m given by _ ! 0 1 X X X ai. xi ^ @ bj. x0jA7-! (ai. bj) . xi^ x0j. i j i,j A unit map S -! HZ is given by the inclusion of generators. With respect to the* *se maps, H becomes a lax symmetric monoidal functor from the category of abelian groups * *to the category of symmetric spectra. As a formal consequence, H turns a ring R into a* * symmetric ring spectrum with multiplication map HR ^ HR - ! H(R R) - ! HR . Similarly, an R-module structure on A gives rise to an HR-module structure on H* *A. The definition of the symmetric spectra HA makes just as much sense when A i* *s a simplicial abelian group; thus the Eilenberg-Mac Lane functor makes simplicial * *rings into symmetric ring spectra, respecting possible commutativity of the multiplication* *s. With a little bit of extra care, the Eilenberg-Mac Lane construction can also be exten* *ded to a differential graded context, compare [SS03, App. B] and [Ri03]. For a fixed ring B, the modules over the Eilenberg-Mac Lane ring spectrum HB* * of a ring B have the same homotopy theory as complexes of B-modules. The first resu* *lts of this kind were obtained by Robinson for A1 -ring spectra [Rob87b ], and later f* *or S-algebras in [EKMM , IV Thm. 2.4]; in both cases, equivalences of triangulated homotopy* * categories are constructed. But more is true: for any ring B, Theorem 5.1.6 of [SS03] pr* *ovides a chain of two Quillen equivalences between the categories of unbounded chain com* *plexes of B-modules and the HB-module spectra. Example 4.15 (Cobordism spectra). We define a commutative symmetric ring spectr* *um MO whose stable homotopy groups are isomorphic to the ring of cobordism classes* * of closed manifolds. We set (MO)n = EO(n)+ ^O(n)Sn . Here O(n) is the n-th orthogonal group consisting of Euclidean automorphisms of* * Rn. The space EO(n) is the geometric realization of the simplicial object of topologica* *l groups which 32 STEFAN SCHWEDE in dimension k is the k-fold product of copies of O(n), and where are face maps* * are projec- tions. Thus EO(n) is a topological group with a homomorphism O(n) -! EO(n) comi* *ng from the inclusion of 0-simplices. The underlying space of EO(n) is contractib* *le and has two commuting actions of O(n) from the left and the right. The right O(n)-actio* *n is used to form the orbit space (MO)n, where we think of Sn as the one-point compactifi* *cation of Rnwith its natural left O(n)-action. Thus the space (MO)n still has a left O* *(n)-action, which we restrict to an action of the symmetric group n, sitting inside O(n) a* *s the coordi- nate permutations. Topologically, (MO)n is nothing but the Thom space of the ta* *utological bundle over the space BO(n). The unit of the ring spectrum MO is given by the maps Sn ~=O(n)+ ^O(n)Sn -! EO(n)+ ^O(n)Sn = (MO)n using the `vertex map' O(n) -! EO(n). There are multiplication maps (MO)p ^ (MO)q -! (MO)p+q which are induced from the identification Sp^Sq ~=Sp+q which is equivariant wit* *h respect to the group O(p) x O(q), viewed as a subgroup of O(p + q). The fact that these mu* *ltiplication maps are associative and commutative uses that o for topological groups G and H, the simplicial model of EG comes with a * *natural, associative and commutative isomorphism E(G x H) ~=EG x EH; o the group monomorphisms O(p)xO(q) -! O(p+q) are strictly associative, an* *d the following diagram commutes O(p) x O(q)_______//_O(p + q) twist|| conj.|byfflp,q| fflffl| fflffl| O(q) x O(p)_______//_O(q + p) where the right vertical map is conjugation by the permutation matrix of* * the block permutation Øp,q. In very much the same way we obtain commutative symmetric ring spectra model fo* *r the oriented cobordism spectrum MSO and the spin cobordism spectrum MSpin. The comp* *lex cobordism ring spectrum MU does not fit in here directly; one has to vary the n* *otion of a symmetric spectrum slightly, and consider only symmetric spectra which are de* *fined `in even dimensions'. 4.3. Characterizing module categories over ring spectra. Several of the example* *s of stable model categories in Section 4.1 already come as categories of modules ov* *er suitable rings or ring spectra. This is no coincidence. In fact, every stable model cate* *gory with a single small generator has the same homotopy theory as the modules over a ring * *spectrum. This is an analog of Theorem 2.5, which characterizes module categories over a * *ring as the cocomplete abelian category with a small projective generator. To an object P in a sufficiently nice stable model category C we can associa* *te a symmetric endomorphism ring spectrum EndC(P ); among other things, this ring spectrum com* *es with an isomorphism of graded rings ß*End C(P ) ~= Ho (C)(P, P )* between the homotopy groups of End C(P ) and the morphism of P in the triangula* *ted ho- motopy category of C. The precise result is as follows: MORITA THEORY 33 Theorem 4.16. Let C be a stable model category which is simplicial, cofibrantly* * generated and proper. If C has a small generator P , then there exists a chain of Quillen* * equivalences between C and the model category of EndC(P )-modules, C 'Q Mod- EndC(P ) . Unfortunately, the theorem is currently only known under the above technical* * hypothesis: the stable model category in question should be simplicial (see [Qui67, II.2], * *[Hov99 , 4.2.18]), cofibrantly generated (see [Hov99 , Sec. 2.1] or [SS00, Sec. 2]) and proper (se* *e [BF78 , Def. 1.2] or [HSS , Def. 5.5.2]). The conditions enter in the construction of `good' endo* *morphism ring spectra. I suspect however, that these hypothesis are not essential and can be * *eliminated with a clever use of framing techniques. For example, in [SS02, Sec. 6], we use* * framings to construct function spectra in a arbitrary stable model category; that construct* *ion does how- ever not yield symmetric spectra, and there is no good composition pairing. The* * condition of being a simplicial model category can be removed by appealing to [RSS ] or [* *Dug ] where suitable model categories are replaced by Quillen equivalent simplicial model c* *ategories. This theorem is a special case of the more general result which applies to s* *table model categories with a set of small generators (as opposed to a single small generat* *or), see [SS03, Thm. 3.3.3]. Spectral model categories. In the algebraic situations which we considered i* *n Sections 2 and 3, the key point is to have a good notion of endomorphism ring or endomor* *phism DG ring together with a `tautological' functor Hom A(P, -) : A - ! Mod- EndA (P ) . (4.17) Then it is a matter of checking that when P is a small generator, the functor H* *om (P, -) is either an equivalence of categories (in the context of abelian categories) o* *r induces an equivalence of derived categories (in the context of DG categories). For abelia* *n categories the situation is straightforward, and the ordinary endomorphism ring does the j* *ob. In the differential graded context already a little complication comes in because the * *categorical hom functor Hom A(P, -) need not preserve quasi-isomorphisms in general. For stable model categories, the key construction is again to have an endomo* *rphism ring spectrum EndC(P ) together with a homotopically well-behaved homomorphism * *functor (4.17)to modules over the endomorphism ring spectrum. This is easy for the fol* *lowing class of spectral model categories where composable function spectra are part o* *f the data. A spectral model category is analogous to a simplicial model category [Qui67, II.* *2], but with the category of simplicial sets replaced by symmetric spectra. Roughly speaking* *, a spectral model category is a pointed model category which is compatibly enriched over th* *e stable model category of spectra. In particular there are `tensors' K ^ X and `cotenso* *rs' XK of an object X of C and a symmetric spectrum K, and function symmetric spectra Hom C(* *A, Y ) between two objects of C. The compatibility is expressed by the following axio* *m which takes the place of [Qui67, II.2 SM7]; there are two equivalent `adjoint' forms * *of this axiom, compare [Hov99 , Lemma 4.2.2] or [SS03, 3.5]. (Pushout product axiom) For every cofibration A -! B in C and every cofibrat* *ion K -! L of symmetric spectra, the pushout product map L ^ A [K^A K ^ B -! L ^ B is a cofibration; the pushout product map is a weak equivalence if in addition * *A -! B is a weak equivalence in C or K -! L is a stable equivalence of symmetric spectra. 34 STEFAN SCHWEDE A spectral Quillen pair is a Quillen adjoint functor pair L : C -! D and R :* * D -! C between spectral model categories together with a natural isomorphism of symm* *etric homomorphism spectra Hom C(A, RX) ~= Hom D (LA, X) which on the vertices of the 0-th level reduces to the adjunction isomorphism. * *A spectral Quillen pair is a spectral Quillen equivalence if the underlying Quillen functo* *r pair is an ordinary Quillen equivalence. A spectral model category is the same as a `Sp -model category' in the sense* * of [Hov99 , Def. 4.2.18]; Hovey's condition 2 of [Hov99 , 4.2.18] is automatic since the un* *it S for the smash product of symmetric spectra is cofibrant. Similarly, a spectral Quillen pair i* *s a `Sp -Quillen functor' in Hovey's terminology. Examples of spectral model categories are modu* *le categories over a ring spectrum, and the category of symmetric spectra over a suitable sim* *plicial model category [SS03, Thm. 3.8.2]. A spectral model category is in particular a simplicial and stable model cat* *egory. More- over, for X a cofibrant and Y a fibrant object of a spectral model category C * *there is a natural isomorphism of graded abelian groups ßs*HomC(X, Y ) ~=Ho(C)(X, Y )*. Th* *ese facts are discussed in Lemma 3.5.2 of [SS03]. For of an object P in a spectral model category, the function spectrum of En* *d C(P ) = Hom C(P, P ) is naturally a ring spectrum; the multiplication is a special cas* *e of the compo- sition product O : Hom C(Y, Z) ^ Hom C(X, Y ) - ! Hom C(X, Z) . Also via the composition pairing, the function symmetric spectrum Hom C(P, X) b* *ecomes a right module over the symmetric ring spectrum End C(P ) for any object X. In * *order for the endomorphism ring spectrum EndC(P ) to have the correct homotopy type, the * *object P should be both cofibrant and fibrant. In that case, the ring of homotopy groups* * ß*End C(P ) is isomorphic to Ho(C)(P, P )*, the ring of graded self maps of P in the homoto* *py category of C. Moreover, the homotopy type of the endomorphism ring spectrum then depend* *s only on the weak equivalence type of the object (see [SS03, Cor. A.2.4]). Note that * *this is not completely obvious since taking endomorphisms is not a functor. If P is a cofibrant object of a spectral model category, then the functor Hom C(P, -) : C -! Mod- EndC(P ) is the right adjoint of a Quillen adjoint functor pair, see [SS03, 3.9.3 (i)]. * *The left adjoint is denoted - ^EndC(P)P : Mod- EndC(P ) - ! C . (4.18) For spectral model categories, the proof of Theorem 4.16 is now straightforward* *, and very analogous to the proofs of Theorem 2.5 and Theorem 3.20; indeed, to obtain the * *following theorem, one applies Proposition 3.10 to the total left derived functor of the * *left Quillen functor (4.18). Theorem 4.19. Let C be a spectral model category and P a cofibrant-fibrant obje* *ct. If P is a small generator for C, then the adjoint functor pair Hom C(P, -) and - ^EndC(* *P)P form a spectral Quillen equivalence. The remaining step is worked out in Theorem 3.8.2 of [SS03], which proves th* *at every simplicial, cofibrantly generated, proper stable model category is Quillen equi* *valent to a spectral model category, namely the category Sp(C) of symmetric spectra over C.* * The proof is technical and we will not go into details here. Theorem 4.16 follows by comb* *ining [SS03, MORITA THEORY 35 Theorem 3.8.2] and Theorem 4.19 to obtain a diagram of model categories and Qui* *llen equivalences (the left adjoints are on top) _____1_____// o-^EndC(P)Po_ C oo__Ev_____ Sp(C) __________//_Mod-End C(P ) . 0 HomC(P,-) 4.4. Morita context for ring spectra. Now we come to `Morita theory for ring sp* *ectra', by which we mean the question when two symmetric spectra have Quillen equivalen* *t module categories. For ring spectra, there is a significant difference between a Quill* *en equivalence of the module categories and an equivalence of the homotopy categories. The for* *mer implies the latter, but not conversely. The same kind of difference already exists for* * differential graded rings, but it is not visible for ordinary rings (see Example 4.5 (5)). We call a symmetric spectrum X flat if the functor X ^ - preserves stable eq* *uivalences of symmetric spectra. If X is cofibrant, or more generally S-cofibrant in the s* *ense of [HSS , 5.3.6], then X is flat, see [HSS , 5.3.10]. Every symmetric ring spectrum has a* * `flat resolution': we may take a cofibrant approximation in the stable model structure of symmetri* *c ring spectra [HSS , 5.4.3]; the underlying symmetric spectrum of the approximation i* *s cofibrant, thus flat. Theorem 4.20. (Morita context) The following are equivalent for two symmetric r* *ing spectra R and S. (1) There exists a chain of spectral Quillen equivalences between the catego* *ries of R- modules and S-modules. (2) There is a small, cofibrant and fibrant generator of the model category * *of S-modules whose endomorphism ring spectrum is stably equivalent to R. Both conditions are implied by the following condition. (3) There exists an R-S-bimodule M such that the derived smash product funct* *or - ^LRM : Ho(Mod- R) -! Ho(Mod- S) is an equivalence of categories. If moreover R or S is flat as a symmetric spectrum, then all three condition* *s are equivalent. Again there is a version of the Morita context 4.20 relative to a commutativ* *e symmetric ring spectrum k. In that case, R and S are k-algebras, condition (1) refers to* * k-linear spectral Quillen equivalences, condition (2) requires a stable equivalence of k* *-algebras, the bimodule M in (3) has to be k-symmetric and in the addendum, one of R or S has * *to be flat as a k-module. Proof of Theorem 4.20.(2)=) (1): Modules over a symmetric ring spectrum form a * *spec- tral model category; so this implication is a special case of Theorem 4.19, com* *bined with the fact that stably equivalent ring spectra have Quillen equivalent module cat* *egories. (1) =) (2): To simplify things we suppose that there exists a single spectra* *l Quillen equivalence __________//_ Mod- R oo_________ Mod-S 36 STEFAN SCHWEDE with the left adjoint. The general case of a chain of such Quillen equivalenc* *es is treated in [SS03, Thm. 4.1.2]. We choose a trivial cofibration ' : (R) -! (R)fof S-mo* *dules such that M := (R)f is fibrant; since M is isomorphic in the homotopy category of S* *-modules to the image of the free R-module of rank one under the equivalence of homotopy* * categories, M is a small generator for the homotopy category of S-modules. It remains to sh* *ow that the endomorphism ring spectrum of M is stably equivalent to R. We define EndS ('), the endomorphism ring spectrum of the S-module map ' : * *(R) -! M, as the pullback in the diagram of symmetric spectra EndS(')_______//_EndS(M) (4.21) | | * | |' fflffl| fflffl| R ___'*__//_HomS( (R), M) The right vertical map '* is obtained by applying Hom S(-, M) to the acyclic co* *fibration '; since M is stably fibrant, '* is acyclic fibration of symmetric spectra. Since * * and form a Quillen equivalence, the adjoint ^': R -! (M) of ' is a stable equivalence of * *R-modules. The lower horizontal map '* is the composite stable equivalence HomR(R,^') R ~= Hom R (R, R) --------! Hom R(R, (M)) ~= Hom S( (R), M) . All maps in the pullback square (4.21)are thus stable equivalences and the morp* *hism con- necting EndS(') to R and EndS(M) are homomorphisms of symmetric ring spectra (w* *hereas the lower right corner Hom S( (R), M) has no multiplication). So R is indeed st* *ably equiv- alent, as a symmetric ring spectrum, to the endomorphisms of M. (3)=) (1): If M happens to be cofibrant as a right S-module, then smashing w* *ith M over R is a left Quillen equivalence from R-modules to S-modules. Since we did * *not assume that M is cofibrant over S, we have to be content with a chain of two Quillen e* *quivalences, which we get as follows. Let M be an R-S-bimodule as in condition (3). We choose a cofibrant approxi* *mation ' : Rc -! R in the stable model structure of symmetric ring spectra and we view* * M as an Rc- S-bimodule by restriction of scalars. Then we choose a cofibrant approximation * *Mc -! M as Rc-S-bimodules. Since the underlying symmetric spectrum of Rc is cofibrant [* *SS00, 4.1 (3)], Rc ^ S is cofibrant as a right S-module, and thus every cofibrant Rc-S-bi* *module is cofibrant as a right S-module. In particular, this holds for Mc, and so we have* * a chain of two spectral Quillen pairs oo_-^RcR___ __-^RcMc___// Mod-R __________//_Mod-Rc oo_________ Mod- S . '* HomS(Mc,-) The left pair is a Quillen equivalence since the approximation map ' : Rc -! R * *is a stable equivalence. For every cofibrant Rc-module X, the map X ^Rc Mc - ! X ^Rc M ~=(X ^Rc R) ^R M MORITA THEORY 37 is a stable equivalence. This means that the diagram of homotopy categories and* * derived functors Ho(Mod- Rc) -^LRcRnnnnn PPP-^LRcMcPPP nnn PPPPP wwnnnn ''P Ho (Mod- R)____________________//_Ho(Mod- S) -^LRM commutes up to natural isomorphism. Thus the right Quillen pair above induces a* *n equiv- alence of homotopy categories, so it is a Quillen equivalence. (2)=)(3), assuming that R or S is flat. Let T be a cofibrant and fibrant sm* *all generator of Ho(Mod- S) such that R is stably equivalent to the endomorphism ri* *ng spectrum of T . We choose a cofibrant approximation Rc '-!R in the model category of sym* *metric ring spectra. Since T is cofibrant and fibrant, its endomorphism ring spectrum * *is fibrant. So any isomorphism between R and End S(T ) in the homotopy category of symmetri* *c ring spectra can be represented by a chain of two stable equivalences R --' Rc -'-! End S(T ) . The module T is naturally an End S(T )-S-bimodule, and we restrict the left act* *ion to Rc and view T as an Rc-S-bimodule. We choose a cofibrant replacement T c-~! T as* * an Rc-S-bimodule. Then we set M = R ^Rc T c, an R-S-bimodule. We have no reason to suppose that M is cofibrant as a right S-* *module, so we cannot assume that the functor - ^R M : Mod-R -! Mod-S is a left Quillen * *func- tor. Nevertheless, smashing with M over R takes stable equivalences between co* *fibrant R-modules to stable equivalences, so it has a total left derived functor - ^LRM : Ho (Mod- R) - ! Ho (Mod- S) ; we claim that this functor is an equivalence. Since Rc is cofibrant as a symmetric ring spectrum, it is also cofibrant as * *a symmetric spectrum [SS00, 4.1 (3)], so Rc^ Sop models the derived smash product of R and * *S. If one of R or S are flat, then R ^ Sop also models the derived smash product, so that* * the map Rc^ Sop -! R ^ Sop is a stable equivalence of symmetric ring spectra. Since T cis cofibrant as an * *Rc^Sop-module, the induced map T c = (Rc^ Sop)Rc^SopT c -! (R ^ Sop)Rc^SopT c ~=R ^Rc T c = M (4.22) is a stable equivalence. We smash the stable equivalence (4.22)from the left wi* *th an Rc- module X to get a natural map of S-modules X ^Rc T c -! X ^Rc M ~=(X ^Rc R) ^R M . (4.23) If X is cofibrant as an Rc-module, then X ^Rc- takes stable equivalences of lef* *t Rc-modules to stable equivalences, so in this case, the map (4.23)is a stable equivalence.* * Thus the 38 STEFAN SCHWEDE diagram of triangulated categories and derived functors Ho(Mod- Rc) (4.24) -^LRcRnnnnn PPPP-^LRcTcPP nnnn PPP wwnnn PP''P Ho (Mod- R)____________________//_Ho(Mod- S) -^LRM commutes up to natural isomorphism. The left diagonal functor in the diagram (4.24)is derived from extensions of* * scalars along stable equivalences of ring spectra; such extension of scalars is a left Quille* *n equivalences, so the derived functor - ^LRcR is an exact equivalence of triangulated categories.* * We argued in the previous implication that any cofibrant Rc ^ Sop-module such as T chas an u* *nderlying cofibrant right S-module. So smashing with T cover Rc is a left Quillen functo* *r. Since T cis isomorphic to T in the homotopy category of S-modules, T cis a small gene* *rator of Ho (Mod- S). So the right diagonal derived functor in (4.24)is an exact equival* *ence by Propo- sition 3.10, applied to the free Rc-module of rank 1. So we conclude that the l* *ower horizontal functor in the diagram (4.24)is also an exact equivalence of triangulated categ* *ories. This establishes condition (3). 4.5. Examples. (1) Matrix ring spectra. As for classical rings (compare Example* * 2.3), matrix ring spectra give rise to the simplest kind of Morita equivalence. Inde* *ed over any a ring spectrum R, the `free module of rank n', i.e., the wedge of n copies of * *R, is a small generator for the homotopy category of R-modules. The endomorphism ring spectru* *m of a (stably fibrant replacement) of R x n is the n x n matrix ring spectrum as we d* *efined it in Example 4.13. So R and Mn(R) = End R((R x n)f) are Morita equivalent as ring spectra. (2) Upper triangular matrices. In Example 3.25 we saw that the upper triangu* *lar 3x3 matrices over a field are derived equivalent, but not Morita equivalent, to its* * sub-algebra of matrices of the form 80 1 9 < x11 x12 x13 = @ 0 x22 0 A | xij2 k . : 0 0 x ; 33 We cannot directly define the algebra of 3 x 3 matrices over a ring spectrum; t* *he problem is that t