"Brave NewÄ lgebraic Geometry and global derived moduli spaces of ring spectra Bertrand To"en Gabriele Vezzosi Laboratoire Emile Picard Dipartimento di Matematica UMR CNRS 5580 Universit`a di Bologna Universit'e Paul Sabatier, Toulouse Italy France September 8, 2003 Abstract We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the s* *pecial context where the base symmetric monoidal model category is that of spectra S, i.e. what* * might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topolo* *gies on the model category of commutative algebras in S, the associated theories of geometri* *c S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Ge* *ometry), and finally show how to define global moduli spaces of associative ring spectra structures * *as geometric S-stacks. Key words: Sheaves, stacks, ring spectra, model categories. MSC-class: 55P43; 14A20; 18G55; 55U40; 18F10. Contents 1 Introduction * * 2 2 Brave new sites * * 4 2.1 The brave new Zariski topology . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . 4 2.2 The brave new 'etale topology . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . 10 2.3 Standard topologies . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . 13 3 S-stacks and geometric S-stacks * * 13 3.1 Some descent theory . . . . . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . 15 3.2 The S-stack of perfect modules . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . 17 3.3 Geometric S-stacks . . . . . . . . . . . . . . . . . . . . . . . . . .* * . . . . . . . . . . . . 17 1 4 Derived moduli spaces in algebraic topology as S-stacks * * 19 4.1 The brave new group scheme RAut_(M) . . . . . . . . . . . . . . . . .* * . . . . . . . . . 20 4.2 Moduli of algebra structures . . . . . . . . . . . . . . . . . . . . . * *. . . . . . . . . . . . 21 1 Introduction Homotopical Algebraic Geometry is a kind of algebraic geometry where the affine* * objects are given by commutative ring-like objects in some homotopy theory (technically speaking,* * in a symmetric monoidal model category); these affine objects are then glued together accordin* *g to an appropri- ate homotopical modification of a Grothendieck topology, and, more generally, w* *e allow ourselves to consider more flexible objects like stacks, in order to deal with appropriate m* *oduli problems. This theory is developed in full generality in [To-Ve 1, To-Ve 2](see also [To-Ve 3]* *). Our motivations for such a theory came from a variety of sources: first of all, on the algebro-geom* *etric side, we wanted to produce a sufficiently functorial language in which the so called Derived Mo* *duli Spaces foreseen by Deligne, Drinfel'd and Kontsevich could really be constructed; secondly, on * *the topological side, we thought that maybe the many recent results in Brave New Algebra, i.e. in (co* *mmutative) algebra over structured ring spectra (in any one of their brave new symmetric monoidal * *model categories, see e.g. [Ho-Sh-Sm, EKMM ]), could be pushed to a kind of Brave New Algebraic Geo* *metry in which one could take advantage of the possibility of gluing these brave new rings tog* *ether into an actual geometric object, much in the same way as commutative algebra is helped (and ge* *neralized) by the existence of algebraic geometry. Thirdly, on the motivic side, following a sug* *gestion of Y. Manin, we wished to have a sufficiently general theory in order to study algebraic geo* *metry over the recent model categories of motives for smooth schemes over a field ([Hu , Ja, Sp]). The purpose of this paper is to present the first steps in the second type * *of applications mentioned above, i.e. a specialization of the general framework of homotopical algebraic * *geometry to the context of stable homotopy theory. Our category S - Affof brave new affine objects will* * therefore be defined as the the opposite model category of the category of commutatve rings in the c* *ategory S of symmetric spectra ([Ho-Sh-Sm]). We first define and study various model topologies defined on S - Aff. They* * are all extensions, to different extents, of the usual Grothendieck topologies defined on the categ* *ory of (affine) schemes, like the Zariski and 'etale ones. With any of these model topologies ø at hand, we define and give the basic * *properties of the cor- responding model category of S-stacks, understood in the broadest sense as non * *necessarily truncated presheaves of simplicial sets on S - Affsatisfying a homotopical descent (i.e. * * sheaf-like) condition with respect to ø-(hyper)covers. As in algebraic geometry one finds useful to s* *tudy those stacks which arise as quotients by smooth groupoids (these are called Artin algebraic stacks* *), we also define a brave new analog of these and call them geometric S-stacks, to emphasize that such S-* *stacks host a rich geometry very close to the geometric intuition learned in algebraic geometry. I* *n particular, given a geometric S-stack F , it makes sense to speak about quasi-coherent and perfect * *modules over F , about the K-theory of F , etc.; various properties of morphisms (e.g. smooth, 'etale,* * proper, etc.) between geometric S-stacks can likewise be defined. Stacks were introduced in algebraic geometry mainly to study moduli problem* *s of various sorts; they provide actual geometric objects (rather than sets of isomorphisms classes* * or coarse moduli schemes) which store all the fine details of the classification problem and on * *which a geometry very similar to that of algebraic varieties or schemes can be developed, the two asp* *ects having a fruitful 2 interplay. In a similar vein, in our brave new context, we give one example of * *a moduli problem arising in algebraic topology (the classification of A1 -ring spectra structures on a g* *iven spectrum M) that can be studied geometrically through the geometric S-stack RAss_M it represents* *. We wish to empha- size that instead of a discrete homotopy type (like the ones studied, for diffe* *rent moduli problems, in [Re, B-D-G, G-H]), we get a full geometric object on which a lot of interest* *ing geometry can be performed. The geometricity of the S-stack RAss_M is actually the main theorem * *of this paper (see theorem 4.2.1). We think that this approach can be extended to other, more inte* *rested and involved, moduli problems algebraic topologists are interested in, and perhaps this riche* *r geometry could be of some help in answering, or at least in formulating in a clearer way, some of th* *e deep questions raised by the recent progress in stable homotopy theory (see [G ]). Acknowledgments. We wish to thank Haynes Miller for his invitation to spea* *k at the INI Workshop on Elliptic Cohomology and Higher Chromatic Phenomena (Cambridge UK, D* *ecember 2003) and Bill Dwyer for his encouraging comments. We would also like to thank * *Michael Mandell, Haynes Miller, Peter May, John Rognes, Stefan Schwede and Neil Strickland for e* *xtremely helpful discussions and suggestions. Notations. To fix ideas, we will work in the category S := Sp of symmetr* *ic spectra (see [Ho-Sh-Sm]), but all the constructions of this paper will also work, possibly w* *ith minor variations (see [Sch]), for other equivalent theories (e.g for the category of S-modules o* *f [EKMM ]). We will consider S as a symmetric monoidal simplicial model category (for the smash pro* *duct - ^ -) with the Shipley-Smith S-model structure (see [Shi, Thm. 2.4]). We define S - Alg as the category of (associative and unital) commutative m* *onoids objects in S, endowed with the S-alg model structure ([Shi, Thm. 3.2]); we will simply cal* *l them commutative S-algebras instead of the more correct but longer, commutative symmetric ring s* *pectra. For any commutative S-algebra A, we will denote by A - Alg the under-category A=S - Alg* *, whose objects will be called commutative A-algebras. Finally, if A is a commutative S-algebra* *, A - Mod will be the category of A-modules with the A-model structure ([Shi, Thm. 2.6 (2)]). This mo* *del category is also a symmetric monoidal model category for the smash product - ^A - over A. For a morphism of commutative S-algebras, f : A -! B one has a Quillen adju* *nction f* : A - Mod -! B - Mod A - Mod - B - Mod : f*, where f*(-) := - ^A B is the base change functor. We will denote by Lf* : Ho (A - Mod ) -! Ho (B - Mod ) Ho (A - Mod ) - Ho (B - Mod ) :* * Rf* the induced derived adjunction on the homotopy categories. Our references for model category theory are [Hi, Ho]. For a model category* * M with equivalences W , the set of morphisms in the homotopy category Ho (M) := W -1M will be denot* *ed by [-, -]M , or simply by [-, -] if the context is clear. The (homotopy) mapping spaces in M* * will be denotedby Map M(-, -). When M is a simplicial model category, the simplicial Hom's (resp* *. derived simplicial Hom's) will be denoted by Hom__M(resp. RHom__M), or simply by Hom__(resp. RHom_* *_) if the context is clear. Recall that in this case one can compute Map M(-, -) as RHom__M(-, -). Finally, for a model category M and an object x 2 M we will often use the c* *oma model categories x=M and M=x. When the model category M is not left proper (resp. is not right p* *roper) we will always assume that x has been replaced by a cofibrant (resp. fibrant) model bef* *ore considering x=M (resp. M=x). More generally, we will not always mention fibrant and cofibrant* * replacements and suppose implicitly that all our objects are fibrant and/or cofibrant when requi* *red. 3 Since we wish to concentrate on applications to stable homotopy theory, som* *e general construc- tions and details about homotopical algebraic geometry will be omitted by refer* *ring to [To-Ve 1]. For a few of the results presented we will only give here sketchy proofs; full proo* *fs will appear in [To-Ve 2]. 2 Brave new sites In this section we present two model topologies on the (opposite) category of c* *ommutative S-algebras. They are brave new analogs of the Zariski and 'etale topologies defined on the * *category of usual com- mutative rings and will allow us to define the brave new Zariski and 'etale sit* *es. We denote by S - Affthe opposite model category of S - Alg. If M is a model category we say that an object x in M is finitely presented* * if, for any filtered direct system of objects {zi}i2Jin M, the natural map colimiMap M(x, zi) -! MapM (x, colimizi) is an isomorphism in the homotopy category of simplicial sets. Definition 2.0.1A morphism A ! B of commutative S-algebras is finitely presente* *d if it is a finitely presented object in the model under-category A=(S - Alg) = A - Alg. An A-module* * E is finitely pre- sented or perfect if it is a finitely presented object in the model category A * *- Mod . 2.1 The brave new Zariski topology Definition 2.1.1 o A morphism f : A -! B in S - Algis called a formal Zariski* * open immer- sion if the induced functor Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is fully * *faithful. oA morphism f : A -! B is a Zariski open immersion if S - Algis it is a for* *mal Zariski open immersion and of finite presentation (as a morphism of commutative S-algeb* *ras). oA family {fi: A -! Ai}i2Iof morphisms in S - Algis called a (formal) Zaris* *ki open covering if it satisfies the following two conditions. - Each morphism A -! Aiis a (formal) Zariski open immersion. - There exist a finite subset J I such that the family of inverse ima* *ge functors {Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism * *if and only if its images by all the Lf*j's are isomorphisms). Example 2.1.2 If A 2 S - Algand E is an A-module such that the associated Bous* *field localization LE is smashing (i.e. the natural transformation LE(-) ! LEA ^LA(-) is an isomo* *rphism), then A ! LEA (which is a morphism of commutative S-algebras by e.g. [EKMM , xVIII.2* *]) is a formal Zariski open immersion. This follows immdiately from the fact that Ho (LEA - Mo* *d) is equivalent to the subcategory of Ho (A - Mod) consisting of LE-local objects, by [Wo ]. 4 It is easy to check that (formal) Zariski open covering families define a m* *odel topology in the sense of [To-Ve 1, x4.3] on the model category S - Aff. For the reader's convenience * *we recall what this means in the following lemma. Lemma 2.1.3 oIf A -! B is an equivalence of commutative S-algebras then th* *e one element family {A -! B} is a (formal) Zariski open covering. oLet {A -! Ai}i2Ibe a (formal) Zariski open covering of S-algebras and A -!* * B a morphism. Then, the family of homotopy push-outs {B -! B ^LAAi}i2Iis also a (formal)* * Zariski open covering. oLet {A -! Ai}i2Ibe a (formal) Zariski open covering of S-algebras, and for* * any i 2 I let {Ai -! Aij}j2Jibe a (formal) Zariski open covering of S-algebras. Then, t* *he total family {A -! Aij}i2I,j2Jiis again a (formal) Zariski open covering. Proof: Left as an exercise. * * 2 Therefore, by definition, Lemma 2.1.3 shows that (formal) Zariski open cove* *rings define a model topology on the model category S - Aff and so, as proved in [To-Ve 1, Prop. 4.* *3.5], induce a Grothendieck topology on the homotopy category Ho (S - Alg). This model topolog* *y is called the brave new (formal) Zariski topology, and endows S - Affwith the structure of a * *model site in the sense of [To-Ve 1, x4]. This model site, denoted by (S - Aff, Zar) for the brav* *e new Zariski topology, and (S - Aff, fZar) for the brave new formal Zariski topology. They will be ca* *lled the brave new Zariski site and the brave new formal Zariski site. Let Alg be the category of (associative and unital) commutative rings. Let * *us recall the existence of the Eilenberg-Mac Lane functor H : Alg -! S - Alg, sending a commutative ring R to the commutative S-algebra HR such that ß0(HR) =* * R and ßi(HR) = 0 for any i 6= 0. This functor is homotopically fully faithful and the followin* *g lemma shows that our brave new Zariski topology does generalize the usual Zariski topology. Lemma 2.1.4 1.Let R -! R0 be a morphism of commutative rings. The induced * *morphism HR -! HR0is a Zariski open immersion of commutative S-algebras (in the sen* *se of Definition 2.1.1) if and only if the morphism Spec R0-! Spec R is an open immersion o* *f schemes. 2.A family of morphisms of commutative rings, {R -! R0i}i2I, induces a Zaris* *ki covering family of commutative S-algebras {HR -! HR0i}i2I(in the sense of Definition 2.1.1* *) if and only if the family {Spec Ri-! Spec R}i2Iis a Zariski open covering of schemes. Proof: Let us start with the general situation of a morphism f : A -! B of* * commutative S- algebras such that the induced functor Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is * *fully faithful. Let L = Rf* O Lf*, which comes with a natural transformation Id -! L. Then, the ess* *ential image of Rf* consist of objects M in Ho (A - Mod ) such that the localization morphism M* * -! LM is an isomorphism. The Quillen adjunction (f*, f*) extends to a Quillen adjunction on the cate* *gory of commutative algebras f* : A - Alg -! B - Alg A - Alg - B - Alg : f*, also with the property that Rf* : Ho (B - Alg) -! Ho (A - Alg) is fully faithfu* *l. Furthermore, the essential image of this last functor consist of all objects C 2 Ho (A-Alg ) suc* *h that the underlying A- module of C satisfies C ' LC (i.e. the underlying A-module of C lives in the im* *age of Ho(B -Mod )). 5 From these observations, we deduce that for any commutative A-algebra C, th* *e mapping space RHom______A-Alg(B, C) is either empty or contractible; it is non-empty if and o* *nly if the underlying A- module of C belongs to the essential image of Rf*. To prove (1), let us first suppose that f : Spec R0- ! Spec R is an open im* *mersion of schemes. Then, clearly the induced functor on the derived categories f* : D(R0) -! D(R) * *is fully faithful. As there are natural equivalences ([EKMM , IV Thm. 2.4]) Ho(HR - Mod ) ' D(R) Ho (HR0- Mod ) ' D(R0) this implies that the functor Rf* : Ho (HR0- Mod ) -! Ho (HR - Mod ) is also fu* *lly faithful. It only remains to show that HR -! HR0is finitely presented in the sense of Defini* *tion 2.0.1. We will first assume that R0 = Rf for some element f 2 R. The essential im* *age of Rf* : Ho (HR0- Mod ) -! Ho (HR - Mod ) then consists of all objects E 2 Ho (HR - Mod* * ) ' D(R) such that f acts by isomorphisms on the cohomology R-module H*(E). By what we * *have seen at the beginning of the proof, this implies that for any commutative HR-algebra C * *the mapping space RHom______HR-Alg(HR0, C) is contractible if f becomes invertible in ß0(C), and * *empty otherwise. From this one easily deduces that RHom__HR-Alg(HR0, -) commutes with filtered colimi* *ts, or in other words that HR0is a finitely presented HR-algebra in the sense of Definition 2.0.1. In the general case, one can write Spec R0 as a finite union of schemes of * *the form Spec Rf for some elements f 2 R. A bit of descent theory (see x2.1) then allows us to reduc* *e to the case where R0= Rf and conclude. Let us now assume that HR -! HR0is a Zariski open immersion of commutative * *S-algebras. By adjunction (between H and ß0 restricted on connective S-algebras) one sees easi* *ly that R -! R0is a finitely presented morphism of commutative rings. The induced functor on (unbounded) derived categories f* : D(R0) ' Ho (HR0- Mod ) -! D(R) ' Ho (HR - Mod ) is fully faithful. Through the Dold-Kan correspondence, this implies that the Q* *uillen adjunction on the model category of simplicial modules (see [G-J]) f* : sR - Mod -! sR0- Mod sR - Mod - sR0- Mod : f* is such that Lf* O f* ' Id. Let sR - Alg and sR0- Alg be the categories of simp* *licial commuta- tive R-algebras and simplicial commutative R0-algebras, endowed with their natu* *ral model structures (equivalences are and fibration are detected in the category of simplicial modu* *les). Then, the Quillen adjunction f* : sR0- Alg -! sR - Alg sR0- Alg - sR - Alg : f* also satisfies Lf* O f* ' Id, as this is true on the level on simplicial module* *s. In particular, for any simplicial R0-module M, the space of derived derivations LDerR(R0, M) := RHom__sR-Alg=R0(R0, R0 M) ' RHom__sR0-Alg=R0(R0, R0 M* *) ' * is acyclic (here R0 M is the simplicial R0-algebra which is the trivial extens* *ion of R0by M). As a consequence one sees that Illusie's contangent complex LR0=R is acyclic, whic* *h implies that the morphism R -! R0is an 'etale morphism of rings. Finally, using that the functor on the category of modules R0-Mod -! R-Mod* * is fully faithful, one sees that Spec R0-! Spec R is a monomorphism of schemes. Therefore, the mor* *phism of schemes 6 Spec R0-! Spec R is an 'etale monomorphism, and so is an open immersion by [EGA* *-IV , Thm. 17.9.1]. Finally, point (2) is clear if one knows (1) and that Ho (HR - Mod ) ' D(R)* *. 2 Remark 2.1.5 The argument at the beginning of the proof of Lemma 2.1.4 shows t* *hat if f : A ! B is a Zariski open immersion, the functor L(f) := Rf*Lf* is a localization funct* *or on the homotopy category of A-modules in the sense of [HPS , Def. 3.1.1]. And it is also clear * *by definition that L(f) is also smashing ([HPS , Def. 3.3.2]). Let us call a localization functor L on * *Ho (A - Mod) a formal Zariski localization functor over A if L ' L(f) for some formal Zariski open im* *mersion f. Let us also say that a localization functor L on Ho (A - Mod) is a smashing algebra Bousfie* *ld localization over A if L ' LB for some A-algebra B such that LB is smashing (over A). Then it is ea* *sy to verify that in the set of equivalence classes of localization functors on Ho (A - Mod), the su* *bset consisting of formal Zariski localization functors over A coincides with the subset consisting of sm* *ashing algebra Bousfield localizations over A. In fact, if f : A ! B is a Zariski open immersion, LB de* *notes the Bousfield localization with respect to the A-module B, and `B=A : A ! LBA the correspondi* *ng morphism of commutative A-algebras, we have L(f) ' LB ' L(`B=A) because all three locali* *zations have the same category of acyclics. Viceversa, if LC is a smashing algebra Bousfield loc* *alization over A, and `C=A : A ! LCA is the corresponding morphism of commutative A-algebras, one has* * LB ' L(`C=A). Let Affbe the opposite category of commutative rings, and (Aff, Zar) the bi* *g Zariski site. The site (Aff, Zar) can also be considered as a model site (for the trivial model struct* *ure on Aff). Lemma 2.1.4 implies in particular that the functor H : Aff- ! S - Affinduces a continuous m* *orphism of model sites ([To-Ve 1, Def. 4.8.4]). In this way, the site (Aff, Zar) becomes a sub-m* *odel site of (S - Aff, Zar). To finish with the Zariski topology we will now describe a general procedur* *e in order to construct interesting open Zariski immersions of commutative S-algebras using the thechni* *ques of Bousfield localization for model categories. Let A be a commutative S-algebra and M be a A-module. We will assume that M* * is a perfect A-module (in the sense of Definition 2.0.1), or equivalently that it is a stron* *gly dualizable object in the monoidal category Ho (A - Mod ). Perfect A-modules are exactly the retract of f* *inite cell A-modules, see [EKMM , Thm. III-7.9]). Let M[n] = Sn L M the n-th suspension A-module of* * M (here n 2 Z). We denote by D(M[n]) the derived dual of M[n], defined as the derived inter* *nal Hom's of A- modules D(M[n]) := RHOM A-Mod(M[n], A). Consider now the (derived) free commutative A-algebra over D(M[n]), LFA(D(M[n])* *), characterized by the usual adjunction [LFA(D(M[n])), -]A-Alg ' [D(M[n]), -]A-Mod . The model category A - Alg is a combinatorial and cellular model category, and * *therefore one can apply the localization techniques (see e.g. [Hi, Sm ]) in order to invert the * *natural augmentations FA(D(M[n])) -! A for all n 2 Z. One checks easily, using that M is strongly du* *alizable, that the local objects for this localization are the commutative A-algebras B such t* *hat M ^LAB ' 0 in Ho (B - Mod ). The local model of A for this localization will be denoted by A* *M . By definition, it is characterized by the following universal property: for any commutative A-alg* *ebra B, the mapping space RHom__A-Alg(AM , B) is contractible if B ^LAM ' 0 and empty otherwise. In* * other words, for any commutative S-algebra B the natural morphism RHom__S-Alg(AM , B) -! RHom__S-Alg(A, B) 7 is equivalent to an inclusion of connected components and its image consists of* * morphisms A -! B in Ho (S - Alg) such that B ^LAM ' 0. Lemma 2.1.6 With the above notations, the morphism A -! AM is a Zariski open* * immersion. Proof: Let us start by showing that AM is a finitely presented commutative * *A-algebra. Let {Bi}i2Ibe a filtered system of commutative A-algebras and B = colimiBi.* * We assume that B ^LAM ' 0, and we need to prove that there exists an i 2 I such that Bi^LAM ' * *0. By assumption, the two points Id and 0 are the same in ß0(REnd_B-Mod (M ^LA* *B)). But, as M is a perfect A-module one has ß0(REnd_B-Mod (M ^LAB)) ' colimi2Iß0(REnd_Bi-Mod(M ^LABi)). This implies that there is some index i 2 I such that Id and 0 are homotopic in* * REnd_Bi-Mod(M^LABi), and therefore that M ^LABiis contractible. It remains to prove that the induced functor Ho (AM - Mod ) -! Ho (A - Mod * *) is fully faithful. To see this, one first notice that by definition the functor Ho (AM - Alg) -! H* *o (A - Alg) is fully faithful, and therefore so is Ho (AM - Alg=AM ) -! Ho (A - Alg=AM ). For any tw* *o AM -modules N and P , one consider the trivial extensions AM _ N and AM _ P of AM by N and* * P (these are AM -augmented commutative AM -algebras). Then, one has a natural equivalence RHom__AM -Alg=AM(AM _ N, AM _ P ) ' RHom__AM -Mod(N, P ). Furthermore one has a natural fiber sequence RHom____A-Mod(N, P_)__//RHom__A-Alg=AM(AM _ N, AM _ P_)_//_RHom__A-Alg=AM(AM ,* * AM _.P ) But, as RHom__A-Alg=AM(AM , AM _ P ) ' RHom__AM -Alg=AM(AM , AM _ P ) ' * this * *shows that the natural morphism RHom__AM -Mod(N, P ) -! RHom__A-Mod(N, P ), is an equivalence and therefore that Ho (AM - Alg) -! Ho (A - Alg) is fully fai* *thful. 2 An important property of the localization A -! AM is the following fact. Lemma 2.1.7 Let A be a commutative S-algebra, and M be a perfect A-module. Th* *en the essential image of the fully faithful functor Ho (AM - Mod ) -! Ho (A - Mod ) consists of all A-modules N such that M ^LAN ' D(M) ^LAN ' 0. Note that since M is perfect, then for any A-module N, M ^LAN ' 0 iff D(M) * *^LAN ' 0, so the two conditions in the lemma are actually one. Moreover, AM ' AD(M) in Ho (A - A* *lg). Proof: As every AM -module N can be constructed by homotopy colimits of fre* *e AM -modules and - ^LAM commutes with homotopy colimits, it is clear that AM ^LAM ' 0 implies N * *^LAM ' 0. Since AM ^LAD(M) ' D(AM ^LAM) ' 0 (here the second derived dual is in the category of* * AM -modules), the same argument shows that N ^LAD(M) ' 0. Conversely, let N be an A-module such that N ^LAM ' N ^LAD(M) ' 0. By defi* *nition, the commutative A-algebra A -! AM is obtained as a local model of A ! A when one in* *verts the set of 8 morphisms LFA(D(M[n])) -! A, for any n 2 Z. It is well known (see e.g. [Hi, x4]* *) that such a local model can be obtained by a transfinite composition of homotopy push-outs of the* * form Aff___________________//_Aff+1OOOO | | | | | | @ p L LFA(D(M[n])) ____//_ p L LFA(D(M[n])) in the category of A-algebras. From this description, and the fact that - ^LAM * *commutes with ho- motopy colimits, one sees that the adjunction morphism N -! N ^LAAM is an equi* *valence because by assumption on N, the natural morphism N ' N ^LAA -! N ^LALFA(D(M[n])) is an * *equivalence. 2 Lemma 2.1.7 allows us to interpret geometrically AM as the open complement* * of the support of the A-module M. Lemma 2.1.7 also has a converse whose proof is left as an exerc* *ise. Lemma 2.1.8 Let f : A -! B be a morphism of commutative S-algebras and M be a* * perfect A- module. We suppose that the functor Rf* : Ho (B -Mod ) -! Ho (A-Mod ) is full* *y faithful and that its essential image consists of all A-modules N such that N ^LAM ' 0. Then, the* * two commutative A-algebras B and AM are equivalent (i.e. isomorphic in Ho (A - Alg)). Remark 2.1.9 One should be careful that even if the Eilenberg-Mac Lane functor* * H embeds (Aff, Zar) in (S - Aff, Zar) as model sites, there exist commutative rings R and Zariski o* *pen coverings HR -! B in S - Affsuch that B is not of the form HR0for some commutative R-algebra R0. * *One example is given by taking R to be C[X, Y ], and considering the localized commutative * *HR-algebra (HR)M (in the sense above), where M is the perfect R-module R=(X, Y ) ' C. If (HR)M * *were of the form HR0for a Zariski open immersion Spec R0-! Spec R, then for any other commutativ* *e ring R00, the set of scheme morphisms Hom(Spec R00, Spec R0) would be the subset of Hom(Spec * *R00, A2) consisting of morphisms factoring through A2 - {0}. This would mean that Spec R0' A2 - {0}* *, which is not possible as A2-{0} is a not an affine scheme. This example is of course the sam* *e as the example given in [To, x2.2] of a 0-truncated affine stack which is not an affine scheme. Thes* *e kind of example shows that there are much more affine objects in homotopical algebraic geometry than * *in usual algebraic geometry. Remark 2.1.10 1.Note that Lemma 2.1.7 shows that the localization process (A, M) /o////ooA* *Mis in some sense ö rthogonal" to the usual Bousfield localization process (A, M) /////oooL* *MiAn that the local objects for the former are exactly the acyclic objects for the latter. To* * state everything in terms of Bousfield localizations, this says that LAM -local objects are ex* *actly LM -acyclic objects (compare with Remark 2.1.5). Note that however, while the Bousfield local* *ization is always defined for any A-module M, the commutative A-algebra AM probably does not* * exist unless M is perfect. 2.Let Sp be the p-local sphere. If f : Sp ! B is any formal Zariski open imm* *ersion then L := Rf*Lf* is clearly a smashing localization functor in the sense of [HPS , x* *3]. Its category C of perfect1 acyclics (i.e. perfect objects X in Ho (Sp - Mod ) such that * *LX is null) is then a localizing thick subcategory of the homotopy category Ho (Sp - Mod perf)* * of the category of perfect Sp-modules, and therefore by [H-S] it is equivalent to the categor* *y Cn of perfect E(n)- acyclics, for some 0 n < 1, where E(n) is the n-th Johnson-Wilson Sp-mod* *ule (see e.g. [Rav]); 1____________________________________ The word finite instead of perfect would be more customary in this setting. 9 in other words L and Ln := LE(n)are both smashing localization functors on* * Ho (Sp - Mod ) having the same subcategory of finite acyclics. Therefore, if we assume (* *one of the form of) the Telescope conjecture (see [Mil]), we get that Ln and L have equivalent* * categories of acyclics and so have equivalent categories of local objects. But the category of l* *ocal objects for L is equivalent to the category Ho (B - Mod ) (since Rf* is fully faithful by h* *ypothesis) and the category of local objects for Ln is equivalent to the category Ho ((LnSp) * *- Mod ), by [Wo ] since Ln is smashing. This easily implies that the two commutative Sp-algebras * *B and LnSp are equivalent (i.e. isomorphic in Ho (Sp- Alg)). In conclusion, one sees that if the Telescope conjecture is true, then, up* * to equivalence of Sp- algebras, the only (non-trivial) formal Zariski open immersions for Sp are* * given by the family U := {Sp ! LnSp}0 n<1. This example shows that the formal Zariski topology might be better suited* * in certain contexts than the Zariski topology itself (e.g. it is not clear that there exists a* *ny non-trivial Zariski open immersion of Sp, i.e. that the morphisms of commutative S-algebras Sp -! L* *nSp are of finite presentation). Note however that the family U is not a formal Zariski cov* *ering according to Definition 2.1.1 because the family of base-change functors n o (-) ^LSpLnSp : Ho (Sp- Mod) -! Ho (LnSp- Mod) 0 n<1 is not conservative; in fact, as Neil Strickland pointed out to us, the Br* *own-Comenetz dual I of Sp is a non-perfect non-trivial Sp-module which is nonetheless Ln-acyclic * *for any n. However, it is true that the family of base-changes above is conservative when restric* *ted to the (homotopy) categories of perfect modules. Therefore, one could modify the second cov* *ering condition in Definition 2.1.1, by only requiring the property of being conservative on * *the subcategories of perfect modules and relaxing the finiteness of J; let us call this modifie* *d covering condition formal Zariski covering-on-finites condition. Then, U is a formal Zariski* * covering-on-finites family and indeed the unique one, up to equivalences of Sp-algebras, if th* *e Telescope conjecture holds. 3.The previous example also shows that the commutative S-algebras LnSp are l* *ocal for the formal Zariski topology (again assuming the Telescope conjecture). Indeed, for an* *y formal Zariski open covering {LnSp -! Bi}i2Ithere is an i such that LnSp -! Biis an equivalenc* *e of commutative S-algebras. 2.2 The brave new 'etale topology Notions of 'etale morphisms of commutative S-algebras has been studied by sever* *al authors ([Ro1, MC-Min ]). In this paragraph we present the definition that appeared in [To-Ve* * 1] and was used there in order to define the 'etale K-theory of commutative S-algebras. We refer to [Ba ] for the notions of topological cotangent spectrum and of * *topological Andr'e-Quillen cohomology relative to a morphism A ! B of commutative S-algebras, except for s* *lightly different notations. We denote by L B=A 2 Ho (B - Mod ) the topological cotangent spectru* *m (denoted as B=A in [Ba ]) and, for any B-module M, by LDerA(B, M) := RHom__A-Alg=B(B, B _ M) the derived space of topological derivations from B to M (B _ M being the trivi* *al extension of B by M). Note that there is an isomorphism LDerA(B, M) ' RHom__B-Mod(L B=A, M), natu* *ral in M. 10 Definition 2.2.1 o Let f : A -! B be a morphism of commutative S-algebras. - The morphism f is called formally 'etale if L B=A ' 0. - The morphism f is called 'etale if it is formally 'etale and of finit* *e presentation (as a mor- phism of commutative S-algebras). oA family of morphisms {fi : A -! Ai}i2Iin S - Alg is called a (formal) 'et* *ale covering if it satisfies the following two conditions. - Each morphism A -! Aiis (formally) 'etale. - There exists a finite subset J I such that the family of inverse im* *age functors {Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism * *if and only if its images by all the Lf*j's are isomorphisms). As shown in [To-Ve 1, x5.2], (formal) 'etale covering families are stable b* *y equivalences, composi- tions and homotopy push-outs, and therefore define a model topology on the mode* *l category S - Aff. Therefore one gets two model topologies called the brave new 'etale topology an* *d the brave new formal 'etale topology. The corresponding model sites will be denoted by (S - Aff, 'et* *) and (S - Aff, f'et), and will be called the brave new 'etale site and the brave new formal 'etale site. As for the brave new Zariski topology one proves that the brave new 'etale * *topology is a general- ization of the usual 'etale topology. Lemma 2.2.2 1.Let R -! R0 be a morphism of commutative rings. The induced * *morphism HR -! HR0is an 'etale morphism of commutative S-algebras (in the sense of * *Definition 2.2.1) if and only if the morphism Spec R0-! Spec R is an 'etale morphism of sche* *mes. 2.A family of morphisms of commutative rings, {R -! R0i}i2I, induces an 'eta* *le covering family of commutative S-algebras {HR -! HR0i}i2I(in the sense of Definition 2.2.1* *) if and only if the family {Spec Ri-! Spec R}i2Iis an 'etale covering of schemes. Proof: This is proved in [To-Ve 1, x5.2]. * * 2 Let Aff be the opposite category of commutative rings, and (Aff, 'et) the b* *ig 'etale site. The site (Aff, 'et) can also be considered as a model site (for the trivial model struct* *ure on Aff). Lemma 2.2.2 shows in particular that the Eilenberg-Mac Lane functor H : Aff -! S - Affinduc* *es a continuous morphism of model sites ([To-Ve 1, x4.8]). In this way, the site (Aff, 'et) bec* *omes a sub-model site of (S - Aff, 'et). Another important fact is that the brave new 'etale topology is finer than * *the brave new Zariski topology. Lemma 2.2.3 1.Any formal Zariski open immersion of commutative S-algebras i* *s a formally 'etale morphism. 2.Any Zariksi open immersion of commutative S-algebras is an 'etale morphism. 3.Any (formal) Zariski open covering of a commutative S-algebra is a (formal* *) 'etale covering. 11 Proof: Only (1) requires a proof, and the proof will be similar to the one* * of Lemma 2.1.4 (2). Let f : A -! B be a formal Zariski open immersion of commutative S-algebras. A* *s the functor Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is a full embedding so is the induced func* *tor Rf* : Ho (B - Alg ) -! Ho (A - Alg). By definition of topological derivations one has for an* *y B-module M, LDerA(B, M) = RHom__A-Alg=B(B, B _ M). This and the fact that Rf* is fully fait* *hful imply that LDerA(B, M) = RHom__A-Alg=B(B, B _ M) ' RHom__B-Alg=B(B, B _ M) ' *, and therefore that L B=A ' 0. * * 2 Lemma 2.2.3 implies that the identity functor of S - Affdefines a continuou* *s morphism between model sites (S - Aff, Zar) -! (S - Aff, 'et), which is a base change functor from the brave new Zariski site to the brave new* * 'etale site. The same is true for the formal versions of these sites. To finish this part, we would like to mention a stronger version of the bra* *ve new 'etale topology, called the thh-'etale topology, which is sometimes more convenient to deal with. Definition 2.2.4 o Let f : A -! B be a morphism of commutative S-algebras. - The morphism f is called formally thh-'etale if for any commutative A* *-algebra C the map- ping space RHom__A-Alg(B, C) is 0-truncated (i.e. equivalent to a dis* *crete space). - The morphism f is called thh-'etale if it is formally thh-'etale and * *of finite presentation (as a morphism of commutative S-algebras). oA family of morphisms {fi: A -! Ai}i2Iin S - Algis called a (formal) thh-'* *etale covering if it satisfies the following two conditions. - Each morphism A -! Aiis (formally) thh-'etale. - There exists a finite subset J I such that the family of inverse im* *age functors {Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism * *if and only if its images by all Lf*jare isomorphisms). It is easy to check that (formal) thh-'etale coverings define a model topol* *ogy on the model cate- gory S - Aff, call the (formal) thh-'etale topology. The model category S - Aff* *together with these topologies will be called the brave new thh-'etale site and the brave new forma* *l thh-'etale site, denoted by (S - Aff, thh-'et) and (S - Aff, fthh-'et), respectively. An equivalent way * *of stating the formal thh- 'etaleness condition for A ! B is to say that the natural map B ! S1 LB in Ho(A* *-Alg ) is an isomor- phism. By [MSV ], this is therefore equivalent to require that the canonical ma* *p B ! THH (B=A, B) is an isomorphism in Ho(A-Alg ), where THH denotes the topological Hochschild coho* *mology spectrum (see e.g. [EKMM , xIX]). This explains the name of this topology and, since as* * observed in [MC-Min ] the Goodwillie derivative of THH is the suspension of the topological Andr'e-Q* *uillen spectrum TAQ (where, for any B-module M, TAQ (B=A; M) is defined as the derived internal Hom* * from L B=A to M in the model category of B-modules) also shows that (formal) thh-'etale morph* *isms are (formal) 'etale morphisms. Therefore the identity functor induces continuous morphisms o* *f model sites (S - Aff, thh-'et) -! (S - Aff, 'et) (S - Aff, fthh-'et) -! (S - * *Aff, f'et). We refer to [MC-Min ] for more details on the notion of thh-'etale morphisms. 12 2.3 Standard topologies Standard model topologies on S - Affare obvious extensions of usual Grothendiec* *k topologies on affine schemes. They are defined in the following way. Let ø be one of the usual Grothendieck topologies on affine schemes (i.e. Z* *ariski, Nisnevich, 'etale or faithfully flat). Definition 2.3.1A family of morphisms of commutative S-algebras {A -! Bi}i2Iis * *a standard ø-covering (also called strong ø-covering) if it satisfies the following two co* *nditions. oThe induced family of morphisms of schemes {Spec ß0(Bi) -! Spec ß0(A)}i2Ii* *s a ø-covering of affine schemes. oFor any i 2 I the natural morphism of ß0(Bi)-modules ß*(A) i0(A)ß0(Bi) -! ß*(Bi) is an isomorphism. Its easy to check that this defines a model topology øs on S - Aff, called * *the standard ø-topology. The model site (S - Aff, øs) may be called the brave new standard-ø site. The i* *mportance of stan- dard topologies is that all øs-coverings of commutative S-algebras of the form * *HR comes from usual ø-coverings of the scheme Spec R. Its behavior is therefore very close to the g* *eometric intuition one gets in Algebraic Geometry. Finally, let us also mention the semi-standard (or semi-strong) model topol* *ogies. A family of mor- phisms of commutative S-algebras {A -! Bi}i2Iis a semi-standard ø-covering (als* *o called semi-strong ø-covering) if the induced family of morphism of commutative graded rings {ß*(A* *) -! ß*(Bi)}i2Iis a ø-covering. This also defines a model topology øss on S - Aff. Both the standard and semi-standard type model sites (and S-stacks over the* *m, see Section 3) could be of some interest in the study of geometry over even, periodic S-algebr* *as (e.g. for elliptic spectra as in [AHS ]). 3 S-stacks and geometric S-stacks Let (M, ø) be a model site (i.e. a model category M endowed with a model topolo* *gy ø in the sense of [To-Ve 1]). Associated to it one has a model category of prestacks M^ and of* * stacks M~,fi. For details concerning these model categories we refer to [To-Ve 1, x4], and for th* *e sake of brevity we only recall the following facts. oThe model category M^ is a left Bousfield localization of the model catego* *ry SSetMop, of simplicial presheaves on M together with the projective levelwise model st* *ructure. The local objects for this Bousfield localization are precisely the simplicial presh* *eaves F : Mop -! SSet which are equivalences preserving. oThe model category M~,fiis a left Bousfield localization ([Hi, x3]) of the* * model category of prestacks M^ , and the localization (left Quillen) functor from M^ to M~,f* *ipreserves (up to equivalences) finite homotopy limits (i.e. homotopy pull-backs). The loc* *al objects for this Bousfield localization are the simplicial presheaves F : Mop- ! SSet which* * satisfy the following two conditions. - The functor F preserves equivalences (i.e. is a local object in M^ ). 13 - For any ø-hypercover U* -! X in the model site (M, ø) ([To-Ve 1, x4.4* *]), the induced morphism F (X) -! F (U*) is an equivalence. There is an associated stack functor a : Ho (M^ ) ! Ho (M~,fi) right adjoi* *nt to the inclusion Ho (M~,fi) ,! Ho (M^ ). oThere is a homotopical variation 2 Rh_: Ho (M) ,! Ho (M^ ) of the Yoneda embedding ([To-Ve 1, x4.2]). Specializing to our present situation, where M = S - Aff, we have one model* * category S - Aff^ of prestacks and zounds of model categories stacks S - Aff~,Zar, S - Aff~,'et, S - Aff~,thh-'et, S - Aff~,fZar, S - Aff~,f'et, S - Aff~,fthh-'et, s ~,'ets ~,ffqcs S - Aff~,Zar, S - Aff , S - Aff , . .e.tc . . . These model categories come with right Quillen functors (the morphism of change* * of sites) S - Aff~,'et__//_S - Aff~,thh-'et//_S -~Aff,Zar//_S -^Aff S - Aff~,f'et_//_S - Aff~,fthh-'et//_S -~Aff,fZar//_S -^Aff S - Aff~,'et__//S - Aff~,f'et//_S - Aff~,fZar//_S -^Aff s___//_ ~,'et___//_ ~,Zar___//_ ^ S - Aff~,'et S - Aff S - Aff S - Aff . .e.tc . . . which allow to compare the various topologies on S - Aff. Definition 3.0.2Let ø be a model topology on the model site S - Aff. oThe model category of S-stacks for the topology ø is S - Aff~,fi. oA simplicial presheaf F 2 SP r(S - Aff) is called an S-stack if it is a lo* *cal object in S - Aff~,fi (i.e. preserves equivalences and satisfies the descent property for ø-hype* *rcovers). oObjects in the homotopy category Ho (S - Aff~,fi) will simply be called S-* *stacks (without refer- ring, unless it is necessary, to the underlying topology). 2____________________________________ If x 2 M, Rh_(x) essentially sends y 2 M to the mapping space MapM(y, x). 14 The category of S-stacks, being the homotopy category of a model category, * *has all kind of ho- motopy limits and colimits. Moreover, one can show that it has internal Hom's. * *Actually, the model category of S-stacks is a model topos in the sense of [To-Ve 1, x3.8] (see also* * [To-Ve 4]), and therefore behaves very much in the same way as a category sheaves (but in a homotopical s* *ense). In practice this is very useful as it allows to use a lot of usual properties of simplicial* * sets in the context of S-stacks (in the same way as a lot of usual properties of sets are true in any topos). The Eilenberg-Mac Lane functor H from commutative rings to commutative S-al* *gebras induces left Quillen functors H!: Aff~,fi0-! S - Aff~,fi, where ø0 is one of the standard topologies on affine schemes (e.g. Zar, 'et, ff* *qc,. . . ), and ø is one of its possible extension to the model category S - Aff(e.g. Zar can be extended to Za* *rsor to Zar, etc.). Here, Aff~,fiis the usual model category of simplicial presheaves on the Grothe* *ndieck site (Aff, ø) (with the projective model structure [Bl]). By deriving on the left one gets a * *functor LH!: Ho (Aff~,fi0) -! Ho (S - Aff~,fi). Therefore, our category of S-stacks receives a functor from the homotopy catego* *ry of simplicial presheaves. In particular, sheaves on affine schemes (and in particular the ca* *tegory of schemes it- self), and also 1-truncated simplicial presheaves (and in particular the homoto* *py category of algebraic stacks) can be all viewed as examples of S-stacks. However, one should be caref* *ul that the functor LH!has no reason to be fully faithful in general, though this is the case for a* *ll the standard extensions (but not semi-standard) described in x2.3 (the reason for this is that all cove* *ring families of some HR are in fact induced from covering families of affine schemes. In particular the* * restriction functor from S - Aff~,fi-! Aff~,fi0will preserve local equivalences.). 3.1 Some descent theory With the notations above, one can compose the Yoneda embedding Rh_: Ho (S - Alg* *)op! Ho (S - Aff^) with the associated stack functor a : Ho (S - Aff^) ! Ho (S - Aff~,fi) and obta* *in the derived Spec functor RSpec : Ho (S - Alg)op= Ho (S - Aff) -! Ho (S - Aff~,fi), for any model topology ø on S - Aff. Definition 3.1.1The topology ø is sub-canonical if the functor RSpec is fully f* *aithful or, equivalently, if for any A 2 S - Alg, Rh_Ais an S-stack. Knowing whether a given model topology ø is sub-canonical or not is known a* *s the descent problem for ø, and in our opinion is a crucial question. Unfortunately, we do not know * *if all the model topologies presented in the previous Section are sub-canonical, and it might be that some * *of them are not. The following lemma gives examples of sub-canonical topologies. Lemma 3.1.2 The (semi-)standard Zariski, Nisnevich, 'etale and flat model top* *ologies of x2.3 are all sub-canonical. Sketch of proof: Let ø be one of these topologies, A be a commutative S-alg* *ebra, and A -! B* be a ø-hypercover ([To-Ve 1, x4.4]). Using that ß*(Bn) is flat over ß*(A) for any * *n, one can check that the cosimplicial ß*(A)-algebra ß*(B*) is again a ø-hypercover of commutative ri* *ngs. By usual descent 15 theory for affine schemes this implies that the cohomology groups of the total * *complex of [n] 7! ß*(Bn) vanish except for H0(ß*(B*)) ' ß*(A). This implies that the spectral sequence f* *or the holim Hp([n] 7! ßq(Bn)) ) ßp-q(holimB*) degenerates at E2 and that ß*(A) -! ß*(holimB*) is an isomorphism. * * 2 Concerning the brave new Zariski topology one has the following partial res* *ult. Lemma 3.1.3 Let {A -! Ai}i2Ibe a finite Zariski covering family of commutativ* *e S-algebras. Let A -! B = _iAibe the coproduct morphism. Let A -! B* be the cosimplicial commuta* *tive A-algebra defined by Bn := B ^LAB ^LA. .^.LAB ________-z_______" (n+1) times (i.e. homotopy co-nerve of the morphism A -! B). Then the induced morphism A -! holimn2Bn is an equivalence. Sketch of proof: By definition of Zariski open immersion it is not hard to * *see that the cosimplicial commutative A-algebra B* is m-coskeletal, where m is the cardinality of I. This* * means the following: let im : m -! be the inclusion functor form the full sub-category of objec* *ts [i] with i m. Then, one has an equivalence of commutative A-algebras B* ' R(im )*i*m(B*) (her* *e (i*m, R(im )*) is the derived adjunction between -diagrams and m -diagrams). From this one deduces* * easily that holimn2 Bn ' holimn2 mBn. In particular, holimn2 Bn is in fact a finite homotopy limit and therefore will* * commute with the base change from A to B, i.e. (holimn2 Bn) ^LAB ' holimn2(Bn ^L B). Now, as the functor Ho (A-Mod ) -! Ho (B -Mod ) is conservative (since the fa* *mily {A -! Ai}i2I is a Zariski covering), one can replace A by B and the Ai by B ^LAAi, and in pa* *rticular one can suppose that A -! B has a section. But, it is well known that any morphism A -!* * B which has a section is such that A ' holimnBn (the section can in fact be used in order to * *construct a retraction). 2 Of course, Lemma 3.1.3 is not enough to check that the brave new Zariski to* *pology is sub-canonical as it only deals with very particular hypercovers, the ~Cech-hypercovers i.e. * *the ones arising as nerves of coverings. However, one can slightly modified our definition of S-stacks by* * asking only descent with respect to ~Cech-hypercovers instead of all hypercovers (this weaker notio* *n of stacks is the one used recently by J. Lurie in [Lu] and also appeared for stacks over Grothendiec* *k sites in [DHI ]). Us- ing this weaker notion of ~Cech S-stacks, Lemma 3.1.3 implies that the brave ne* *w Zariski topology is then sub-canonical. What will be said from here on, will also be correct for th* *is weaker notion of stacks. 16 3.2 The S-stack of perfect modules Let ø be a model topology on S - Aff. One defines the S-prestack Perf_of perfe* *ct modules in the following way. For any commutative S-algebra A, we consider the category Perf(A* *), whose objects are perfect and cofibrant A-modules, and whose morphisms are equivalences of A-* *modules. The pull back functors define a pseudo-functor Perf: S - Alg -! Cat A 7-! Perf(A) (A ! B) 7-! (- ^A B : Perf(A) ! Perf(B)). Making this pseudo-functor into a strict functor from S - Algto Cat ([May , Th.* * 3.4]), and applying the classifying space functor Cat ! SSet, we get a simplicial presheaf denoted * *by Perf_. The following theorem relies on the so called strictification theorem ([To-* *Ve 1, A.3.2]), and its proof will appear in [To-Ve 2]. Theorem 3.2.1 The object Perf_is an S-stack (i.e. satisfies the descent condit* *ion for all ø-hypercovers) iff the model topology ø is subcanonical. Another way to state Theorem 3.2.1 is by saying that ø is subcanonical iff,* * for any commutative S-algebra A, the natural morphism Hom__S-Aff~,ø(Spec A, Perf_) ' Perf_(A) -! RHom__S-Aff~,ø(Spec A, * *Perf_) is an equivalence of simplicial sets. The S-stack of perfect complexes is a brave new analog of the stack of vect* *or bundles, and is of fundamental importance in brave new algebraic geometry. 3.3 Geometric S-stacks In this paragraph we will work with a fixed sub-canonical model topology ø on t* *he model site S - Aff. We will define the notion of geometric S-stack, which roughly speaking are quot* *ients of affine S-stacks by a smooth affine groupoid. They will be brave new generalizations of Artin a* *lgebraic stacks (see [La-Mo ]). In order to state the precise definition, one first needs a notion o* *f smoothness for morphisms of commutative S-algebras. For any perfect S-module M one has the (derived) free commutative S-algebra* * over M, S -! LFS(M). For any commutative S-algebra A, one gets a morphism A -! A ^LSLFS(M) ' LFA(A ^LSM). Any morphism A -! B in Ho (A - Alg) which is isomorphic to such a morphism will* * be called a perfect morphism of commutative S-algebras (and we will also say that B is a pe* *rfect commutative A-algebra). Definition 3.3.1A morphism of commutative S-algebras f : A -! B is called smoot* *h if it satisfies the following two conditions. oThe A-algebra B is finitely presented. 17 oThere exists an 'etale covering family {vi: B ! B0i}i2Iand, for any i 2 I,* * a homotopy commu- tative square of commutative S-algebras f A _____//B u|| vi|| |fflffl fflffl| A0_f0_//_B0i, i where f0iis a perfect morphism, and u is an 'etale morphism. One checks easily that smooth morphisms are stable by compositions and homo* *topy base changes. Furthermore, any 'etale morphism is smooth, and therefore so is any Zariski ope* *n immersion. Assumption: At this point we will assume that the notion of smooth morphism* *s is local with respect to the chosen model topology ø. This assumption will insure that the notion of geometric S-stack, to be def* *ined below, behaves well. Some terminology: oLet us come back to our homotopy category Ho (S - Aff~,fi) of S-stacks, an* *d the Yoneda em- bedding (or derived Spec) RSpec : Ho (S - Alg)op-! Ho (S - Aff~,fi). The essential image of RSpec is called the category of affine S-stacks, wh* *ich is therefore anti- equivalent to the homotopy category of commutative S-algebras. We will als* *o call affine S-stack any object in S - Aff~,fiwhose image in Ho (S - Aff~,fi) is an affine S-st* *ack. Clearly, affine S-stacks are stable by homotopy limits (indeed holimi(RSpec Ai) ' RSpec (h* *ocolimiAi)). oA morphism of affine S-stacks is called smooth (over S) (resp. 'etale, a Z* *ariski open immersions . . . ) if the corresponding one in Ho (S - Alg) is so. oA Segal groupoid object in S - Aff~,fiis a simplicial object X* : op-! S - Aff~,fi which satisfies the following two conditions. - For any n 1, the n-th Segal morphism Xn -! X1xhX0X1xhX0. .X.1 _________-z________" n times is an equivalence (in the model category S - Aff~,fiof S-stacks). Wh* *en this condition is satisfied, it is well known that one can define a composition law * *(well defined up to homotopy) ~ : X1xhX0X1 -! X1. - The induced morphism (~, pr2) : X1xhX0X1 -! X1xhX0X1 is an equivalence (i.e. the composition law is invertible up to homot* *opy). 18 oFor any simplicial object X* : op-! S - Aff~,fi, we will denote by |X*| t* *he homotopy colimit of X* in the model category S - Aff~,fi. We are now ready to define geometric S-stacks. Definition 3.3.2An S-stack F is called geometric if it is equivalent to some |X* **|, where X* is a Segal groupoid in S - Aff~,fisatisfying the following two additional conditions. oThe S-stacks X0 and X1 are affine S-stacks. oThe morphism d0 : X1 -! X0 is a smooth morphism of affine S-stacks. The theory of geometric S-stacks can then be pursued along the same lines a* *s the theory of al- gebraic stacks (as done in [La-Mo ]). For example, one can define the notions o* *f quasi-coherent and perfect modules on a geometric S-stack, K-theory of a geometric S-stack (using * *perfect modules on it), higher geometric S-stacks (such as 2-geometric S-stacks), etc. We refer th* *e reader to [To-Ve 2] for details. We will finish this paragraph with the definition of the tangent S-stack an* *d its main properties. First of all, one defines a commutative S-algebra S["] := S _ S, which is t* *he trivial extension of S by S. The S-algebra S["] can be thought as the brave new algebra of dual numb* *ers, i.e. the analog of Z["]. For any commutative S-algebra A, one has A["] := A ^LSS["] ' A _ A, t* *he commutative A-algebra of dual numbers over A. For any S-stack F 2 S - Aff~,fi, one defines the tangent S-stack of F as T F : S - Alg -! SSet A 7! F (A["]). The tangent S-stack T F comes equipped with a natural projection p : T F -!* * F . One first notice that if F is a geometric S-stack (over any base A), then so is T F . Furthermor* *e, the homotopy fibers of the projection p are linear S-stacks in the following sense. Let A be a comm* *utative S-algebra and x : RSpec A -! F be a morphism of S-stacks, i.e. an A-point of F . One consider* *s the homotopy pull back Fx ______//_T F | | | | fflffl| fflffl| RSpec A__x__//F. Then, one can show that there exists an A-module M, such that Fx is equivalent * *(as a stack over RSpec A) to RSpec(LFA(M)). In other words, one has a natural equivalence Fx(B) ' RHom__A-Mod(M, B) for any commutative A-algebra B. The A-module M is called the cotangent complex* * of F at the point x, and denoted by L F,x. Its derived dual A-module D(L F,x) is called the tange* *nt space of F at x. 4 Derived moduli spaces in algebraic topology as S-stacks In this last Section we present an example of a geometric S-stack that arises f* *rom a classification problem in Algebraic Topology. This example shows that moduli spaces in Algebra* *ic Topology are 19 not only discrete homotopy types (as e.g. in [B-D-G ]), but might also have som* *e additional rich geo- metric structures very similar to the moduli spaces one studies in Algebraic Ge* *ometry. The example presented below seems to us the simplest one, and several more involved and int* *eresting moduli prob- lems can also be constructed and studied as geometric S-stacks. We will work with a fixed subcanonical model topology ø on S - Aff. 4.1 The brave new group scheme RAut_(M) We fix a perfect S-module M, and we are going to define a group S-stack RAut_(M* *), of auto- equivalences of M. This group S-stack will be a generalization of the group sc* *heme GLn, since it will be shown to be an affine and smooth group S-stack. Like many algebraic st* *acks in Algebraic Geometry are quotients of affine schemes by GLn, our example of a geometric S-s* *tack in x4.2 will be a quotient of an affine S-stack by RAut_(M) for some S-module M. For any commutative S-algebra A, one first defines REnd_(M)(A) := RHom__A-Mod(A ^L M, A ^L M). Using for example the Dwyer-Kan simplicial localization techniques ([D-K1 , D-K* *2]), one can make A 7! REnd_(M)(A) into a functor from S - Algto the category SMon of simplicial* * monoids REnd_(M) : S - Alg -! SMon A 7-! REnd_(M)(A). This defines REnd_(M) as a monoid object in S - Aff~,fi. As its underlying * *object in S - Aff~,fi is an S-stack (for example using Theorem 3.2.1), we will say that REnd_(M) is a* * monoid S-stack. Lemma 4.1.1 The S-stack REnd_(M) is affine and the structural morphism REnd_(* *M) -! RSpec S is perfect (hence smooth). Proof: This is clear as REnd_(M) ' RSpec (LFS(M ^LSD(M))). * * 2 For any commutative S-algebra A, one defines RAut_(M)(A) to be the sub-mono* *id of REnd_(M)(A) consisting of auto-equivalences. In other words, RAut_(M)(A) is defined by the * *following homotopy pull-back diagram in SSet RAut_(M)(A)________//_REnd_(M)(A) | | | | fflffl| " fflffl| [M ^L A, M ^L A]0Ø__//_[M ^L A, M ^L A] where [-, -]0is the subset of isomorphisms in Ho (SSet). This defines a functor RAut_(M) : S - Alg -! SMon A 7-! RAut_(M)(A). Once again the underlying object in S - Aff~,fiis an S-stack, and therefore RAu* *t_(M) is a monoid S-stack. Furthermore, the monoid law on RAut_(M) is invertible up to homotopy, * *and we will therefore say that RAut_(M) is a group S-stack. 20 Lemma 4.1.2 The S-stack RAut_(M) is affine and the structural morphism RAut_(* *M) -! RSpec S is smooth. In other words, RAut_(M) is an affine and smooth group S-stack. Proof: The following proof is inspired by the proof of [EGA-I , I.9.6.4]. L* *et B be the commutative S-algebra LFS(M ^LSD(M)) corresponding to the affine S-stack REnd_(M). There ex* *ists a universal endomorphism of B-modules u : M ^LSB -! M ^LSB such that for any commutative B-algebra C, the endomorphism u ^LBidC : M ^LSC -! M ^LSC is equal (in Ho (B - Mod )) to the corresponding point in REnd_(M)(C) ' RHom__S-Alg(B, C). Consider now the homotopy cofiber K 2 Ho (B -Mod ) of the universal endomo* *rphism u. Clearly, K is a perfect B-module, and one can therefore consider the open Zariski immers* *ion B -! BK (Lemma 2.1.6). It is easy to check by construction that RAut_(M) ' RSpec BK , which proves that RAut_(M) is an affine S-stack. Finally, as the morphism RSpec* * BK -! RSpec B is smooth (being a Zariski open immersion), one sees (using that RSpec B is perfec* *t hence smooth) that RAut_____(M) -! RSpec S is also smooth. * * 2 4.2 Moduli of algebra structures In this paragraph, we fix a perfect S-module M. We will define an S-stack Ass_M* *, classifying associative and unital algebras whose underlying module is M. For any commutative S-algebra A, we have the category A - Ass, of associati* *ve and unital A- algebras (i.e. associative monoids in the monoidal category (A - Mod , ^A)); th* *ese are new versions of the old A1 -ring spectra. The category A - Asshas a model category structure* * for which fibrations and equivalences are detected on the underlying objects in A - Mod . We denote * *by A - AsscofMthe subcategory of A - Ass whose objects are cofibrant objects B such that there ex* *ists a ø-covering family {A -! Ai}i2Isuch that each Ai-module B ^LAAi is equivalent to M ^LAAi (w* *e say that the underlying A-module of B is ø-locally equivalent to M), and whose morphisms are* * equivalences of A-algebras. The base change functors define a lax functor AssM : S - Alg - ! Cat A 7! A - AsscofM (A ! B) 7! - ^A B. Strictifying this functor (([May , Th. 3.4])) and then applying the classifying* * space functor, one gets a simplicial presheaf Ass_M: S - Alg -! SSet A 7! B(A - AsscofM). For the following theorem, let us recall that for any commutative S-algebra* * A, any associative and unital A-algebra B and any B-bimodule M, one has an A-module of A-derivations D* *erA(B, M) from B to M. This can be derived on the left (in the model category of associative a* *nd unital A-algebras !) to LDerA(B, M). 21 Theorem 4.2.1 1. The object Ass_M2 S - Aff~,fiis an S-stack. 2.The S-stack Ass_Mis geometric. 3.Let A be a commutative S-algebra, B be an associative A-algebra whose unde* *rlying A-module is ø-locally equivalent to M ^LSA and x : RSpec A -! Ass_M the correspondi* *ng point. Then, the tangent space of Ass_M at the point x is equivalent (as an A-module, s* *ee the end of x3.3) to the suspension LDerA(B, B)[1] of the A-module of derived A-derivations * *of the associative A-algebra B into the B-bimodule B. Sketch of proof: Point (1) can be proved with the same techniques used in T* *heorem 3.2.1 and will not be proved here. We refer to [To-Ve 2] for details. Point (3) seems to be a well-known fact. We again refer to [To-Ve 2] for de* *tails. Let us prove part (2) which is in fact a corollary of one of the main resul* *t of C. Rezk thesis [Re]. Let us start by considering the full sub-S-stack of Perf_(see x3.2) consist* *ing of perfect modules which are ø-locally equivalent to M. By the result of Dwyer and Kan [D-K3 , 2.* *3], this S-stack is clearly equivalent as an object in S - Aff~,fito BRAut_(M), the classifying sim* *plicial presheaf of the group S-stack RAut_(M). Forgetting the algebra structure gives a morphism of S-* *stacks f : Ass_M-! BRAut_(M). Using the techniques of equivariant stacks developed in [Ka-Pa-To1] (or more pr* *ecisely their straight- forward extensions to the present context of S-stacks), one sees that the S-sta* *ck Ass_Mis equivalent to the quotient S-stack [X=RAut_(M)], where X is the homotopy fiber of the morphism f and RAut_(M) acts on X. By Lemm* *a 4.1.2, RAut_(M) is an affine smooth group S-stack, so we only need to show that X is an affine * *S-stack (because the classifying Segal groupoid for the action of RAut_(M) on X will then satisf* *ies the conditions of Definition 3.3.2). Using [Re, Thm. 1.1.5], one sees that the homotopy fiber X is equivalent to* * the S-stack RHom__Oper(ASS , End_(M)) : A 7-! RHom__Oper(ASS , End(M ^LSA)), where RHom__Oper(ASS , End(M ^LSA)) is the derived Hom (or mapping space) of un* *ital operad mor- phisms from the final operad ASS (classifying associative and unital algebras) * *to the endomorphisms operad End(M ^LSA) of the A-module M ^LSA (here operads are in the symmetric mo* *noidal category S of S-modules). This means that, for any commutative S-algebra A, there is an * *equivalence X(A) ' RHom__Oper(ASS , End_(M))(A), functorial in A. Now, writing the operad ASS as a homotopy colimit ASS ' hocolimn2 opOn, where each On is a free operad, one sees that X ' holimn2RHom__Oper(On, End_(M)). 22 Since affine S-stacks are stable under homotopy limits, it is therefore enough * *to check that the S-stack RHom______Oper(O, End_(M)) is affine for any free operad O. But, saying that an* * operad O is free means that there is a family {Pm }m>0of S-modules, and functorial (in A 2 S - Alg) eq* *uivalences Y L RHom__Oper(O, End_(M))(A) ' RHom__S-Mod(Pm ^LSM^ m ^LSD(M), A), m where the funny notation M^Lm stands for the derived smash product M ^L . .^.LM* * of M with itself m times. So it is enough to show that for any S-module P , the (pre)stack Lm L A 7-! RHom__S-Mod(P ^LSM^ ^S D(M), A) is affine. But this is clear since this stack is equivalent to RSpec B where B* * is the derived free commutative S-algebra Ln L B := LFS(P ^LSM^ ^S D(M)). This implies that X is an affine S-stack and completes the proof. * * 2 Theorem 4.2.1 has also generalizations when one consider algebra structures* * over a given operad (for example commutative algebra structures). 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