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"Brave New"Algebraic Geometry and global derived moduli spaces of ring spectra

We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topological modular forms as geometric S-stacks.


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 according to an appropriate homotopical modification of a Grothendieck topology (a model topology, see 4.3]). More generally, we allow ourselves to consider more flexible objects like stacks, in order to deal with appropriate moduli problems. This theory is developed in full generality in  (see also ). Our motivations for such a theory came from a variety of sources: first of all, on the algebro-geometric side, we wanted to produce a sufficiently functorial language in which the so called Derived Moduli 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 (commutative) 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 Geometry in which one could take advantage of the possibility of gluing these brave new rings together into an actual geometric object, much in the same way as commutative algebra is helped (and generalized) by the existence of algebraic geometry. Thirdly, on the motivic side, following a suggestion of Y. Manin, we wished to have a sufficiently general theory in order to study algebraic geometry 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 − Aff of brave new affine objects will therefore be defined as the the opposite model category of the category of commutative rings in the category 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 category of (affine) schemes, like the Zariski andétale ones.
With any of these model topologies τ at our disposal, we define and give the basic properties of the corresponding model category of S-stacks, understood in the broadest sense as not necessarily truncated presheaves of simplicial sets on S − Aff satisfying a homotopical descent (i.e. sheaf-like) condition with respect to τ -(hyper)covers. A model topology on S − Aff is said to be subcanonical if the representable simplicial presheaves, i.e. those of the form Map(A, −), for some commutative ring A in S, Map being the mapping space in S − Aff op , are S-stacks.
As in algebraic geometry one finds it useful to study those stacks defined 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. In 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,étale, proper, etc.) between geometric S-stacks can likewise be defined.
Stacks were introduced in algebraic geometry mainly to study moduli problems 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 aspects having a fruitful 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 A ∞ -ring spectrum structures on a given spectrum M ) that can be studied geometrically through the geometric S-stack RAss M it represents. We wish to emphasize that instead of a discrete homotopy type (like the ones studied, for different moduli problems, in [Re, B-D-G, G-H]), we get a full geometric object on which a lot of interesting geometry can be performed. The geometricity of the S-stack RAss M , with respect to any fixed subcanonical model topology, is actually the main theorem of this paper (see Theorem 4.2.1).
We also wish to remark that the approach presented in this paper can be extended to other, more interested and involved, moduli problems algebraic topologists are interested in, and perhaps this richer geometry could be of some help in answering, or at least in formulating in a clearer way, some of the deep questions raised by the recent progress in stable homotopy theory (see [G]). In this direction, we will explain in §4.3 how topological modular forms give rise to a natural geometric S-stack which is an extension in the brave new direction of the moduli stack of elliptic curves (see Theorem 4.3.1). This fact seems to us a very important remark (probably much more interesting than our Theorem 4.2.1), and we think it could be the starting point of a very interesting research program.
We also present a brave new analog of the stack of vector bundles on a scheme, called the S-(pre)stack Perf of perfect modules (Section 3.2), and we expect it to be a key tool in brave new algebraic geometry. The prestack Perf is a stack if and only if the model topology we are working with is subcanonical (Thm. 3.2.1 whose proof is postponed to ). This is another instance of the relevance of the descent problem, i.e. the question whether a given model topology is subcanonical or not (see Section 3.1). Though we prove that some of the model topologies we introduce (namely the standard and the semi-standard ones, Section 2.3) are subcanonical, at present we are not able to settle (nor in the positive nor in the negative) the descent problem for the three most promising model topologies we define, namely the Zariski,étale and thh-étale ones. Though this is at the moment quite unsatisfactory, we believe that the descent problem for these topologies is a very interesting question in itself even leaving outside its crucial role in brave new algebraic geometry. For the Zariski model topology, we have a partial positive result in this direction. By definition, for a model topology τ the property of being subcanonical depends on the notion of stacks we consider; if instead of defining a stack as a prestack (i.e. a simplicial presheaf) satisfying homotopical descent with respect to all homotopy τ -hypercovers, we simply require descent with respect to allČech τhypercovers (i.e. those arising as nerves of τ -covers), we obtain a notion ofČech stacks, recently considered by J. Lurie ([Lu]) and Dugger-Hollander-Isaksen ( [DHI]). We prove (Corollary 3.1.4) that the Zariski model topology is in fact subcanonical with respect to the notion ofČech stacks. Moreover, by replacing in Theorem 4.2.1 the word "stack" with the weaker "Čech stack", the statement remains true for any model topology which is subcanonical with respect to the notion ofČech stacks.
It is therefore natural to ask why we did not choose to formulate everything only in terms ofČech stacks. We believe that at this early stage of development of homotopical algebraic geometry and, in particular, of brave new algebraic geometry, it is not advisable to make choices that could prevent some applications or obscure some of the properties of the objects involved, while it is more useful to keep in mind various options, some of which can be more useful in one context than in others. For example, it is clear that knowing that a given, geometrically meaningful, simplicial presheaf is a stack and not only aČech stack adds a lot more informations, in fact exactly the descent property with respect to unbounded hypercovers ( [DHI,Thm. A.6]). Moreover,Čech stacks fail in general to satisfy an analog of Whitehead theorem: a pointedČech stack may have vanishing π i sheaves for any i ≥ 0 without being necessarily contractible. This last fact is a very inconvenient property of Č ech stacks, that makes Postnikov decompositions and spectral sequences arguments uncertain. On the other hand, theČech descent condition is usually much easier to establish than the full descent condition, and as we have already remarked, some natural model topologies are easily seen to be subcanonical with respect to the notion ofČech stacks while it might be tricky to show that they are actually subcanonical. Finally, we would like to mention that in our experience we have never met serious troubles by using one or the other of the two notions, and in many interesting contexts it does not really matter which notion one uses, as the rather subtle differences actually tend not to appear in practice.

Notations.
To fix ideas, we will work in the category S := Sp Σ of symmetric spectra (see [Ho-Sh-Sm]), but all the constructions of this paper will also work, possibly with minor variations (see [Sch]), for other equivalent theories (e.g for the category of S-modules of [EKMM]). We will consider S as a symmetric monoidal simplicial model category (for the smash product − ∧ −) with the Shipley-Smith positive S-model structure (see [Shi,Prop. 3.1]).
We define S − Alg as the category of (associative and unital) commutative monoids objects in S, endowed with the S-model structure of [Shi,Thm. 3.2]; we will simply call them commutative S-algebras instead of the more correct but longer, commutative symmetric ring spectra. 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,Prop. 3.1]). This model 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 adjunction where f * (−) := − ∧ A B is the base change functor. We will denote by 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 −1 M will be denoted 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 ∈ M we will often use the coma model categories x/M and M/x. When the model category M is not left proper (resp. is not right proper) we will always assume that x has been replaced by a cofibrant (resp. fibrant) model before 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 required.
Since we wish to concentrate on applications to stable homotopy theory, some general constructions and details about homotopical algebraic geometry will be omitted by referring to . For a few of the results presented we will only give here sketchy proofs; full proofs will appear in .

Brave new sites
In this section we present two model topologies on the (opposite) category of commutative S-algebras. They are brave new analogs of the Zariski andétale topologies defined on the category of usual commutative rings and will allow us to define the brave new Zariski andétale sites.
We denote by S − Aff the 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 {z i } i∈J in M , the natural map is an isomorphism in the homotopy category of simplicial sets.
Definition 2.0.1 A morphism A → B of commutative S-algebras is finitely presented if it is a finitely presented object in the model under-category A/(S − Alg) = A − Alg; in this case, we also say that B is a finitely presented commutative A-algebra. An A-module E is finitely presented or perfect if it is a finitely presented object in the model category A − Mod.
Perfect A-modules can also be characterized as retracts of finite cell A-modules (see [EKMM,]); in particular, there are plenty of them. If A is a commutative S-algebra, then the free commutative A-algebra on a finite number of generators (or, more generally, on any perfect A-module) is a finitely presented A-algebra. The reader will find other examples of finitely presented morphisms of commutative S-algebras in Lemma 2.1.6. Example 2.1.2 If A ∈ S − Alg and E is an A-module such that the associated Bousfield localization L E is smashing (i.e. the natural transformation L E (−) → L E A ∧ L A (−) is an isomorphism), then A → L E A (which is a morphism of commutative S-algebras by e.g. [EKMM,§VIII.2]) is a formal Zariski open immersion. This follows immdiately from the fact that Ho(L E A − M od) is equivalent to the subcategory of Ho(A − M od) consisting of L E -local objects, by [Wo].

The brave new Zariski topology
It is easy to check that (  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 π 0 (HR) = R and π i (HR) = 0 for any i = 0. This functor is homotopically fully faithful and the following lemma shows that our brave new Zariski topology does generalize the usual Zariski topology.  The Quillen adjunction (f * , f * ) extends to a Quillen adjunction on the category of commutative algebras also with the property that Rf * : Ho(B − Alg) −→ Ho(A − Alg) is fully faithful. Furthermore, the essential image of this last functor consist of all objects C ∈ Ho(A − Alg) such that the underlying Amodule of C satisfies C ≃ LC (i.e. the underlying A-module of C lives in the image of Ho(B −Mod)). ¿From these observations, we deduce that for any commutative A-algebra C, the mapping space this implies that the functor Rf * : Ho(HR ′ − Mod) −→ Ho(HR − Mod) is also fully faithful. It only remains to show that HR −→ HR ′ is finitely presented in the sense of Definition 2.0.1. We will first assume that R ′ = R f for some element f ∈ R. The essential image of Rf * : 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 (HR ′ , C) is contractible if f becomes invertible in π 0 (C), and empty otherwise. ¿From this one easily deduces that RHom HR−Alg (HR ′ , −) commutes with filtered colimits, or in other words that HR ′ is a finitely presented HR-algebra in the sense of Definition 2.0.1.
In the general case, one can write Spec R ′ as a finite union of schemes of the form Spec R f for some elements f ∈ R. A bit of descent theory (see §3.1) then allows us to reduce to the case where R ′ = R f and conclude.
Let us now assume that HR −→ HR ′ is a Zariski open immersion of commutative S-algebras. By adjunction (between H and π 0 restricted on connective S-algebras) one sees easily that R −→ R ′ is a finitely presented morphism of commutative rings.
The induced functor on (unbounded) derived categories is fully faithful. Through the Dold-Kan correspondence, this implies that the Quillen adjunction on the model category of simplicial modules (see [G-J]) Let sR − Alg and sR ′ − Alg be the categories of simplicial commutative R-algebras and simplicial commutative R ′ -algebras, endowed with their natural model structures (equivalences are and fibration are detected in the category of simplicial modules). Then, the Quillen adjunction as this is true on the level on simplicial modules. In particular, for any simplicial R ′ -module M , the space of derived derivations . As a consequence one sees that Quillen's cotangent complex L R ′ /R is acyclic, which implies that the morphism R −→ R ′ is anétale morphism of rings. Finally, using the fact that the functor on the category of modules R ′ − Mod −→ R − Mod is fully faithful, one sees that Spec R ′ −→ Spec R is a monomorphism of schemes. Therefore, the morphism of schemes Spec R ′ −→ Spec R is anétale monomorphism, and so is an open immersion by [EGA-IV, Thm. 17.9.1].
Finally, point (2)  Let Aff be the opposite category of commutative rings, and (Aff , Zar) the big Zariski site. The site (Aff , Zar) can also be considered as a model site (for the trivial model structure on Aff ). Lemma 2.1.4 implies in particular that the functor H : Aff −→ S − Aff induces a continuous morphism of model sites ( Def. 4.8.4]). In this way, the site (Aff , Zar) becomes a sub-model site of (S − Aff , Zar).
To finish with the Zariski topology we will now describe a general procedure in order to construct interesting open Zariski immersions of commutative S-algebras using the techniques 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 strongly dualizable object in the monoidal category Ho(A − Mod). As already noticed, perfect A-modules are exactly the retracts of finite cell A-modules, see [EKMM, We denote by D ( 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 One checks easily that, since M is strongly dualizable, the local objects for this localization are the commutative A-algebras B such that M ∧ L A B ≃ 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-algebra B, the map- A M ≃ 0 and empty otherwise. In other words, for any commutative S-algebra B the natural morphism is equivalent to an inclusion of connected components and its image consists of morphisms Lemma 2.1.6 With the above notations, the morphism A −→ A M is a Zariski open immersion.
Proof: Let us start by showing that A M is a finitely presented commutative A-algebra.
Let {B i } i∈I be a filtered system of commutative A-algebras and B = colim i B i . We assume that B ∧ L A M ≃ 0, and we need to prove that there exists an i ∈ I such that B i ∧ L A M ≃ 0. By assumption, the two points Id and 0 are the same in π 0 This implies that there is some index i ∈ I such that Id and 0 are homotopic in For this, one uses that for any commutative S-algebra B, the morphism is an inclusion of connected components. Therefore, the natural morphism is an isomorphism in Ho(SSet). This implies that the natural morphism is an equivalence of commutative S-algebras. In particular, one has for any showing that the base change functor is fully faithful. ✷ An important property of the localization A −→ A M is the following fact.
Lemma 2.1.7 Let A be a commutative S-algebra, and M be a perfect A-module. Then the essential image of the fully faithful functor Proof: As every A M -module N can be constructed by homotopy colimits of free A M -modules and − ∧ L A M commutes with homotopy colimits, it is clear that It is well known (see e.g. [Hi,§4]) that such a local model can be obtained by a transfinite composition of homotopy push-outs of the form in the category of A-algebras. ¿From this description, and the fact that − ∧ L A M commutes with homotopy colimits, one sees that the adjunction morphism Lemma 2.1.7 allows us to interpret geometrically A M 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 exercise. Remark 2.1.9 One should note carefully that even though if the Eilenberg-Mac Lane functor H embeds (Aff , Zar) in (S − Aff , Zar) as model sites, there exist commutative rings R and Zariski open coverings HR −→ B in S − Aff such that B is not of the form HR ′ for some commutative R-algebra R ′ . One example is given by taking R to be C[X, Y ], and considering the localized commutative HRalgebra (HR) M (in the sense above), where M is the perfect R-module R/(X, Y ) ≃ C. If (HR) M were of the form HR ′ for a Zariski open immersion Spec R ′ −→ Spec R, then for any other commutative ring R ′′ , the set of scheme morphisms Hom(Spec R ′′ , Spec R ′ ) would be the subset of Hom(Spec R ′′ , A 2 ) consisting of morphisms factoring through A 2 − {0}. This would mean that Spec R ′ ≃ A 2 − {0}, which is not possible as A 2 − {0} is a not an affine scheme. This example is of course the same as the example given in [To,§2.2] of a 0-truncated affine stack which is not an affine scheme. These kind of example shows that there are many more affine objects in homotopical algebraic geometry than in usual algebraic geometry. L M A in 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 L A M -local objects are exactly L M -acyclic objects (compare with Remark 2.1.5). Note that however, while the Bousfield localization is always defined for any A-module M , the commutative A-algebra A M probably does not exist unless M is perfect.
2. Let S p be the p-local sphere. If f : S p → B is any formal Zariski open immersion then L := Rf * Lf * is clearly a smashing localization functor in the sense of [HPS,§3]. Its category C of perfect 1 acyclics (i.e. perfect objects X in Ho(S p − Mod) such that LX is null) is then a localizing thick subcategory of the homotopy category Ho(S p − Mod perf ) of the category of perfect S p -modules, and therefore by [H-S] it is equivalent to the category C n of perfect E(n)acyclics, for some 0 ≤ n < ∞, where E(n) is the n-th Johnson-Wilson S p -module (see e.g. [Rav]); in other words L and L n := L E(n) are both smashing localization functors on Ho(S p − 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 L n and L have equivalent categories of acyclics and so have equivalent categories of local objects. But the category of local objects for L is equivalent to the category Ho(B − Mod) (since Rf * is fully faithful by hypothesis) and the category of local objects for L n is equivalent to the category Ho((L n S p ) − Mod), by [Wo] since L n is smashing. This easily implies that the two commutative S p -algebras B and L n S p are equivalent (i.e. isomorphic in Ho(S p − Alg)).
In conclusion, one sees that if the Telescope conjecture is true, then, up to equivalence of S palgebras, the only (non-trivial) formal Zariski open immersions for S p are given by the family 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 any non-trivial Zariski open immersion of S p , i.e. that the morphisms of commutative S-algebras S p −→ L n S p are of finite presentation). Note however that the family U is not a formal Zariski covering according to Definition 2.1.1 because the family of base-change functors is not conservative; in fact, as Neil Strickland pointed out to us, the Brown-Comenetz dual I of S p is a non-perfect non-trivial S p -module which is nonetheless L n -acyclic for any n. However, it is true that the family of base-changes above is conservative when restricted to the (homotopy) categories of perfect modules. Therefore, one could modify the second covering 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 modified 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 S p -algebras, if the Telescope conjecture holds.
3. The previous example also shows that the commutative S-algebras L n S p are local for the formal Zariski topology (again assuming the Telescope conjecture). Indeed, for any formal Zariski open covering {L n S p −→ B i } i∈I there is an i such that L n S p −→ B i is an equivalence of commutative S-algebras.

The brave newétale topology
Notions ofétale morphisms of commutative S-algebras has been studied by several authors ( [Ro1,). In this paragraph we present the definition that appeared in  and was used there in order to define theétale K-theory of commutative S-algebras.
We refer to [Ba] for the notions of topological cotangent spectrum and of topological André-Quillen cohomology relative to a morphism A → B of commutative S-algebras, except for slightly different notations. We denote by LΩ B/A ∈ Ho(B − Mod) the topological cotangent spectrum (denoted as Ω B/A in [Ba]) and, for any B-module M, by -The morphism f is called formallyétale if LΩ B/A ≃ 0.
-The morphism f is calledétale if it is formallyétale and of finite presentation (as a morphism of commutative S-algebras).
• A family of morphisms {f i : A −→ A i } i∈I in S − Alg is called a (formal)étale covering if it satisfies the following two conditions.
-There exists a finite subset J ⊂ I such that the family of inverse image functors is an isomorphism if and only if its images by all the Lf * j 's are isomorphisms).
As shown in [To-Ve 1, §5.2], (formal)étale covering families are stable by equivalences, compositions and homotopy push-outs, and therefore define a model topology on the model category S − Aff . Therefore one gets two model topologies called the brave newétale topology and the brave new formaĺ etale topology. The corresponding model sites will be denoted by (S − Aff ,ét) and (S − Aff , fét), and will be called the brave newétale site and the brave new formalétale site.
As for the brave new Zariski topology one proves that the brave newétale topology is a generalization of the usualétale topology. Another important fact is that the brave newétale topology is finer than the brave new Zariski topology. which is a base change functor from the brave new Zariski site to the brave newétale 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 brave newétale topology, called the thh-étale topology, which is sometimes more convenient to deal with.

Definition 2.2.4
• Let f : A −→ B be a morphism of commutative S-algebras.
-The morphism f is called formally thh-étale if for any commutative A-algebra C the mapping space RHom A−Alg (B, C) is 0-truncated (i.e. equivalent to a discrete space).
-The morphism f is called thh-étale if it is formally thh-étale and of finite presentation (as a morphism of commutative S-algebras).
• A family of morphisms {f i : A −→ A i } i∈I in S − Alg is called a (formal) thh-étale covering if it satisfies the following two conditions.
-Each morphism A −→ A i is (formally) thh-étale. It is easy to check that (formal) thh-étale coverings define a model topology on the model category S − Aff , call the (formal) thh-étale topology. The model category S − Aff together with these topologies will be called the brave new thh-étale site and the brave new formal thh-étale site, denoted by (S − Aff , thh-ét) and (S − Aff , fthh-ét), respectively. An equivalent way of stating the formal thh-étaleness condition for A → B is to say that the natural map B → S 1 ⊗ L B in Ho(A − Alg) is an isomorphism, or equivalently (by [MSV]), that the canonical map B → THH(B/A, B) is an isomorphism in Ho(A − Alg), where THH denotes the topological Hochschild cohomology spectrum (see e.g. [EKMM, §IX]). This equivalent characterization follows from the observation that, for any morphism f : B → C of commutative A-algebras, one has an isomorphism (in Ho(SSet)) of mapping spaces: where Ω f denotes the loop space at f . Therefore, the canonical map 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 André-Quillen spectrum TAQ (where, for any B-module M , TAQ(B/A; M ) is defined as the derived internal Hom from LΩ B/A to M in the model category of B-modules) also shows that (formal) thh-étale morphisms are (formal)étale morphisms. Therefore the identity functor induces continuous morphisms of model sites We refer to [MC-Min] for more details on the notion of thh-étale morphisms.

Standard topologies
Standard model topologies on S − Aff are obvious extensions of usual Grothendieck topologies on affine schemes. They are defined in the following way. Let τ be one of the usual Grothendieck topologies on affine schemes (i.e. Zariski, Nisnevich,étale or faithfully flat). Definition 2.3.1 A family of morphisms of commutative S-algebras {A −→ B i } i∈I is a standard τ -covering (also called strong τ -covering) if it satisfies the following two conditions.
• The induced family of morphisms of schemes {Spec π 0 (B i ) −→ Spec π 0 (A)} i∈I is a τ -covering of affine schemes.
• For any i ∈ I the natural morphism of π 0 (B i )-modules is an isomorphism.
Its easy to check that this defines a model topology τ s on S − Aff , called the standard τ -topology. The model site (S − Aff , τ s ) may be called the brave new standard-τ site. The importance of standard topologies is that all τ s -coverings of commutative S-algebras of the form HR comes from usual τ -coverings of the scheme Spec R. Its behavior is therefore very close to the geometric intuition one gets in Algebraic Geometry.
Finally, let us also mention the semi-standard (or semi-strong) model topologies. A family of morphisms of commutative S-algebras {A −→ B i } i∈I is a semi-standard τ -covering (also called semi-strong τ -covering) if the induced family of morphism of commutative graded rings {π * (A) −→ π * (B i )} i∈I is a τ -covering. This also defines a model topology τ ss on S − Aff .
Both the standard and semi-standard type model sites (and S-stacks over them, see Section 3) could be of some interest in the study of geometry over even, periodic S-algebras (e.g. for elliptic spectra as in [AHS]).

S-stacks and geometric S-stacks
Let (M, τ ) be a model site (i.e. a model category M endowed with a model topology τ in the sense of ). Associated to it one has a model category of prestacks M ∧ and of stacks M ∼,τ . For details concerning these model categories we refer to §4], and for the sake of brevity we only recall the following facts. Specializing to our present situation, where M = S − Aff , we have one model category S − Aff ∧ of prestacks and zounds of model categories stacks These model categories come with right Quillen functors (the morphism of change of sites) which allow to compare the various topologies on S − Aff .
Definition 3.0.2 Let τ be a model topology on the model site S − Aff .
• The model category of S-stacks for the topology τ is S − Aff ∼,τ .
• A simplicial presheaf F ∈ SP r(S − Aff ) is called an S-stack if it is a local object in S − Aff ∼,τ (i.e. preserves equivalences and satisfies the descent property for τ -hypercovers).
• Objects in the homotopy category Ho(S − Aff ∼,τ ) will simply be called S-stacks (without referring, unless it is necessary, to the underlying topology).
The category of S-stacks, being the homotopy category of a model category, has all kind of homotopy 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 §3.8] (see also ), and therefore behaves very much in the same way as a category sheaves (but in a homotopical sense). 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-algebras induces left Quillen functors where τ 0 is one of the standard topologies on affine schemes (e.g. Zar,ét, ffqc,. . . ), and τ is one of its possible extension to the model category S − Aff (e.g. Zar can be extended to Zar s or to Zar, etc.).
Here, Aff ∼,τ is the usual model category of simplicial presheaves on the Grothendieck site (Aff , τ ) (with the projective model structure [Bl]). By deriving on the left one gets a functor Therefore, our category of S-stacks receives a functor from the homotopy category of simplicial presheaves. In particular, sheaves on affine schemes (and in particular the category of schemes itself), and also 1-truncated simplicial presheaves (and in particular the homotopy category of algebraic stacks) can be all viewed as examples of S-stacks. However, one should be careful that the functor LH ! has no reason to be fully faithful in general, though this is the case for all the standard extensions (but not semi-standard) described in §2.3 (the reason for this is that all covering families of some HR are in fact induced from covering families of affine schemes. In particular the restriction functor from S − Aff ∼,τ −→ Aff ∼,τ 0 will preserve local equivalences.).
If instead of requiring descent with respect to all τ -hypercovers, we only require descent with respect to those τ -hypercovers which arise as homotopy nerves of τ -covers, we obtain the following weaker notion of stack.
Definition 3.0.3 Let τ be a model topology on S − Aff . A simplicial presheaf F : S − Aff op → SSet is said to be aČech S-stack with respect to τ if it preserves equivalences and satisfies the followinǧ Cech descent condition. For any τ -cover U = {U i → X}, denoting byČ(U ) * its homotopy nerve, the canonical map is an isomorphism in Ho(SSet).
This weaker notion of stacks is the one used recently by J. Lurie in [Lu] and has also appeared for stacks over Grothendieck sites in [DHI].
Note that, similarly to the case of S-stacks, there is a model category S − Aff ∼,τ C ofČech Sstacks, defined as the left Bousfield localization of S − Aff ∧ with respect to all theČech-nerves, whose homotopy category is equivalent to the full subcategory of Ho(S − Aff ∧ ) consisting ofČech S-stacks. In general the inclusion of S-stacks intoČech S-stacks is proper; however, one can prove (by adapting Prop. 6.1] to the context of model sites), that given any n-truncated (equivalence preserving) simplicial presheaf F on S − Aff , F is an S-stack iff it is aČech S-stack (regardless of the model topology τ ). The reader might wish to read Appendix A of [DHI] for more comparison results between stacks andČech stacks over usual Grothendieck sites. Definition 3.1.1 The topology τ is sub-canonical (respectively,Čech-subcanonical) if for any A ∈ S − Alg, Rh A is an S-stack (resp., aČech S-stack).

Some descent theory
Note that τ is sub-canonical iff the functor RSpec is fully faithful, and that sub-canonical implieš Cech-subcanonical.
Knowing whether a given model topology τ is sub-canonical or not is known as the descent problem for τ , and in our opinion is a crucial question. At present, 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. Sketch of proof: Let τ be one of these topologies, A be a commutative S-algebra, and A −→ B * be a τ -hypercover ([To-Ve 1, §4.4]). Using the fact that π * (B n ) is flat over π * (A) for any n, one can check that the cosimplicial π * (A)-algebra π * (B * ) is again a τ -hypercover of commutative rings. By usual descent theory for affine schemes this implies that the cohomology groups of the total complex of [n] → π * (B n ) vanish except for H 0 (π * (B * )) ≃ π * (A). This implies that the spectral sequence for the holim degenerates at E 2 and that π * (A) −→ π * (holimB * ) is an isomorphism. ✷ Concerning the brave new Zariski topology one has the following partial result. 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 i m : ∆ ≤m −→ ∆ be the inclusion functor form the full sub-category of objects [i] with i ≤ m. Then, one has an equivalence of commutative A-algebras B * ≃ R(i m ) * i * m (B * ) (here (i * m , R(i m ) * ) is the derived adjunction between ∆-diagrams and ∆ ≤m -diagrams). From this one deduces easily that holim n∈∆ B n ≃ holim n∈∆ ≤m B n .
In particular, holim n∈∆ B n is in fact a finite homotopy limit and therefore will commute with the base change from A to B, i.e.

Now, as the functor
is a Zariski covering), one can replace A by B and the A i by B ∧ L A A i , and in particular 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 ≃ holim n B n (the section can in fact be used in order to construct a retraction). ✷ Corollary 3.1.4 The Zariski topology on S − Aff isČech subcanonical.
The results of the next sections, though stated for S-stacks will also be correct by replacing "Sstack" with "Čech S-stack", and "subcanonical" with "Čech subcanonical".

The S-stack of perfect modules
Let τ be a model topology on S − Aff . One defines the S-prestack Perf of perfect 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 Making this pseudo-functor into a strict functor from S − Alg to 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 .
Theorem 3.2.1 The object Perf is an S-stack (i.e. satisfies the descent condition for all τ -hypercovers) iff the model topology τ is subcanonical.
Another way to state Theorem 3.2.1 is by saying that τ is subcanonical iff, for any commutative S-algebra A, the natural morphism is an equivalence of simplicial sets.
The S-stack of perfect complexes is a brave new analog of the stack of vector bundles, and is of fundamental importance in brave new algebraic geometry.

Geometric S-stacks
In this paragraph we will work with a fixed sub-canonical model topology τ on the model site S − Aff . We will define the notion of geometric S-stack, which roughly speaking are quotients of affine S-stacks by a smooth affine groupoid. They will be brave new generalizations of Artin algebraic stacks (see [La-Mo]). In order to state the precise definition, one first needs a notion of smoothness for morphisms of commutative S-algebras.
For any perfect S-module M one has the (derived) free commutative S-algebra over M , S −→ LF S (M ). For any commutative S-algebra A, one gets a morphism 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 perfect commutative A-algebra).
Definition 3.3.1 A morphism of commutative S-algebras f : A −→ B is called smooth if it satisfies the following two conditions.
• The A-algebra B is finitely presented.
• There exists anétale covering family {v i : B → B ′ i } i∈I and, for any i ∈ I, a homotopy commutative square of commutative S-algebras where f ′ i is a perfect morphism, and u is anétale morphism.
One checks easily that smooth morphisms are stable by compositions and homotopy base changes. Furthermore, anyétale morphism is smooth, and therefore so is any Zariski open immersion.
Assumption: At this point we will assume that the notion of smooth morphisms is local with respect to the chosen model topology τ .
This assumption will insure that the notion of geometric S-stack, to be defined below, behaves well. Some terminology: • Let us come back to our homotopy category Ho(S − Aff ∼,τ ) of S-stacks, and the Yoneda embedding (or derived Spec) The essential image of RSpec is called the category of affine S-stacks, which is therefore antiequivalent to the homotopy category of commutative S-algebras. We will also call affine S-stack any object in S − Aff ∼,τ whose image in Ho(S − Aff ∼,τ ) is an affine S-stack. Clearly, affine S-stacks are stable by homotopy limits (indeed holim i (RSpec A i ) ≃ RSpec (hocolim i A i )).
• A morphism of affine S-stacks is called smooth (over S) (resp.étale, a Zariski open immersions . . . ) if the corresponding one in Ho(S − Alg) is so.
• A Segal groupoid object in S − Aff ∼,τ is a simplicial object which satisfies the following two conditions.
-For any n ≥ 1, the n-th Segal morphism is an equivalence (in the model category S − Aff ∼,τ of S-stacks). When this condition is satisfied, it is well known that one can define a composition law (well defined up to homotopy) µ : is an equivalence (i.e. the composition law is invertible up to homotopy).
We are now ready to define geometric S-stacks.
Definition 3.3.2 An S-stack F is called geometric if it is equivalent to some |X * |, where X * is a Segal groupoid in S − Aff ∼,τ satisfying the following two additional conditions.
• The S-stacks X 0 and X 1 are affine S-stacks.
• The morphism d 0 : X 1 −→ X 0 is a smooth morphism of affine S-stacks.
The theory of geometric S-stacks can then be pursued along the same lines as the theory of algebraic stacks (as done in [La-Mo]). For example, one can define the notions of 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 the reader to  for details.
We will finish this paragraph with the definition of the tangent S-stack and its main properties.
First of all, one defines a commutative S-algebra S[ε] := S ∨ S, which is the trivial extension of S by S. The S-algebra S[ε] can be thought as the brave new algebra of dual numbers, i.e. the analog of Z[ε]. For any commutative S-algebra A, one has A[ε] := A ∧ L S S[ε] ≃ A ∨ A, the commutative A-algebra of dual numbers over A.
For any S-stack F ∈ S − Aff ∼,τ , one defines the tangent S-stack of F as 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 . Furthermore, the homotopy fibers of the projection p are linear S-stacks in the following sense. Let A be a commutative S-algebra and x : RSpec A −→ F be a morphism of S-stacks, i.e. an A-point of F . One considers the homotopy pull back Then, one can show that there exists an A-module M , such that F x is equivalent (as a stack over RSpec A) to RSpec(LF A (M )). In other words, one has a natural equivalence for any commutative A-algebra B. The A-module M is called the cotangent complex of F at the point x, and denoted by LΩ F,x . Its derived dual A-module D(LΩ F,x ) is called the tangent space of F at x.

Some examples of geometric S-stacks
In this last Section we present two examples of geometric S-stacks. The first one arises from a classification problem in Algebraic Topology, whereas the second one is directly related to topological modular forms. The first of these examples (see §4.2) shows that moduli spaces in Algebraic Topology are not only discrete homotopy types (as e.g. in [B-D-G]), but have some additional rich geometric structures very similar to the moduli spaces one studies in Algebraic Geometry. It seems to us one of the most simple non trivial example of brave new moduli stack, and a relevant test for the whole theory.
The second example (see §4. 3) seems to us much more intriguing, and might give new insights on the construction and the study of topological modular forms. We think that this research direction is definitely worth being investigated in the future, and therefore we present a key open question that could be the starting point of such an investigation.
We will work with a fixed subcanonical model topology τ on S − Aff .

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 autoequivalences of M . This group S-stack will be a generalization of the group scheme GL n , since it will be shown to be an affine and smooth group S-stack. Like many algebraic stacks in Algebraic Geometry are quotients of affine schemes by GL n , our example of a geometric S-stack in §4.2 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 This defines REnd(M ) as a monoid object in S − Aff ∼,τ . As its underlying object in S − Aff ∼,τ is an S-stack (for example using Theorem 3.2.1), we will say that REnd(M ) is a monoid S-stack.
where [−, −] ′ is the subset of isomorphisms in Ho(SSet). This defines a functor Once again the underlying object in S − Aff ∼,τ is an S-stack, and therefore RAut(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. Consider now the homotopy cofiber K ∈ Ho(B − Mod) of the universal endomorphism u. Clearly, K is a perfect B-module, and one can therefore consider the open Zariski immersion B −→ B K (Lemma 2.1.6). It is easy to check by construction that which proves that RAut(M ) is an affine S-stack. Finally, as the morphism RSpec B K −→ RSpec B is smooth (being a Zariski open immersion), one sees (using the fact that RSpec B is perfect hence smooth) that RAut(M ) −→ RSpec S is also smooth. ✷

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 associative and unital Aalgebras (i.e. associative monoids in the monoidal category (A − Mod, ∧ A )); these are new versions of the old A ∞ -ring spectra. The category A − Ass has a model category structure for which fibrations and equivalences are detected on the underlying objects in A − Mod. We denote by A − Ass cof M the subcategory of A − Ass whose objects are cofibrant objects B such that there exists a τ -covering we say that the underlying A-module of B is τ -locally equivalent to M ), and whose morphisms are equivalences of A-algebras. The base change functors define a lax functor Strictifying this functor ( [May,Th. 3.4]) and then applying the classifying space functor, one gets a simplicial presheaf  1. The object Ass M ∈ S − Aff ∼,τ is an S-stack.
2. The S-stack Ass M is geometric.
Sketch of proof: Point (1) can be proved with the same techniques used in Theorem 3.2.1 and will not be proved here. We refer to  for details.
Let us prove part (2)  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 satisfies 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 where RHom Oper (ASS, End(M ∧ L S A)) is the derived Hom (or mapping space) of unital operad morphisms from the final operad ASS (classifying associative and unital algebras) to the endomorphisms operad End(M ∧ L S A) of the A-module M ∧ L S A (here operads are in the symmetric monoidal category S of S-modules). This means that, for any commutative S-algebra A, there is an equivalence Now, writing the operad ASS as a homotopy colimit

Topological modular forms and geometric S-stacks
In this final section we will be working with the standardétale topology on S − Aff . The corresponding model site will be denoted by (S − Aff ,ét s ), and its model category of stacks by S − Aff ∼,ét s . We recall that the topologyét s is known to be subcanonical (see Lem. 3.1.2). As explained right after Def. 3.0.2, the Eilenberg-MacLane spectrum construction gives rise to a fully faithful functor where the left hand side is the homotopy category of simplicial presheaves on the usualétale site of affine schemes. This functor has a right adjoint, called the truncation functor Let us denote by E the moduli stack of generalized elliptic curves with integral geometric fibers, which is the standard compactification of the moduli stack of elliptic curves by adding the nodal curves at infinity (see e.g. [Del-Rap, IV], where it is denoted by M (1) ); recall that E is a Deligne-Mumford stack, proper and smooth over Spec Z ([Del-Rap, Prop. 2.2]).
As shown by recent works of M. Hopkins, H. Miller, P. Goerss, N. Strickland, C. Rezk and M. Ando, there exists a natural presheaf of commutative S-algebras on the smallétale site Eé t of E. We will denote this presheaf by tmf. Recall that by construction, if U = Spec A −→ E is anétale morphism, corresponding to an elliptic curve E over the ring A, then tmf(U ) is the (connective) elliptic cohomology theory associated to the formal group of E (in particular, one has π 0 (tmf(U )) = A). Recall also that the (derived) global sections RΓ(E, tmf), form a commutative S-algebra, well defined in Ho(S − Alg), called the spectrum of topological modular forms, and denoted by tmf.
Let U −→ E be a surjectiveétale morphism with U an affine scheme, and let us consider its nerve U * : ∆ op −→ Aff [n] → U n := U × E U × E · · · × E U n times . This is a simplicial object in Eé t , and by applying tmf we obtain a co-simplicial object in S − Alg tmf(U * ) : ∆ −→ S − Alg [n] → tmf(U n ).
The following result is just a remark as there is essentially nothing to prove; however, we prefer to state it as a theorem to emphasize its importance.
Theorem 4.3.1 The stack E S defined above is a geometric S-stack. Furthermore, there exists a natural isomorphism in Ho(Aff ∼,ét ) h 0 (E S ) ≃ E.
Proof: To prove that E S is geometric, it is enough to check that the simplicial object RSpec (tmf(U * )) is a Segal groupoid satisfying the conditions of Def. 3.3.2. For this, recall that for any morphism U = Spec B → V = Spec A in Eé t , the natural morphism π * (tmf(V )) ⊗ π 0 (tmf(V )) π 0 (tmf(U )) ≃ π * (tmf(V )) ⊗ A B −→ π * (tmf(U )) is an isomorphism. This shows that the functor RSpec (tmf(−)) : Eé t −→ S − Aff ∼,ét s preserves homotopy fiber products and therefore sends Segal groupoid objects to Segal groupoid objects. This shows in particular that RSpec (tmf(U * )) is a Segal groupoid object. The same fact also shows that for any morphism U = Spec B → V = Spec A in Eé t , the induced map tmf(V ) −→ tmf(U ) is a strongétale morphism in the sense of Def. 2.3.1, and therefore is anétale and thus smooth morphism. This implies that RSpec (tmf(U * )) satisfies the conditions of 3.3.2 and therefore shows that E S is indeed a geometric S-stack.
Therefore, we get h 0 (E S ) ≃ hocolim n∈∆ op h 0 (RSpec (tmf(U n ))) ≃ hocolim n∈∆ op U n ≃ E, as U * is the nerve of anétale covering of E. ✷ Remark 4.3.2 The proof of Theorem 4.3.1 shows not only that E S is geometric, but also that it is a strong Deligne-Mumford geometric S-stack, in the sense that one can replace the word smooth by stronglyétale in Definition 3.3.2. See  for further details.
Theorem 4.3.1 tells us that the presheaf of topological modular forms tmf provides a natural geometric S-stack E S whose truncation is the usual stack of elliptic curves E. Furthermore, as the small strongétale topoi of E S and E coincides (this is a general fact about strongétale model topologies, see ), we see that tmf := RΓ(E, tmf) ≃ RΓ(E S , O), and therefore that topological modular forms can be simply interpreted as functions on the geometric S-stack E S . Of course, our construction of E S has essentially been done to make this true, so this is not a surprise. However, we have gained a bit from the conceptual poit of view: since after all E is a moduli stack, now that we know the existence of the geometric S-stack E S we can ask for a modular interpretation of it, or in other words for a direct geometric description of the corresponding simplicial presheaf on S − Aff . An answer to this question not only would provide a direct construction of tmf, but would also give a conceptual interpretation of it in a geometric language closer the usual notion of modular forms. Essentially, we ask which are the brave new "objects" that the S-stack E S classifies. Of course we do not know the answer to this question, though some progress are being made by J. Lurie and the authors. It seems that the S-stack E S itself is not really the right object to look at, and one should rather consider the non-connective version of it (defined using the non-connective version of tmf) for which a modular interpretation seems much more accessible. Though this modular interpretation is still conjectural and not completely achieved, it does use some very interesting notions of brave new abelian varieties, brave new formal groups and their geometry. We think that the achievement of such a program could be the starting point of a rather new and deep interaction between stable homotopy theory and algebraic geometry, involving many new questions and objects, but probably also new insights on classical objects of algebraic topology.