1 Introduction

These are expanded notes of some talks given during Fall 2002 about homotopical algebraic geometry with special emphasis on its applications to derived algebraic geometry and derived deformation theory. We have omitted proofs that will appear mainly in [HAG-II]. The main purpose of this work is to present in a somehow informal way the category of D-stacks and to give some examples of derived moduli spaces as D-stacks.

We would like to thank the organizers of the conferences Axiomatic and enriched homotopy theory (Cambridge, September 2002) and Intersection theory and moduli (Trieste September 2002), in which some of the material in the present note has been reported. We would also like to thank S. Müller-Stach, H. Esnault and E. Viehweg for inviting us to lecture on our work at a DFG-Schwerpunkt "Globale Methoden in der Komplexen Geometrie" in Essen and the Research in Pairs program at the Matematisches Forschunginstitut Oberwolfach for providing the excellent working conditions in which this paper was written.

1.1 What's HAG ?

Homotopical Algebraic Geometry (or HAlgebraic Geometry, or simply HAG) was conceived as a framework to talk about schemes in a context where affine objects are in one-to-one correspondence with commutative monoid-like objects in a base symmetric monoidal model category.

This general definition might seem somewhat obscure, so we'd rather mention the most important examples of base symmetric monoidal model category, and the corresponding notion of commutative monoid-like objects. In each of the following situations, HAG will provide a context in which one can do algebraic geometry (and in particular, talk about schemes, algebraic spaces, stacks ...), hence giving rise to various geometries.

1. The model category Ab of abelian groups (with its trivial model structure) and the tensor product of abelian groups. Commutative monoid objects are commutative rings. The corresponding geometry is the usual, Grothendieck-style algebraic geometry.
2. The model category Mod(O) of O-modules over some ringed site (with the trivial model structure) and the tensor product of O-modules. Commutative monoid objects are sheaves of commutative O-algebras. The corresponding geometry is called relative algebraic geometry, and was introduced and studied in [Ha, De].
3. The model category C(k) of complexes over some ring k and the tensor product of complexes (see [Ho, §2.3]). Commutative monoid-like objects are commutative E_¥-algebras over k ([Kr-Ma]). The corresponding geometry is the so-called derived algebraic geometry that we are going to discuss in details in this paper, and for which one possible avatar is the theory of dg-schemes and dg-stacks of [Ci-Ka1, Ci-Ka2].
4. The model category Sp of symmetric spectra and the smash product (see [Ho-Sh-Sm]), or equivalently the category of -modules (see [EKMM]). Commutative monoid-like objects are E_¥-ring spectra, or commutative -algebras. We call the corresponding geometry brave new algebraic geometry, quoting the expression brave new algebra introduced by F. Waldhausen (for more details on the subject, see e.g. [Vo]).
5. The model category Cat of categories and the direct product (see, e.g. [Jo-Ti]). Commutative monoid-like objects are symmetric monoidal categories. The corresponding geometry does not have yet a precise name, but could be called 2-algebraic geometry, since vector bundles in this setting will include both the notion of 2-vector spaces (see [Ka-Vo]) and its generalization to 2-vector bundles.

For the general framework, we refer the reader to [HAG-I, HAG-II]. The purpose of the present note is to present one possible incarnation of HAG through a concrete application to derived algebraic geometry (or "DAG" for short).

1.2 What's DAG ?

Of course, the answer we give below is our own limited understanding of the subject.

As far as we know, the foundational ideas of derived algebraic geometry (whose infinitesimal theory is also referred to as derived deformation theory, or "DTT" for short) were introduced by P. Deligne, V. Drinfel'd and M. Kontsevich, for the purpose of studying the so-called derived moduli spaces. One of the main observation was that certain moduli spaces were very singular and not of the expected dimension, and according to the general philosophy this was considered as somehow unnatural (see the hidden smoothness philosophy presented in [Ko1]). It was therefore expected that these moduli spaces are only truncations of some richer geometric objects, called the derived moduli spaces, containing important additional structures making them smooth and of the expected dimension. In order to illustrate these general ideas, we present here the fundamental example of the moduli stack of vector bundles (see the introductions of [Ci-Ka1, Ci-Ka2, Ka1] for more motivating examples as well as philosophical remarks).

Let C be a smooth projective curve (say over ), and let us consider the moduli stack V ect_n(C) of rank n vector bundles on C (here V ect_n(C) classifies all vector bundles on C, not only the semi-stable or stable ones). The stack V ect_n(C) is known to be an algebraic stack (in the sense of Artin). Furthermore, if E Î V ect _n(C)() is a vector bundle on C, one can easily compute the stacky tangent space of V ect_n(C) at the point E. This stacky tangent space is actually a complex of -vector spaces concentrated in degrees [-1,0], which is easily seen to be quasi-isomorphic to the complex C^*(C_Zar,End (E))[1] of Zariski cohomology of C with coefficient in the vector bundle End(E) = E Ä E^*. Symbolically, one writes

Let now S be a smooth projective surface, and V ect_n(S) the moduli stack of vector bundles on S. Once again, V ect_n(S) is an algebraic stack, and the stacky tangent space at a point E Î V ect _n(S)() is easily seen to be given by the same formula

The main idea of derived algebraic geometry is that V ect_n(S) is only the truncation of a richer object V ect_n(S), called the derived moduli stack of vector bundles on S. This derived moduli stack, whatever it may be, should be such that its tangent space at a point E is the whole complex C^*(S,End (E))[1], or in other words,

In the above example of the stack of vector bundles, the tangent space of V ect_n(S) is expected to be a complex concentrated in degree [-1,1]. More generally, one can get convinced that tangent spaces of derived moduli (1-)stacks should be complexes concentrated in degree [-1,¥[ (see [Ci-Ka1]). It is therefore pretty clear that in order to make sense of an object such as V ect_n(S), schemes and algebraic stacks are not enough, and one should look for a more general definition of spaces. This leads to the following general question.

Problem: Provide a framework in which derived moduli stacks can actually be constructed. In particular, construct the derived moduli stack of vector bundles V ect(S) discussed above.

Several construction of formal derived moduli spaces have appeared in the litterature (see for example [Ko-So, So]), a general framework for formal DAG have been developed by V. Hinich in [Hin2], and pro-representability questions were investigated by Manetti in [Man]. So, in a sense, the formal theory has already been worked out, and what remains of the problem above is an approach to global DAG.

A first approach to the global theory was proposed by M. Kapranov and I. Ciocan-Fontanine, and is based on the theory of dg-schemes or more generally of dg-stacks (see [Ci-Ka1, Ci-Ka2]). A dg-scheme is, roughly speaking, a scheme together with an enrichement of its structural sheaf into commutative differential graded algebras. This enriched structural sheaf is precisely the datum encoding the derived information.

This approach has been very successful, and many interesting derived moduli spaces (or stacks) have already been constructed as dg-schemes (e.g. the derived version of the Hilbert scheme, of the Quot scheme, of the stack of stable maps, and of the stack of local systems on a space have been defined in [Ka2, Ci-Ka1, Ci-Ka2]). However, this approach have encountered two major problems, already identified in [Ci-Ka2, 0.3].

1. The definition of dg-schemes and dg-stacks seems too rigid for certain purposes. By definition, a dg-scheme is a space obtained by gluing commutative differential graded algebras for the Zariski topology. It seems however that certain constructions really require a weaker notion of gluing, as for example gluing differential graded algebras up to quasi-isomorphisms.
2. The notion of dg-schemes is not very well suited with respect to the functorial point of view, as representable functors would have to be defined on the derived category of dg-schemes (i.e. the category obtained by formally inverting quasi-ismorphisms of dg-schemes), which seems difficult to describe and to work with. As a consequence, the derived moduli spaces constructed in [Ka2, Ci-Ka1, Ci-Ka2] do not arise as solution to natural derived moduli problems, and are constructed in a rather ad-hoc way.

The first of these difficulties seems of a technical nature, whereas the second one seems more fundamental. It seems a direct consequence of these two problems that the derived stack of vector bundles still remains to be constructed in this framework (see [Ka1] and [Ci-Ka1, Rem. 4.3.8]).

It is the purpose of this note to show how HAG might be applied to provide a framework for DAG in which problems (1) and (2) hopefully disappear. We will show in particular how to make sense of various derived moduli functors whose representability can be proved in many cases.