3 First examples of D-stacks

Before going further with the geometric properties of D-stacks, we would like to present some examples. More examples will be given in the Section 5.

3.1 Representables

The very first examples of schemes are affine schemes. In the same way, our first example of D-stacks are representable D-stacks2.

We start by fixing a fibrant resolution functor G on the model category CDGA. Recall that this means that for any cdga B, G(B) is a simplicial object in CDGA, together with a natural morphism B-®G(B) that makes it into a fibrant replacement for the Reedy model structure on simplicial objects (see [Ho, §5.2]). In the present situation, one could choose the following standard fibrant resolution functor

Now, for any cdga A, we define a functor

A fundamental property of this functor is the following lemma.

Lemma 3.1 The functor Spec is fully faithful. More generally, for two cdga's A and B, it induces a natural equivalence on the mapping spaces

The above lemma contains two separated parts. The first part states that Spec is fully faithful when considered to have values in Ho(D - Aff^Ù) (i.e. when one forgets about the topology). This first part is a very general result that we call Yoneda lemma for model categories (see [HAG-I, §4.2]). The second part of the lemma states that for a cofibrant cdga A, the object Spec(A) is a D-stack (see Definition 2.10). This is not a general fact, and of course depends on the choice of the topology. Another way to express this last result is to say that the étale topology is sub-canonical.

In particular, Lemma 3.1 implies that the full subcategory of Ho(D -Aff) consisting of representable D-stacks is equivalent to the homotopy category of cdga's.

3.2 Stacks vs. D-stacks

Our second example of D-stacks are simply stacks. In other words, any stack defined over the category of affine schemes with the étale topology gives rise to a D-stack.

Let Alg be the category of commutative -algebras, and Aff = Alg^op its opposite category. Recall that there exists a model category of simplicial presheaves on Aff for the étale topology (see [Ja1]). We will consider its projective version described in [Bl], and denote it by Aff. This model category is called the model category of stacks for the étale topology. Its homotopy category Ho(Aff) contains as full subcategories the category of sheaves of sets and the category of stacks in groupoids (see e.g. [La-Mo]). More generally, one can show that the full subcategory of n-truncated objects in Ho(Aff) is naturally equivalent to the homotopy category of stacks in n-groupoids (unfortunately there are no references for this last result until now but the reader might consult [Hol] for the case n = 1). In particular, Ho(Aff) contains as a full subcategory the category of schemes, and more generally of Artin stacks.

There exists an adjunction

Lemma 3.3 The functor

is fully faithful.

The important consequence of the previous lemma is that Ho(D - Aff) contains schemes, algebraic stacks ..., as full sub-categories.

Warning: The full embedding i does not commute with homotopy pull-backs, nor with internal Hom-D-stacks.

This warning is the real heart of DAG: the category of D-stacks contains usual stacks, but these are not stable under the standard operations of D-stacks. In other words, if one starts with some schemes and performs some constructions on these schemes, considered as D-stacks, the result might not be a scheme anymore. This is the main reason why derived moduli spaces are not schemes, or stacks in general !

Notations. In order to avoid confusion, a scheme or a stack X, when considered as a D-stack will always be denoted by i(X), or simply by iX.

The full emdedding i = j_! has a right adjoint j^* = j^*. It will be denoted by

Terminology. Points with values in commutative algebras will be called classical points.

We just saw that a D-stack F and its truncation h⁰(F) always have the same classical points.

Given two stacks F and G in Aff, there exists a stack of morphisms HOM(F,G), that is the derived internal Hom's of the model category Aff (see [HAG-I, §4.7]). As remarked above, the two objects iHOM(F,G) and HOM(iF,iG) are different in general. However, one has

3.3 dg-Schemes

We have just seen that the homotopy category of D-stacks Ho(D - Aff) contains the categories of schemes and algebraic stacks. We will now relate the notion of dg-schemes of [Ci-Ka1, Ci-Ka2] to D-stacks.

Recall that a dg-scheme is a pair (X,A_X), consisting of a scheme X and a sheaf of O_X-cdga's on X such that A_X⁰ = O_X (however, this last condition does not seem so crucial). For the sake of simplicity we will assume that X is quasi-compact and separated. We can therefore take a finite affine open covering U = {U_i}_i of X, and consider its nerve N(U) (which is a simplicial scheme)

For each integer n, let A(n) be the cdga of global sections of A_X on the scheme N(U)_n. In other words, one has

The simplicial structure on N(U) makes [n]A(n) into a co-simplicial diagram of cdga's. By applying levelwise the functor Spec, we get a simplicial object [n]SpecA(n) in D -Aff. We define the D-stack Q(X,A_X) Î Ho(D - Aff) to be the homotopy colimit of this diagram

One can check, that (X,A_X)Q(X,A_X) defines a functor

Question: Is the functor Q fully faithful ?

We do not know the answer to this question, and there are no real reasons for this answer to be positive. As already explained in the Introduction, the difficulty comes from the fact that the homotopy category of dg-schemes seems quite difficult to describe. In a way, it might not be so important to know the answer to the above question, as until now morphisms in the homotopy category of dg-schemes have never been taken into account seriously, and only the objects of the category Ho(dg - Sch) have been shown to be relevant. More fundamental is the existence of the functor Q which allows to see the various dg-schemes constructed in [Ka2, Ci-Ka1, Ci-Ka2] as objects in Ho(D - Aff).

3.4 The D-stack of G-torsors

As our last example, we present the D-stack of G-torsors where G is a linear algebraic group G. As an object in Ho(D -Aff) it is simply iBG (where BG is the usual stack of G-torsors), but we would like to describe explicitly the functor CDGA-®SSet it represents.

Let H := O(G) be the Hopf algebra associated to G. By considering it as an object in the model category of commutative differential graded Hopf algebras, we can take a cofibrant model QH of H, as a dg-Hopf algebra. It is not very hard to check that QH is also a cofibrant model for H is the model category of cdga's. Using the co-algebra structure on QH, one sees that the simplicial presheaf

Next, we consider the category iG-Mod, of objects in D -Aff together with an action of iG. If one sees iG as a monoid in D - Aff, the category iG - Mod is simply the category of modules over iG. The category iG - Mod is equipped with a notion of weak equivalences, that are defined through the forgetful functor iG-Mod-®D -Aff (therefore a morphism of iG-modules is a weak equivalence if the morphism induced on the underlying objects is a weak equivalence in D - Aff). More generally, there is a model category structure on iG - Mod, such that fibrations and equivalences are defined on the underlying objects. For any object F Î iG - Mod, we also get an induced model structure on the comma category iG - Mod/F. In particular, it makes sense to say that two objects G-®F and G'-®F in iG - Mod are equivalent over F, if the corresponding objects in Ho(iG - Mod/F) are isomorphic.

Let Q be a cofibrant replacement functor in the model category CDGA. For any cdga A, we have SpecQA Î D - Aff, the representable D-stack represented by A Î D - Aff, that we will consider as iG-module for the trivial action. A G-torsor over A is defined to be a iG-module F Î iG - Mod, together with a fibration of iG-modules F-®SpecQA, such that there exists an étale covering A-®B with the property that the object

For a cdga A, G-torsors over A form a full sub-category of iG-Mod/SpecQA, that will be denoted by G-Tors(A). This category has an obvious induced notion of weak equivalences, and these equivalences form a subcategory denoted by wG - Tors(A). Transition morphisms wG - Tors(A)-®wG - Tors(B) can be defined for any morphism A-®B by sending a G-torsor F-®SpecQA to the pull-back F ×_SpecQASpecQB-®SpecQB. With a bit of care, one can make this construction into a (strict) functor

We are now ready to define our functor

An important case is G = Gl_n, for which we get that the image under i of the stack V ect_n of vector bundles of rank n is equivalent to BGl_n as defined above.