4 The geometry of D-stacks

We are now ready to start our geometric study of D-stacks. We will define in this Section a notion of (1)-geometric D-stack, analogous to the notion of algebraic stack (in the sense of Artin). We will also present the theory of tangent D-stacks, as well as its relations to the cotangent complex.

4.1 Geometricity

A 1-geometric D-stack is a quotient of a disjoint union of representable D-stacks by the action of a smooth affine groupoid. In order to define precisely this notion, we need some preliminaries.

1. Let f : F-®F' be a morphism in Ho(D - Aff). We say that f is a representable morphism, if for any cdga A, and any morphism SpecA-®F', the homotopy pull-back F ×_F'^hSpecA is a representable D-stack (see Definition 3.2).
2. We say that a D-stack F has a representable diagonal if the diagonal morphism D : F-®F ×F is representable. Equivalently, F has a representable diagonal if any morphism SpecA-®F from a representable D-stack is a representable morphism.
3. Let u : A-®B be a morphism of cdga's. We say that u is strongly smooth3 if there exists an étale covering B-®B', and a factorization with A Ä[X₁,...,X_n]-®B' formally étale; here [X₁,...,X_n] is the usual polynomial ring, viewed as a cdga concentrated in degree zero. This is an extension of one of the many equivalent characterizations of smoothness for morphisms of schemes (see [Mil, Prop. 3.24 (b)]); we learn it from [MCM] in which smooth morphisms (called there thh-smooth) between -algebras are defined.
4. A representable morphism of D-stacks f : F-®F' is called strongly smooth, if for any morphism from a representable D-stack SpecA-®F', the induced morphism is induced by a strongly smooth morphism of cdga's.
5. A morphism f : F-®F' in Ho(D - Aff) is called a covering (or an epimorphism), if the induced morphism p₀(F)-®p₀(F') is an epimorphism of sheaves.

Note that definition (4) above makes sense because of (1) and because the functor ASpecA is fully faithful on the homotopy categories.

Using these notions, we give the following

The following proposition collects some of the basic properties of strongly geometric D-stacks.

In particular, Proposition 3.5 and point (3) above, tell us that the derived stack BG of G-torsors is a stongly geometric D-stack for any linear algebraic group G.

We are not going to present the theory in details in this work, but we would like to mention that standard notions in algebraic geometry (e.g. smooth or flat morphisms, sheaves, cohomology ...) can be extended to strongly geometric D stacks. We refer to [La-Mo] and [S1] for the main outline of the constructions. The reader will find all details in [HAG-II].

4.2 Modules, linear D-stacks and K-theory

Let _a be the additive group scheme (over ) and consider the object i_a Î Ho(D - Aff). It has a nice model in D - Aff which is Spec[T] that we will denote by O (note that [T] as a cdga in degree 0 is a cofibrant object). The D-stack O is actually an object in commutative -algebras, explicitly given by

Let us now fix a D-stack F, and consider the comma category D - Aff/F of D-stacks over F; this category is again a model category for the obvious model structure. We define the relative structural D-stack by

Then we can consider the category O_F -Mod, of objects in O_F -modules in the category D -Aff/F. If one defines equivalences and fibrations through the forgetful functor D - Aff/F-®D - Aff, the category O_F - Mod becomes a model category. It has moreover a natural tensor product structure Ä_OF. The model category O_F - Mod is called the model category of O-modules on F.

Let A be a cdga and M be an (unbounded) A-dg module. We define a O_SpecA-module in the following way.

Let G be a fibrant resolution functor on the model category CDGA. For any cdga B, and any integer n, we define (B)_n as the set of pairs (u,m), where u is a morphism of cdga's A-®G_n(B) (i.e. u Î SpecA(B)), and m is a degree 0 element in M Ä_AG_n(B) (i.e. m is a morphism of complexes of -vector spaces m : -®M Ä_AG_n(B)). This gives a simplicial set [n] (B)_n, and therefore defines an object in D - Aff

Clearly, the projection (u,m)u in the notation above induces a morphism -®SpecA. Finally, this object is endowed in an obvious way with a structure of O_SpecA-module.

This construction, M induces a functor

The construction M described above also has a dual version, denoted by MSpel(M) and defined in a similar way.

Let A be a cdga and M be an (unbounded) dg-A-module. For a cdga B and an integer n, we define Spel(M)(B)_n to be the set of pairs (u,a), where u : A-®G_n(B) is a morphism of cdga, and a : M-®G_n(B) is a morphism of dg-A-modules. This defines a D-stack BSpel(M)(B) which has a natural projection (u,a)u, to the SpecA. Once again, Spel(M) comes equipped with a natural structure of O_SpecA-module. Also, this Spel construction can be derived, to get a functor

One can prove that the homotopy category of perfect O-modules on SpecA is naturally equivalent to the full sub-category of Ho(A - Mod) consisting of strongly dualizable modules, or equivalently of dg-A-modules which are retracts of finite cell modules (in the sense of [Kr-Ma, §III.1]). In particular, if A is concentrated in degree 0, then the homotopy category of perfect O-modules on SpecA is naturally equivalent to the derived category of bounded complexes of finitely generated projective A-modules.

This notion of perfect O-modules can be used in order to define the K-theory of D-stacks. For any D-stack F, one can consider the homotopy category of perfect O-modules on F, that we denote by D_Perf(F). This is a triangulated category having a natural Waldhausen model WPerf(F), from which one can define the K-theory spectrum on the D-stack F, as K(F) := K(WPerf(F)). The tensor product of O-modules makes K(F) into an E_¥-ring spectrum. Of course, when X is a scheme K(iX) is naturally equivalent to the K-theory spectrum of X as defined in [TT].

A related problem is that of defining reasonable Chow groups and Chow rings for strongly geometric D-stacks, receiving Chern classes from the K-theory defined above. We are not aware of any such constructions nor we have any suggestion on how to approach the question. It seems however that an intersection theory over D-stacks would be a very interesting tool, as it might for example give new interpretations (and probably extensions) of the notion of virtual fundamental class defined in [Be-Fa]. For this case, the idea would be that for any strongly geometric D-stack F, there exists a virtual fundamental class in the Chow group of its truncation h⁰F. The structural sheaf of F should give rise, in the usual way, to a fundamental class in its Chow group, such that integrating against it over the entire F is the same thing as integrating on its truncation h⁰F against the virtual fundamental class. However, even if there is not yet a theory of Chow groups for D-stacks, if one is satisfied with working with K-theory instead of Chow groups, the obvious class 1 =: [O_F ] Î K₀(F), will correspond exactly to the class of the expected virtual structure sheaf.

4.3 Tangent D-stacks

Let Spec[e] the spectrum of the dual numbers, and let us consider iSpec[e] Î Ho(D - Aff).

Definition 4.8 The tangent D-stack of a D-stack F is defined to be

Note that the zero section morphism Spec-®Spec[e] and the natural projection Spec[e]-®Spec induces natural morphisms

An important remark is that for any D-stack F, the truncation h⁰TF is equivalent to the tangent stack of h⁰F (in the sense of [La-Mo, §17]). In other words, one has

Definition 4.9 If x : iSpec-®F is a point of a D-stack F, then the tangent D-stack of F at x is the homotopy fiber of p : TF-®F at the point x. It is denoted by

Let us now suppose that F is a strongly geometric D-stack. One can show that TF is also strongly geometric. In particular, for any point x in F() the D-stack TF_x is strongly geometric.

Actually much more is true. For any strongly geometric D-stack F, and any point x in F(), the D-stack TF_x is a linear D-stack (over iSpec) as defined in 4.7. Let us recall that this implies the existence of a natural complex W_F,x¹ of -vector spaces (well defined up to a quasi-isomorphism and concentrated in degree ] -¥,1]), with the property that, for any cdga A, there exists a natural equivalence

C()

Definition 4.10 If x : iSpec-®F is a point of a strongly geometric D-stack, then we say that the dimension of F at x is defined if the complex W_F,x¹ has bounded and finite dimensional cohomology. If this is the case, the dimension of F at x is defined by

This linear description of TF_x has actually a global version. In fact, one can define a cotangent complex W_F ¹ of a strongly geometric D-stack, which is in general an O-module on F in the sense of Definition 4.5, which is most of the times quasi-coherent. One then shows that there exists an equivalence of D-stacks over F

An already interesting application of this description, is to the case F = iX, for X a scheme or even an algebraic stack. Indeed, the cotangent complex W_iX¹ mentioned above is precisely the cotangent complex _X of [La-Mo, §17]. The equivalence

[...] Il est très probable que cette théorie pourra s'étendre de façon à donner une correspondance entre complexes de chaines de longeur n, et certaines "n-catégories" cofibrées sur C; et il n'est pas exclus que par cette voie on arrivera également à une "interprétation géométrique" du complexe cotangent relatif de Quillen.

4.4 Smoothness

To finish this part, we investigate various non-equivalent natural notions of smoothness for geometric D-stacks.

Strong smoothness. We have already defined the notion of a strongly smooth morphisms of cdga's in §4.1. We will therefore say that a morphism

Strong smoothness is not very interesting for D-stacks, as a strongly geometric D-stack F will be strongly smooth if and only if it is of the form iF', for F' a smooth algebraic stack.

Standard smoothness. A more interesting notion is that of standard smooth morphisms, or simply smooth morphisms. On the level of cdga's they are defined as follows.

A morphism of cdga's A-®B is called standard smooth (or simply smooth), if there exists an étale covering B-®B', and a factorization

This notion is more interesting than strong smoothness, as a strongly geometric D-stack can be smooth without being an algebraic stack. However, one can check that if F is a smooth strongly geometric D-stack in this sense, then h⁰(F) is also a smooth algebraic stack. In particular, the derived version of the stack of vector bundles on a smooth projective surface, discussed in the Introduction (see also conjecture 5.4), will never be smooth in this sense as its truncation is the stack of vector bundles on the surface which is singular in general).

Nevertheless, smooth morphisms can be used in order to define the following more general notion of geometric D-stacks.

Essentially all we have said about strongly geometric D-stacks is also valid for geometric D-stacks in the above sense. The typical example of a geometric D-stack which is not strongly geometric is BG, where G is a representable group D-stack which is not a scheme. For example, one can take G to be of the from Spel(M) for a non-positively graded bounded complex of finite dimensional vector spaces. Then, G is a representable D-stack (it is precisely SpecL(M), where L(M) is the free cdga on M), and BG is naturally equivalent to Spel(M[-1]). When M[-1] has non-zero H¹ then BG is not representable anymore but is 1-geometric for the above definition.

More generally, the definition above allows one to consider quotient D-stacks [X/G], where X is a representable D-stack and G is a smooth representable group D-stack acting on X.

fp-smoothness. The third notion of smoothness is called fp-smoothness and is the weakest of the three and it seems this is the one which is closer to the smoothness notion referred to in the Derived Deformation Theory program in general. It is also well suited in order for the derived stack of vector bundles to be smooth.

Recall that a morphism of cdga's, A-®B is finitely presented if it is equivalent to a retract of a finite cell A-algebra, or equivalently if the mapping space Map_A/CDGA(B,-) commutes with filtered colimits (this is the same as saying that SpecA commutes with filtered colimits). We will then say that a morphism of geometric D-stacks, F-®F' is locally finitely presented if for any morphism SpecA-®F' there exists a smooth atlas

Of course, one has strongly smooth Þsmooth Þfp-smooth, but each of these implications is strict. For example, a smooth scheme is strongly smooth. Let E be a complex in non-positive degrees which is cohomologically bounded and of finite dimension. Then Spel(E) is smooth but not strongly smooth as it is not a scheme in general. Finally, any scheme which is a local complete interesection is fp-smooth, but not smooth in general.