Could someone give an overview, or just some examples, of “finiteness conditions” for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, but I am interested in particular in the following two things:
1) I once heard Toen say something about this, and that one can express some kind of condition on simplicial sheaves/presheaves/schemes in terms of convergence of some power series, or something along these lines. What kind of power series is this, and what are the definitions/statements?
2) As explained for example in Deligne’s classical papers on Hodge theory, a cohomology theory for varieties (defined as hypercohomology of a complex of sheaves, say) can be extended to simplicial varieties using a spectral sequence. I suspect (but am not sure) that this works well in some sense only under some kind of condition on the simplicial variety, but what would such a condition be?
Some background: I am interested primarily in simplicial sheaves/presheaves on a site coming from algebraic geometry, which would typically be some category of schemes equipped with the Zariski/Nisnevich/etale/flat topology. A simplicial sheaf/presheaf should be thought of a “generalized space”, some of the most important examples being the “motivic spaces” in A1-homotopy theory, and stacks in the sense of Toen.
Cisinski on Math Overflow:
I don’t know what Toën was talking about, but I suspect that it was about finiteness conditions for Artin stacks: the problem is that the usual finiteness conditions we look at for schemes (like the notion of constructibility for l-adic sheaves) do not extend to stacks in a straightforward way, which gives some trouble if one wants to count points (i.e. to define things like Euler characteristics). Some notions of finiteness are developed to define Grothendieck rings of Artin stacks (e.g. in Toën’s paper arXiv:0509098 and in Ekedahl’s paper arXiv:0903.3143), which can be realized by our favourite cohomologies (l-adic, Hodge, etc), but the link with a good notion of finiteness for categories of coefficients over Artin stacks (l-adic sheaves, variation of mixed Hodge structures) does not seem to be fully understood yet, at least conceptually (and by myself).
As for finiteness conditions for sheaves (in some homotopical context), the kind of properties we might want to look at are of the following shape.
Consider a variety of you favourite kind X, and a derived category D(X) of sheaves over some site S associated to X (e.g. open, or étale, or smooth subvarieties over X etc.). For instance, D(X) might be the homotopy category of the model category of simplicial sheaves, or the derived category of sheaves of R-modules. Important finiteness properties can be expressed by saying that for any U in the site S, we have
(1) hocolimᵢ RΓ(U,Fᵢ)= RΓ(U,hocolimᵢ Fᵢ)
where {Fᵢ} is a filtered diagram of coefficients. If you are in such a context, then you can look at the compact objects in D(X), i.e. the objects A of D(X) such that
(2) hocolimᵢ RHom(A,Fᵢ)= RHom(A,hocolimᵢ Fᵢ)
for any filtered diagram {Fᵢ}. In good situations, condition (1) will imply that the category of compact objects will coincide with constructible objects (i.e. the smallest subcategory of D(X) stable under finite homotopy colimits (finite meaning: indexed by finite posets) which contains the representable objects).
Sufficient conditions to get (1) are the following:
a) For simplicial sheaves (as well as sheaves of spectra or R-modules…), a sufficient condition is that the topology on S is defined by a cd-structure in the sense of Voevodsky (see arXiv:0805.4578). These include the Zariski topology, the Nisnevich topology, as well as the cdh topology (the latter being generated by Nisnevich coverings as well as by blow-ups in a suitable sense), at least if we work with noetherian schemes of finite dimension. Note also that topologies associated to cd structures define what Morel and Voevodsky call a site of finite type (in the language of Lurie, this means that, for such sites, the notion of descent is the same as the notion of hyperdescent: descent for infinity-stacks over S can be tested only using truncated hypercoverings; this is the issue discussed by David Ben Zvi above). In practice, the existence of a cd structure allows you to express (hyper)descent using only Mayer-Vietoris-like long exact sequences (the case of Zariski topology was discovered in the 70’s by Brown and Gersten, and they used it to prove Zariski descent for algebraic K-theory).
b) For complexes of sheaves of R-modules, a sufficient set of conditions are
i) the site S is coherent (in the sense of SGA4).
ii) any object of the site S is of finite cohomological dimension (with coefficients in R).
The idea to prove (1) under assumption b) is that one proves it first when all the Fᵢ’s are concentrated in degree 0 (this is done in SGA4 under assumption b)i)). This implies the result when the Fᵢ’s are uniformly bounded. Then, one uses the fact, that, under condition b)ii), the Leray spectral sequence converges strongly, even for unbounded complexes (this done at the begining of the paper of Suslin and Voevodsky “Bloch-Kato conjecture and motivic cohomology with finite coefficients”).
This works for instance for étale sheaves of R-modules, where R=Z/n, with n prime to the residual characteristics. Note moreover that, in the derived category of R-modules, the compact objects (i.e. the complexes A satisfying (2)) are precisely the perfect complexes. The fact that the six Grothendieck operations preserves constructibility can then be translated into the finiteness of cohomology groups (note however that the notion of constructiblity is more complex then this in general: if we work with l-adic sheaves (with Ekedahl’s construction, for instance), then the notion of constructiblity does not agree with compactness anymore). However, condition (1) is preserved after taking the Verdier quotient of D(X) by any thick subcategory T obtained as the smallest thick subcategory which is closed under small sums and which contains a given small set of compact objects of D(X) (this is Thomason’s theorem). This is how such nice properties survive in the context of homotopy theory of schemes for instance. Note also that, in a stable (triangulated) context, condition (2) for A implies that we have the same property, but without requiring the diagrams {Fᵢ} to be filtering.
For your second question, the extension of a cohomology theory to simplicial varieties is automatic (whenever the cohomology is given by a complex of presheaves), at least if we have enough room to take homotopy limits, which is usually the case (and not difficult to force if necessary). The only trouble is that you might lose the finiteness conditions, unless you prove that your favorite simplicial object A satisfies (2). The fact that Hironaka’s resolution of singularities gives the good construction (i.e. gives nice objects for open and/or singular varieties) can be expained by finiteness properties related to descent by blow-ups (i.e. cdh descent), but the arguments needed for this use strongly that we work in a stable context (I don’t know any argument like this for simplicial sheaves). The fuzzy idea is that if a cohomology theory satisfies Nisnevich descent and homotopy invariance, then it satisfies cdh descent (there is a nice very general proof of this in Voevodsky’s paper arXiv:0805.4576 (thm 4.2, where you will see we need to be able to desuspend)); then, thanks to Hironaka, locally for the cdh topology, any scheme is the complement of a strict normal crossing divisor in a projective and smooth variety. As cdh topology has nice finiteness properties (namely a)), and as any k-scheme of finite type is coherent in the cdh topos, this explains, roughly, why we get nice extensions of our cohomology theories (as far as you had a good knowledge of smooth and projective varieties). If we work with rational coefficients, the same principle applies for schemes over an excellent noetherian scheme S of dimension lesser or equal to 2, using de Jong’s results instead of Hironaka’s, and replacing the cdh topology by the h topology (the latter being obtained from the cdh topology by adding finite surjective morphisms): it is then sufficient to have a good control of proper regular S-schemes.
nLab page on Finiteness conditions for simplicial sheaves