Suppose that A is an abelian category with Ab4*, i.e., products exists and are exact. Then a product of quasi-isomorphisms is a quasi-isomorphism and we can define products in D(A) just by taking the product of underlying complexes. If A has just Ab3* (i.e., products exist) then this doesn’t work.

Let A be a Grothendieck abelian category. Then A has Ab3* (this does not follow directly from the definitions, but rather is an example of what Akhil was referring to here). In a nice short paper entitled Resolution of unbounded complexes in Grothendieck categories, C. Serpé shows that the category of unbounded complexes over A has enough K-injectives. There are other references; I like this one because its proof is a modification of Spaltenstein’s argument in his famous paper Resolutions of unbounded complexes. Combining these results we can show products exist in D(A).

In fact, I claim that products exist in D(A) if A has Ab3* and enough K-injective complexes. Namely, suppose that we have a collection of complexes K^*_λ in A parametrized by a set Λ. Choose quasi-isomorphisms K^*_λ —> I^*_λ into K-injective complexes I^*_λ and consider the termwise product

Π_{λ ∈ Λ} I^*_λ

I claim this is a product of the objects K^*_λ in D(A). Namely, it is a result in the Spaltenstein paper that the product of K-injective complexes is K-injective. Hence to check our assertion we need only check this on the level of maps up to homotopy, where it is clear.

OK, now what I want to know is this: Let A be a Grothendieck abelian category and let B ⊂ A be a subcategory such that D_B(A) makes sense. When does D_B(A) have products? Are there some reasonable assumptions we can make to guarantee this?

At the end of a post on crystalline cohomology I asked a question which was answered the same day by Bhargav Bhatt. It turns out that all cohomology groups of the sheaf Ω^1 (differentials compatible with divided powers) on the crystalline site of a scheme in characteristic p are zero! As a consequence Bhargav and I get a short proof of Berthelot’s comparison theorem relating crystalline and de Rham cohomology. If you think you’re confused, note that the de Rham cohomogy is computed on the scheme and not on the crystalline site. Here is a link to a recent version of the write-up — it should appear on the arxiv soon.

As an example, let’s consider an algebraically closed field k and the power series ring A = k[[t]]. It turns out that A has a p-basis, namely {t}. This simply means that every element a of A can be uniquely written as ∑_{i = 0,1,…,p-1} a_i^pt^i. Let W = W(k) be a Cohen ring for k (i.e., the Witt ring). By a result of Berthelot and Messing the category of crystals in quasi-coherent modules on (Spec(A)/Z_p)_{cris} is equivalent to the category of pairs (M, ∇) where M is a p-adically complete W[[t]]-module and ∇ : M —> Mdt is a topologically quasi-nilpotent connection. Given F corresponding to (M, ∇) the comparison theorem (in this special case) states

the complex ∇ : M —> Mdt is quasi-isomorphic to RΓ(F).

You can generalize this to power series rings in more variables. In fact, you can’t find exactly this statement in the preprint linked to above; it is just that the method of the proof works in this case too. Upshot: comparison with the de Rham complex works for rings with p-bases.

Computing crystalline cohomology over a power series ring is relevant in situations where one wants to do deformation theory. For example, I was recently asked by Davesh Maulik if there is an explanation of Artin’s result on specialization of Picard lattices of supersingular K3 surfaces which avoids the formal Brauer group. What Artin proves is that the Neron-Severi rank doesn’t jump in a family of supersingular K3 surfaces. It turns out that, using crystalline cohomology, given a family of K3’s X/k[[t]], you can split this question into two parts:

1. When can you lift elements of H^2_{cris}(X_0/W) to elements of H^2_{cris}(X/W)?
2. Can you lift an invertible sheaf on X_0 to X if its crystalline c_1 lifts to X?

Of course then you generalize (also Artin’s result is more general) and you can ask these questions for any smooth proper X/k[[t]]. It turns out that both questions have a positive answer under some conditions. I have written a short note with a discussion. Enjoy!

This is a follow-up to Akhil Mathew’s blog post which explains that the category of quasi-coherent sheaves on a scheme X is a Grothendieck abelian category. The key is a result of Gabber: given a scheme X there exists a cardinal κ such that every quasi-coherent sheaf is the directed colimit of its κ-generated quasi-coherent subsheaves. It follows by a standard argument that the embedding QCoh(O_X) —> Mod(O_X) has a right adjoint, whence limits exist in QCoh(O_X) and QCoh(O_X) has enough injectives.

Earlier today I wrote this up for the stacks project (it is in the chapter on properties of schemes) and it occurred to me that the exact same results hold for algebraic stacks, with the exact same proof. (The proof is one of these “randomly pick elements and see what happens” arguments, kinda like this post.) I’ll check the details and write out the proof some time later this week; keep watching this feed to see it appear.

Anyway, I guess it is just one of those general facts… easy to prove but hard to use.

Edit 10/17/2011. Beware of the following facts on quasi-coherent modules:

• It isn’t true that a product of quasi-coherent modules is quasi-coherent.
• An injective object in QCoh(O_X) is not always injective O_X-module.
• Cohomology using resolutions in QCoh(O_X) does not agree with cohomology.
• There exists a ring A and an injective A-module I such that the quasi-coherent sheaf I~ associated to I isn’t flasque, I~ isn’t an injective O_X-module, and there exists an open U of Spec(A) such that I~|_U isn’t an injective object of QCoh(O_U).
• D^+_{QCoh}(O_X) isn’t equivalent to D^+(QCoh(O_X)) in general.
• The coherator Q : Mod(O_X) —> QCoh(O_X) isn’t exact in general.
• And so on.

This post is a follow-up on the post on adequate modules. There I described a construction of the higher direct images of a quasi-coherent sheaf in terms of the morphism of big fppf sites associated to a quasi-compact and quasi-separated morphism of schemes. As I mentioned in my last post, this is now implemented (in the stacks project) for quasi-compact and quasi-separated morphisms of algebraic stacks, with the slight modification that we work with locally quasi-coherent modules with the flat base change property. (In this post, ‘module’ means fppf O-module.)

Given an algebraic stack X let’s denote M_X either the abelian category of locally quasi-coherent modules with the flat base change property, or the abelian category of adequate modules. Since M_X is a weak Serre subcategory of Mod(O_X) we have the derived category D_M(X) := D_{M_X}(O_X) of complexes of O_X-modules whose cohomology sheaves are objects of M_X. The category of parasitic objects in M_X is a Serre subcategory. We define

D_{QCoh}(X) = D_M(X) / complexes with parasitic cohomology sheaves

If X is a scheme, then this definition recovers the usual notion (see chapter on adequate modules for the adequate case). So now let’s think about the derived pullback of a quasi-coherent module along a morphism f : X —> Y of algebraic stacks. It is clear that we have an induced functor

f^* : D_M(Y) —> D_M(X)

In fact f^* : Mod(O_Y) —> Mod(O_X) is exact (big sites!) and transforms objects of M_Y into objects of M_X. But f^* does not preserve parasitic modules if f isn’t a flat morphism of algebraic stacks. We define Lf^* : D_{QCoh}(Y) —> D_{QCoh}(X) as the left derived functor (in the sense of Deligne, see Definition Tag 05S9) of the displayed functor f^* above! What could be more natural?

Thus the question isn’t “What is derived pullback?” but it is “When is derived pullback everywhere defined?”

However, a better question is: “Does there exist a functor L^* : D_{QCoh}(Y) —> D_M(Y) which is left adjoint to the quotient functor q_Y : D_M(Y) —> D_{QCoh}(Y)?” If it exists, then Lf^* = q_X o f^* o L^* where q_X : D_M(X) —> D_{QCoh}(X) is the quotient functor for X, so derived pullback exists for any morphism with target Y. The existence of L^* is equivalent to asking the quotient map q_Y to be a Bousfield colocalization: for every E in D_M(Y) there should be a distinguished triangle

E’ —> E —> C —> E'[1]

in D_M(Y) where C is parasitic and Hom(E’, C’) = 0 for every parasitic object C’ of D_M(Y). Formulated in this way, there is lots of general theory we can rely on to (dis)prove the existence.

For example if the subcategory of parasitic objects of D_M(Y) has products and they agree with products in D_M(Y) we are through I think; this isn’t as crazy as it sounds, e.g., the category of quasi-coherent sheaves on a scheme has products, see Akhil Mathew’s post. (In any case being parasitic is preserved under products.) Hmm? I’ll think more.

Since the last updateon July 1st we have added

1. an introductory chapter on algebraic stacks,
2. a short chapter on Brauer groups of fields,
3. a chapter on cohomology of sheaves on algebraic spaces,
4. a chapter on adequate modules on schemes as discussed in this and this post,
5. a chapter on sheaves on stacks, following the layout suggested in this post,
6. a chapter on cohomology on algebraic stacks which contains a discussion of functoriality for quasi-coherent sheaves on algebraic stacks including higher direct images for quasi-compact and quasi-separated morphisms.

Let’s discuss the last topic a bit. We use locally quasi-coherent sheaves (sheaves that we called “quasi quasi-coherent” in this post) as an essential technical tool to prove the results. We also think about parasitic modules, which was a hint in an email of Martin Olsson. It turns out that the category of quasi-coherent modules is the quotient of the category of locally quasi-coherent modules satisfying the flat base change condition by the subcategory of parasitic ones. Then one can proceed as discussed in the post on adequate modules. This is not precisely how the results are stated, since the description of the category of a quasi-coherent sheaves as a quotient category isn’t needed. The main result at the moment is Proposition Tag 077A.

Enjoy!

On Mathoverflow Anton Geraschenko asks the following question:

Suppose F : Sch^{opp}→Set is a sheaf in the fpqc topology, has quasi-compact representable diagonal, and has an fpqc cover by a scheme. Must F be an algebraic space? That is, must F have an étale cover by a scheme?

The question isn’t well posed as the question does not specify _exactly_ what is meant by an “fpqc cover by a scheme”. Does it mean a morphism which is a surjection of sheaves in the fpqc topology? In that case you get counter examples by looking at ind schemes (for example the functor of morphisms A^1 —> A^1). Does it mean a flat morphism which is a surjection of sheaves in the fpqc topology? In that case, I think there is a counter example by taking an ind-scheme where the transition morphisms are flat closed immersions (I can explain but it isn’t interesting). Does it mean a flat, surjective, quasi-compact morphism? In this case it is more difficult to give a counter example, but I think I have one. Before I get into it, note that algebraic spaces in general do not have such coverings, so that the resulting category of “fpqc-spaces” does not contain the category of algebraic spaces.

Here is my idea for a counter example. (I’m having a kind of deja vu here, so it is perhaps somebody else’s idea? Please let me know if so.) Consider the functor F = (P^1)^∞, i.e., for a scheme T the value F(T) is the set of f = (f_1, f_2, f_3, …) where each f_i : T —> P^1 is a morphism. A product of sheaves is a sheaf, so F is a sheaf. The diagonal is representable: if f : T —> F and g : S —> F, then T ×_F S is the scheme theoretic intersection of the closed subschemes T ×_{f_i, P^1, g_i} S inside the scheme T × S. Consider U = (SL_2)^∞ with its canonical morphism U —> F. Note that U is an affine scheme. OK, and now you can show that the morphism U —> F is flat, surjective, and even open. Without giving all the details, if f : T —> F is a morphism, then you show that Z = T &times_F U is the infinite fibre product of the schemes Z_i = T ×_{f_i, P^1} SL_2 over T. Each of the morphisms Z_i —> T is surjective, smooth, and affine which implies the assertions. In particular, if F where an algebraic space it would be a quasi-compact and separated (by our description of fibre products over F) algebraic space. Hence cohomology of quasi-coherent sheaves would vanish above a certain cutoff (see Proposition Tag 072B and remarks preceding it). But clearly by taking O(-2,…,-2,0,…) on F = (P^1)^∞ we get a quasi-coherent sheaf whose cohomology is nonzero in an arbirary positive degree.

A long time ago I attended a semester course by Faltings on crystalline cohomology. This was when I was visiting Princeton with Frans Oort as a graduate student. I learned a lot in his course and it really helped me with my thesis (I eventually used a crystalline ext group to define a Dieudonne module for group schemes in characteristic p). Faltings never used any notes, except during the lecture where he explained the crystalline cohomology of an abelian variety (and then it was a tiny piece of paper he pulled out of his breast pocket). Of course, yours truly can’t even teach a calculus course without notes…!

Dumbed down as much as possible here are some ingredients of crystalline cohomology.

Sheaf theory. Let C be a site. Suppose there is an object X in C such that (1) every object T of C has a map T —> X and (2) the products X^n exist in C. Then X —> * is surjective and we obtain a Cech-to-cohomology spectral sequence H^m(X^n, F) => H^{n + m}(F) for any abelian sheaf F. If H^m(X^n, F) = 0 for m > 0 then the Cech complex

0 —> F(X) —> F(X^2) —> F(X^3) —> …

computes cohomology. Sometimes X is an ind-object of C and not a real object. Then the above still works, except that you have to clarify what the values F(X^n) and H^m(X^n, F) are.

Thickenings: Let A be a finite type F_p-algebra. Set S = Spec(A). Consider the site C consisting of finite order thickenings S —> T where T is a scheme over Z_p. We denote an object just T with the immersion S —> T understood. Coverings are jointly surjective families (T_i —> T). Choose a surjection Z_p[x_1, …, x_r] —> A with kernel J. Let B be the J-adic completion of Z_p[x_1, …, x_r]. Then X = Spec(B) is an ind-object of C such that every T has a morphism to X (because of the universal property of polynomial rings). The products X^n = Spec(B(n + 1)) exist in the category of thickenings with B(n + 1) defined as the completion of a polynomial ring in r(n + 1) variables. Looking at the structure sheaf on this site we get that its cohomology is computed by the Cech complex

0 —> B —> B(1) —> B(2) —> …

We’d like to rewrite this complex in another way, but that’s hard to do without divided powers.

Divided power thickenings: Here we consider S —> T as above where the ideal defining S in T is endowed with a divided power structure. In this case the universal ring isn’t the J-adic completion of the polynomial ring, but it’s (a suitably completed) divided power envelope D of J in Z[x_1, …, x_r]. Similarly X^n corresponds to a divided power envelope D(n + 1) of a polynomial ring in r(n + 1) variables. The cohomology of the structure sheaf is computed by the complex

0 —> D —> D(1) —> D(2) —> …

just as before.

Crystalline Poincare lemma: There is a module of differentials Ω_D^1 where the differentials are compatible with the divided powers. It turns out that this is free on the elements dx_i over D. We get a de Rham complex Ω_D^*. A version of the Poincare lemma states that the complex displayed above is canonically quasi-isomorphic to Ω_D^* (as complexes of abelian groups). The usual method for proving this, very roughly, is to consider a double complex with terms Ω_{D(q + 1)/D(q)}^p, use spectral sequences. One concludes using some homological algebra (analogous to Grothendieck’s thing with Amitsur’s complex) and a more classical Poincare lemma for a divided power polynomial algebra.

Upshot. It’s easier and often convenient to think of crystalline cohomology in terms of de Rham cohomology of suitable algebras. In this approach you prove the independence of the choice of the particular algebra directly. In particular, you don’t have to consider the crystalline site at all. This works for nonaffine schemes as well, but you then you have to consider affine open coverings, a double complex, etc.

Question: Suppose you look at the sheaf Ω^1 which associates to an object T of the crystalline site the sections of Ω_T^1 (differentials compatible with divided powers). Does anybody know what should be H^i(Ω^1)? How about H^0?

Just yesterday, upon some prodding from Michael Thaddeus, I added a two short sections comparing algebraic spaces and algebraic stacks in the fppf and étale topology (see the corresponding sections of chapters Bootstrap and Criteria for Representability). Let me just tell you what the statement is. For algebraic spaces the result is that

If F is a sheaf on (Sch) in the étale topology whose diagonal is representable by schemes and which has an étale covering by a scheme, then F is also a sheaf in the fppf topology hence an algebraic space (as defined in the stacks project).

For algebraic stacks the result is that

If X is a stack in groupoids over (Sch) with the étale topology whose diagonal is representable by algebraic spaces and which has a smooth covering by a scheme, then X is also a stack for the fppf topology hence an algebraic stack (as defined in the stacks project).

Till yesterday I had filed away this material under the heading: “Things that have to go into the stacks project at some point but which are not as interesting as other material I am working on now.” However, I probably should have worked it out sooner as some related remarks in the stacks project were misleading (I have now removed these remarks, see the red text in this commit).

This is a follow-up on the previous post with the same title. So this morning my inbox contained a short email from Bhargav about a typo in the stacks project. I recorded the change here. As you can see there I (finally) figured out how to tell git who authored this commit. So from now on, if you email an improvement here, then you’ll end up showing up as the author in the git logs. (Apologies for those who’ve sent me typos etc in the past before I figured this out.)