This post is another attempt to explain how incredibly useful the notion of a cocontinuous functor of sites really is. I already tried once here.

Let u : C —> D be a functor between sites. We say u is cocontinuous if for every object U of C and every covering {V_j —> u(U)} in D there exists a covering {U_i —> U} in C such that {u(U_i) —> U} refines {V_j —> u(U)}. This is the direct translation of SGA 4, II, Defintion 2.1 into the language of sites as used in the stacks project and in Artin’s notes on Grothendieck topologies. Note that we do not require that u transforms coverings into coverings, i.e., we do not assume u is continuous. Often the condition of cocontinuity is trivial to check.

Lemma Tag 00XO A cocontinuous functor defines a morphism of topoi g : Sh(C) —> Sh(D) such that g^{-1}G is the sheaf associated to U |—> G(u(U)).

The reader should contrast this with the “default” which is morphisms of topoi associated to continuous functors (where one has to check the exactness of the pull back functor explicitly in each case!). Let’s discuss some examples where the lemma applies.

The standard example is the functor Sch/X —> Sch/Y associated to a morphism of schemes X —> Y for any of the topologies Zariski, etale, smooth, syntomic, fppf. This defines functoriality for the big topoi. This also works to give functoriality for big topoi of algebraic spaces and algebraic stacks. In exactly the same way we get functoriality of the big crystalline topoi.

Another example is any functor u : C —> D between categories endowed with the chaotic topology, i.e., such that sheaves = presheaves. Then u is cocontinuous and we get a morphism of topoi Sh(C) —> Sh(D).

Finally, an important example is localization. Let C be a site and let K be a sheaf of sets. Let C/K be the category of pairs (U, s) where U is an object of C and s ∈ K(U). Endow C/K with the induced topology, i.e., such that {(U_i, s_i) —> (U, s)} is a covering in C/K if and only if {U_i —> U} is a covering in C. Then C/K —> C is cocontinuous (and continuous too) and we obtain a morphism of topoi Sh(C/K) —> Sh(C) whose pullback functor is restriction.

What I am absolutely not saying is that the lemma above is a “great” result. What I am saying is that, in algebraic geometry, the lemma is easy to use (no additional conditions to check) and situations where it applies come up frequently and naturally.

PS: Warning: In some references a cocontinuous functor is a functor between categories (not sites) is defined as a functor that commutes with colimits. This is a different notion. Too bad!

Since the last update we have added a new chapter. This chapter explains the Popescu-Ogoma-Andre-Swan proof of general Neron desingularization (GND). As explained here there is a way to reduce to the case of a base field. This does simplify the rest of the arguments somewhat, but not as much as I’d have liked.

The heart of the proof of GND is in the proof of Lemma Tag 07FJ. For some reason working through this proof made me think of playing chess, in that you have to think ahead several moves and the steps you take early in the proof almost don’t seem to make sense. I have a hard time explaining it, even to myself. But then, I was never any good at chess.

Well actually 3011 pages at this very moment. Also

• 290489 lines of tex,
• 9300 tags, and
• 2226 commits since we started using git on May 20, 2008.

Enjoy!

Since the last updateon October 12 we have added the following material

1. Gabber’s argument that categories of quasi-coherent modules form a Grothendieck abelian category (for schemes, spaces, and algebraic stacks),
2. an example of an fpqc space which is not an algebraic space,
3. an example of a quasi-compact non-quasi-separated morphism of schemes such that pushforward does not preserve quasi-coherency,
4. some material related to my course on commutative algebra: exercise, lemmas, shorten proof of ZMT, etc
5. introduced lisse-etale (and flat-fppf) sites,
6. functoriality of lisse-etale topos for smooth morphisms (and flat-fppf for flat morphisms),
7. material on Grothendieck abelian categories, incuding existence of injectives and existence of enough K-injective complexes (following Spaltenstein and Serp\’e),
8. cohomology of unbounded complexes and adjointness of Lf^* and Rf_*,
9. a lot of material on D_{QCoh}(X) for an algebraic stack X, including Rf_* (on bounded below for quasi-compact and quasi-separated morphisms) and Lf^* (unbounded for general f).

In particular my suggestion in this post worked out exactly as advertised. The existence of Rf_* is straightforward. It turns out that once you prove that the category D_{QCoh}(X) as defined in the blog post is equivalent to the version of D_{QCoh}(X) in L-MB or Martin Olsson’s paper (i.e. defined using the lisse-etale site), then you immediately obtain the existence of Lf^*. Namely, the existence of the lisse-etale site is used to prove that the Verdier quotient used to define D_{QCoh}(X) is a Bousfield colocalization (technically it is easier to use the flat-fppf site to do this, because we use the fppf topology as our default topology, but one can use either).

A bit of care is needed when working with the lisse-etale site and the lisse-etale topos. As discussed elsewhere, one reason is that the lisse-etale topos isn’t functorial for morphisms of algebraic stacks. Here is a another. There is a comparison morphism of topoi

g : Sh(X_{lisse,etale}) —-> Sh(X_{etale})

The functor g^{-1} has a left adjoint denoted g_! (on sheaves of sets) and we have g^{-1}g_! = g^{-1}g_* = id. This means that Sh(X_{lisse,etale}) is an essential subtopos of Sh(X_{etale}), see SGA 4, IV, 7.6 and 9.1.1. Let K be a sheaf of sets on X_{lisse,etale}. Let I be an injective abelian sheaf on X_{etale}. Question: H^p(K, g^{-1}I) = 0? In other words, is the pullback by g of an injective abelian sheaf limp? If true this would be a convenient way to compare cohomology of sheaves on X_{etale} with cohomology of sheaves on the lisse-etale site. Unfortunately, we think this isn’t true (Bhargav made what is likely a counter example — but we haven’t fully written out all the details).

Here is a simple example that shows that in order to obtain a derived functor Rf_* on unbounded complexes with quasi-coherent cohomology sheaves we need some additional hypothesis beyond just requiring f to be quasi-compact and quasi-separated.

Let k be a field of characteristic p > 0. Let G = Z/pZ be the cyclic group of order p. Set S = Spec(k[x]) and let X = [S/G] be the stacky quotient where G acts trivially on S. Consider the morphism f : X —> S. Then Rf_*O_X is a complex with cohomology sheaves isomorphic to O_S for all p >= 0. In fact Rf_*O_X is quasi-isomorphic to ⊕ O_S[-n] where n runs over nonnegative integers.

Now consider the complex K = ⊕ O_X[m] where m runs over the nonnegative integers. This is an object of D_{QCoh}(X) but it isn’t bounded below. So we have to pay attention if we want to compute Rf_*K. Namely, in D(O_X) the complex K is also K = ∏ O_X[m]. Since cohomology commutes with products, we see that

Rf_*K = ∏ Rf_*O_X[m] = ∏ (⊕ O_S[m – n]).

In degree 0 we get an infinite product of copies of O_S which isn’t quasi-coherent.

Conclusion: Rf_* does not map D_{QCoh}(X) into D_{QCoh}(S).

Of course if f is a quasi-compact and quasi-separated morphism between algebraic spaces, then this kind of thing doesn’t happen.

In Groupes de Brauer II, Remark 1.11(b) Grothendieck notes that results of Mumford’s paper “The topology of normal singularities of an algebraic surface and a criterion for simplicity” gives one an example of a normal surface Y over the complex numbers such that H^2(Y, G_m) isn’t torsion and does not inject into H^2(C(Y), G_m). Grothendieck even references a page number, namely 16. To explain this in the graduate student seminar on Brauer groups this semester I came up with the following, which may be what Grothendieck had in mind.

Let E ⊂ P^2 be a smooth degree 3 curve. Let P ∈ E be a flex point. Blow up P exactly 10 times on E, i.e., blow up P in P^2, then blow up P on the strict transform of E, etc. The result is a surface X with an embedding E ⊂ X such that

1. the self square of E in X is -P, and
2. the image of the map Pic(X) —> Pic(E) is contained in ZP.

This means you can blow down E on X to get a normal projective surface Y with a unique singular point y. Part 2 implies that the local ring of O_{Y, y} is factorial (this is one of Grothendieck’s claims — in fact we won’t need it). Now look at the Leray Spectral Sequence for G_m and the morphism f : X —> Y. You get something like

Pic(X) —> H^0(Y, R^1f_*G_m) —> H^2(Y, G_m) —> H^2(X, G_m)

We have R^1f_*G_m = Pic(E) placed at y and H^2(X, G_m) = 0 as X is a smooth projective rational surface. Using 1 and 2 above we conclude that H^2(Y, G_m) = E as abelian groups. By Gabber’s result on Brauer groups of quasi-projective schemes it follows that Br(Y) = E_{tors}. Of course both H^2(Y, G_m) and Br(Y) map to zero in the Brauer group of the generic point.

This is just a quick note on the paper Brown representability does not come for free by Casacuberta and Neeman. This is going to be completely bare bones as you can read more details in the paper.

We are going to define a “big” abelian category A as follows. An object of A consists of a pair (M, α, s_β) where M is an abelian group and α is an ordinal and s_β : M —> M is a commuting family of homomorphisms parametrized by β ∈ α. A morphism (M, α s_β) —> (N, γ, t_δ) is given by a homomorphism of abelian groups f : M —> N such that f(s_β(m)) = t_β(f(m)) for any ordinal β where the rule is that we set s_β equal to zero if β is not in α and similarly we set t_β equal to zero if β is not an element of γ.

A special object is Z = (Z, 0, ∅), i.e., all the operators are zero. The observation is that computed in A the “group” Ext^1_A(Z, Z) is a proper class and not a set. Namely, for each ordinal β we can find an extension M of Z by Z whose underlying group is M = Z ⊕ Z and where s_β acts by a nonzero operator s_β, e.g. via the matrix (0, 1; 0, 0). This clearly produces a proper class of isomorphism classes of extensions.

In my world forming the category D(A) doesn’t make sense because the Hom’s aren’t sets. Another conclusion is that in K(A) the subcategory of acyclic complexes does not give rise to a Bousfield localization or colocalization.

Scarier than Halloween?

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.