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.)

Here are two definitions as currently in the stacks project:

1. A Serre subcategory of an abelian category is a strictly full subcategory closed under taking subquotients and closed under taking extensions.
2. A weak Serre subcategory of an abelian category is a strictly full subcategory which is abelian, which has an exact inclusion functor, and which is closed under taking extensions.

Here the subquotients and extensions are taken in the bigger abelian category. The formal definitions can be found here.

Yesterday I realized I had confused these two notions. In some situations the first is more appropriate (e.g., the kernel of an exact functor is a Serre subcategory) and in others the second is better (e.g., given a weak Serre subcategory B of A the derived category D_B(A) makes sense).

Nomenclature: I think the notion of a Serre subcategory is pretty standard, in the sense that all of the definitions of a Serre subcategory of an abelian category that I have seen are equivalent to the one above (single exception: nlab). Serre used the same definition (in the case that the ambient category is the category of abelian groups). On the other hand, the notion of a “weak Serre subcategory” is nonstandard. In some papers/books the terminology “thick subcategory” is used for this, but unfortunately in many texts “thick subcategory” is synonymous with “Serre subcategory”. In fact, it seems that the notion of a “thick subcategory” is very malleable — there is no real agreement on what this term should mean, and, googling, I found at least one instance where this confusion led to a mathematical error. In the case of subcategories of a triangulated categories I decided to avoid using “thick” and I have used “saturated” just like Verdier does in his thesis. (Unfortunately, some authors use “saturated” to mean “closed under isomorphism”, but they seem in the minority.)

Is there a word, other than “thick”, we can use to describe weak Serre subcategories?

Here is a picture of my collaborator Jason Starr, who seems like a content, but a little bit nerdy person (notice the tiny bit of chalk on his nose):

But who is this person here? It looks like this guy is both happy and crazy, always a dangerous combination.

These pictures are copyright C. J. Mozzochi, Princeton N.J. For more see this web site.

In Shioda cycles, II and Shioda cycles, I we discussed how almost any arithmetic family of surfaces produces an infinite family of instances of the Tate conjecture for divisors of surfaces over finite fields. In this post we’ll see how to produce explicit equations for surfaces where the Tate conjecture is open.

Namely, a while back I wrote a computer program that computes the matrix of geometric frobenius on H^2_{cris, prim}(X) for a quasi-smooth hypersurface in a weighted projective space over a prime field using an algorithm due to Kiran Kedlaya. It is quite usable, except that only I can parse its output since I didn’t bother to write documentation. One of the things I like about it is that it works for any quasi-smooth hypersurface in any weighted projective 3-space (of course in most cases the run time is too large). For example, look at the degree 92 hypersurface

X : x^11y + x^5z^2w + x^2yzw^2 + xy^3zw + xz^5 + y^5z + w^4 = 0

in the weighted projective space P(7, 15, 17, 23) over F_2. The characteristic polynomial of Frobenius on the primitive middle cohomology is

x^9 + 2x^8 + x^7 – 2x^6 – 4x^5 + 8x^4 + 16x^3 – 32x^2 – 256x – 512

which has x = 2 as a root. Hence the Picard group of this surface should have rank 2 according to the Tate conjecture.

One of the things I tried was to write a bunch of scripts running through all possible quasi-smooth surfaces for a given collection of combinatorial data. Moreover, I wrote a program that looked for (very) low degree curves lying on the surfaces; this often finds a cycle if the Tate conjecture predicts one (presumable because I only looked at cases of low degree and with few monomials). But the example above is a case where my (silly) cycle finder didn’t find one. Can you find one?

Another explicit example is the hypersurface of degree 91 in P(7, 11, 16, 25) over F_2 defined by

x^13 + x^6y^3z + x^4y^2zw + x^3y^2z^3 + x^2yzw^2 + y^6w + yz^5 + zw^3 = 0

which has characteristic polynomial of frobenius on the primitive part of H^2 given by

x^12 + 2x^10 – 4x^9 – 14x^8 + 12x^7 – 64x^6 + 48x^5 – 224x^4 – 256x^3 + 512x^2 + 4096

and 2 is a root of this polynomial with multiplicity 2. Hence the Picard group should have rank 3. My cycle finder program suggests looking at the curve defined by

x^5y^2 + x^2yz^2 + z^2w = 0 and x^3z^3 + x^2y^5 + y^4w = 0

but even if this works (i.e., is independent of the hyperplace class) we still have to find yet another cycle in order to finish the proof of the Tate conjecture for this surface.

In other words, with current technology, there is no effort involved in making explicit examples where we know the Tate conjecture predicts something nontrivial. Even if we assume the Tate conjecture, we don’t know how to get our hands on these cycles. When you compute the matrix of Frobenius on the crystalline cohomology (as in Kiran’s algorithm) you are actually performing some polynomial operations such as raising to the pth power, taking derivatives, and (occasionally) dividing by p. In some sense these computations are “proving” cycles should exist. This motivates the idea, explained by previous two posts, that similar computations could provide hints as to _where_ to find the cycles too.

This post won’t make sense if you haven’t read Shioda cycles, I.

Let X be a hypersurface of even degree d in P^3_{F_p} such that the determinant of the geometric frobenius acting on H^2 has a positive sign. Assuming the Tate conjecture (which we will do throughout this post), we can find our Shioda cycle by listing all the low degree curves in P^3_{F_p} and for each of them checking whether the curve lies on X and if so whether it gives a Shioda cycle. Now although this is a common recipe for finding a Shioda cycle if it should exist, it isn’t the kind of pattern I am looking for. (Moreover, you’d be hard pressed to argue that this recipe is uniform over all primes p because after all the lists will change with p.)

Now, I have a suggestion for a recipe that could work (which only means I can’t prove it doesn’t work). I am not saying or conjecturing that it does work, although I do have some very special cases where I can show that it works (basically families of surfaces related to families of abelian surfaces). A while ago I wrote a preprint about this (you can find it on my web page), but I think I can explain it here in a few sentences.

Namely, suppose that F = F(X_0, …, X_3) ∈ Z[X_0, X_1, X_2, X_3; A_I] is the universal polynomial of degree d, i.e., the coefficients A_I of F are variables where I = (i_0, i_1, i_2, i_3) with i_0 + i_1 + i_2 + i_3 = d. For every collection of values a = (a_I), a_I ∈ F_p we obtain a hypersurface X(a) in P^3_{F_p} by setting A_I equal to a_I in F. Now, suppose that we have a polynomial

G(X_0, …, X_3, Y_0, …, Y_3) ∈ Z[X_i; Y_j; A_I]

For each a = (a_I) as above we can consider the intersection of X(a) with

G(X_0, …, X_3, X_0^p, …, X_3^p)|_{A_I = a_I} = 0

i.e., we replace Y_j by X_j^p and A_I by a_I. Let’s call this intersection Z(a). Then my suggestion is to look for a Shioda cycle among the irreducible components of Z(a). In other words, given the even integer d, is there a polynomial G as above, such that, if X(a) is a surface which should have a Shioda cycle, then one of the irreducible components of Z_a is a Shioda cycle?

Actually in my write-up I (a) only require this to work in most of the cases where we expect a Shioda cycle, and (b) allow G also to depend on more variables which get replaced by X_j^{p^n}.

You might think it would be more natural to consider a system of polynomials such as G and ask them, after being mangled as above, to actually cut out a Shioda cycle. It seemed to me at the time of writing the preprint that this might be too strong a requirement, but I actually do not know how to disprove even this statement.

There are variant constructions we could use, e.g., we could allow variables Z_{ij} that get replaced by

(1/p)[(X_i + X_j)^p – X_i^p – X_j^p] mod p

if you know what I mean. The _meta_ question I have is whether anything like this can be true? Can you think of a (heuristic) argument showing this cannot work?

For example, if you could show that the (minimal) degrees of Shioda cycles tends to infinity rapidly with p then we would get a contradiction. However one can prove, assuming the Tate conjecture is true, an upper bound of the degree of Shioda cycles occuring in the family (unfortunately I don’t remember the shape of the formula I got when I worked it out) which shows this kind of argument won’t contradict my suggestion.

Let X be a smooth projective geometrically irreducible surface over F_p where p is a prime. (For example a smooth hypersurface in P^3_{F_p}.) The second betti number of X is the dimension over Q_l of the \’etale cohomology group H^2 = H^2(\bar{X}, Q_l) where l is a prime different from p and where \bar{X} is the base change of X to the algebraic closure of F_p. Since X is projective the first chern class of an ample divisor is a nonzero element of H^2 which is an eigenvector for the action of the geometric frobenius of X with eigenvalue p. Assume that the second betti number of X is even. (For example a smooth hypersurface in P^3_{F_p} of even degree.) Since the geometric frobenius of X acts by multiplication with p^2 on H^4 and since the nondegenerate pairing H^2 x H^2 —> H^4 is compatible with the action of frobenius we see that the eigenvalues of geometric frobenius on H^2 which do not equal to +p or -p have to pair up: if λ occurs so does p^2/λ (with the same multiplicity). Since the second betti number is even, we conclude that, besides the eigenvalue p we found above, there is at least one more eigenvalue p or -p. Moreover, which of the two cases occurs depends on the sign of the determinant of the geometric frobenius acting on H^2.

In the situation above the Tate conjecture predicts the rank of the Picard group of X is equal to the multiplicity of p as a generalized eigenvalue of the geometric frobenius acting on H^2. (Of course this is just a very special case of the Tate conjecture.) Thus in the situation above there is a “50% chance” that the Picard group is strictly bigger than Z. In fact, in a paper of Nick Katz and myself we proved a precise version of this statement in some cases. Perhaps the most straightforward of these is the case of hypersurfaces:

Fix d even, d ≥ 4. In this case the percentage of smooth hypersurfaces X of degree 4 in P^3 over F_p where the multiplicity of p of an eigenvalue of geometric frobenius on H^2(X) is 2 tends to 50% as p tends to infinity.

One concludes that for p large enough at least 49% of the smooth hypersurfaces X have Picard number 2 (according to the Tate conjecture). Similarly, in at least 49% of the cases the Picard number is (proveably) 1. Finally, there is a remaining 2% of cases where the Picard number could be larger than 2.

Let’s say a Shioda cycle is an effective divisor on one of our X’s above which is independent of the hyperplane class in Pic and whose existence is predicted via the Tate conjecture by the parity considerations above. Now, if you assume the Tate conjecture, then there exist a lot of Shioda cycles (they exist on roughly 50% of all hypersurfaces over F_p of even degree). What I want to know is this:

Is there a pattern in these cycles?

More precisely, as we vary X in the family, could it be that there is a common recipe for the construction of a Shioda cycle for most of the 50% of the X’s where we expect one? I’ll say a bit more about this in a future post.

So Bhargav and I were, just earlier today, thinking about the alternating Cech complex in the setting of etale cohomology and this is what we came up with. Caveat: This may be wrong in which case it is my fault (I’m a little worried because the final result of this blog post seems to contradict a throwaway comment in some preprint). Also: it may be in the literature; if you know a reference for this construction please email, thanks.

Let X be an algebraic space. Let U be a separated scheme and let f : U —> X be a surjective etale morphism. Assume that there exists an integer d such that every geometric fibre of f has at most d points. (This is true if U is quasi-compact and X is quasi-separated.) Consider the trace map

f_!Z —> Z

and consider the Koszul complex on this

… —> ∧^2 f_!Z —> f_!Z —> Z

Looking at stalks we see that this is exact. Thus we obtain a quasi-isomorphism K^* —> Z[0] where the complex K^* has as terms K^i = ∧^{i + 1} f_!Z. Moreover, K^i = 0 for i ≥ d. Thus for any abelian sheaf F on X_{etale} we obtain a spectral sequence with E_1-page

E_1^{p, q} = Ext^q(K^p, F)

converging to H^{p + q}(X_{etale}, F). The complex E_1^{*, 0} is our alternating Cech complex.

Now, we want to make explicit the groups Ext^q(K^p, F). These are the right derived functors of Hom(K^p, F). To describe Hom(K^p, F) we introduce some notation. Namely, let W_p be the complement of ALL diagonals in U^{p + 1} = U ×_X … ×_X U. Since f is separated and etale W_p is both open and closed in U^{p + 1}. Moreover, the group S_{p + 1} has a free action on W_p. We claim that

Hom(K^p, F) = S_{p + 1}-anti-invariants in F(W_p)

To see this look at (W_p —> X)_!Z. The stalk of this sheaf at a geometric point x of X is the free Z-module with basis the set of injective maps {0, …, p} —> U_x. Hence ∧^{p + 1}f_!Z is the maximal S_{p + 1}-anti-invariant quotient of (W_p —> X)_!Z. This proves the displayed formula. Since S_{p + 1} acts freely on W_p over X the quotient U_p = W_p/S_{p + 1} is an algebraic space etale over X. There is a way to “twist” F|_{U_p} by the sign character S_{p + 1} —> {+1, -1} giving a sheaf F_p on U_p. With a little bit of work we obtain

Ext^q(K^p, F) = H^q(U_p, F_p).

Why is this useful? Suppose that F is a quasi-coherent O_X-module, X is quasi-compact, X is separated, and U is affine. Then each W_p is affine too, and so is U_p. Moreover, the sheaves F_p are still quasi-coherent. Thus we see that the E_1^{p, q} are nonzero only when q = 0 and we obtain vanishing of H^n(X, F) for all n >= d! This is exactly the vanishing you traditionally obtain from the alternating Cech complex associated to a finite affine open covering of a scheme.

For a quasi-compact, quasi-separated algebraic space X we can redo the argument with U an affine scheme. We find (because we can apply the previous result to the separated quasi-compact algebraic spaces U_p) that X has finite cohomological dimension for quasi-coherent sheaves. And that’s the thing I was stuck on in the stacks project yesterday…

[Edit Aug 19, 2011: This material is now in the stacks project. The spectral sequence is Lemma Tag 0728. The application to algebraic spaces is Proposition Tag 072B and Lemma Tag 072C. Note that the first vanishing result is interesting for schemes also.]

Some types of questions in algebra immediately reduce to the “countable” case. Simple example: Let A be a ring and let φ : A —> A be an automorphism. Then for every finite subset E of A there exists a countable subring A’ ⊂ A containing E such that φ induces an automorphism of A’. The proof is to let A’ be the subring of A generated by φ^n(e) for all e in E and all n ∈ Z.

Another example of this phenomenon is that any projective module is a direct sum of countably generated (projective) modules (Kaplansky’s theorem).

The technique also applies to the following problem: Let A ⊂ B be an integral extension of rings with A Noetherian. Let M be an A-module such that M ⊗_A B is flat over B. Problem: Show M is flat over A. (This is equivalent to the direct summand conjecture by a 1 page paper of Ohi.) A key case is to show M ⊗_A B = 0 implies M = 0. Picking suitable families of elements this reduces to the case where B is countably generated over A and M is a countably generated A-module.

Here are two examples involving algebraic stacks: (1) Suppose X is a quasi-compact algebraic stack with affine diagonal. I claim you can write X as a filtered limit of stacks X’ of the form X’ = [U’/R’] with U’ and R’ spectra of countable rings. I haven’t written out the details but it seems to me one can do this by just “adding elements” as above. (2) A quasi-coherent module F on a quasi-separated and quasi-compact algebraic stack is a filtered colimit of countably generated quasi-coherent modules.

I wonder if this type of argument can ever be used to bootstrap? Do some general arguments become easier if you assume all rings/modules in question are countable? Are there some _useful_ properties that hold for countable rings? Things like “the topology on Spec has a countable basis” aren’t really useful, or are they?

Before you say “No!” let me just point out that Kaplansky’s theorem is used in the proof of faithfully flat descent for projectivity of modules, so sometimes…

Let X = Spec(A) be an affine scheme. Let M, N be A-modules. Let F, G be the sheaves of O_{big}-modules on the big fppf site of X associated to M and N, e.g., F(Spec(B)) = B ⊗_A M and similarly for G. As a by-product of the material on adequate modules I proved the following formula

Ext^i_{O_{big}}(F, G) = Pext^i_A(M, N)

The ext group on the left is the ext group in the category of all O_{big}-modules. The ext group on the right is the ith pure extension group of M by N over A. This group is computed by taking a universally exact resolution 0 -> N -> I^0 -> I^1 -> … with each I^j pure injective and taking the ith cohomology group of Hom_A(M, I^*). An A-module I is pure injective if for any universally injective map M_0 -> M_1 the map Hom_A(M_1, I) -> Hom_A(M_0, I) is surjective.

There seems to be a lot of papers on pure modules, pure injectivity, etc. Gruson and Jensen characterized pure injective modules as those modules such that the functor – ⊗_A M : mod-A —> Ab is injective in the functor category (mod-A, Ab). Here mod-A is the category of finitely presented A-modules. It follows that

Ext^i_{(mod-A, Ab)}(- ⊗_A M, – ⊗_A N) = Pext^i_A(M, N)

Our formula above is about O_{big}-modules, which in terms of functors means functors F : Alg_A —> Ab such that F(B) has the structure of a B-module for every A-algebra B and such that B —> B’ gives a B-linear map F(B) —> F(B’). These are called module-valued functors (terminology due to Jaffe). Then we can rewrite the first equality above as

Ext^i_{module-valued functors}(F, G) = Pext^i_A(M, N)

where F(B) = B ⊗_A M and G(B) = B ⊗_A N. In this formula you can let Alg_A be any sufficiently large category of A-algebras, e.g., the category of finitely presented A-algebras.

The two results seem related. But there is a big difference between the functor categories (mod-A, Ab) and (Alg_A, Ab). Namely, if we look at Ext^i_{(Alg_A, Ab)}(F, G) then we get a completely different animal. For example suppose that G_a(B) = B for all A-algebras B and suppose that A is an F_p algebra. Then we see that Hom_{(Alg_A, Ab)}(G_a, G_a) contains the frobenius map frob : G_a —> G_a which on values over B raises every element to the pth power. In fact, the work of Breen on ext groups of abelian sheaves on the fppf-site (warning: this is not exactly what he studies there) implies some of the higher ext groups Ext^i_{(Alg_A, Ab)}(G_a, G_a) are nonzero also (lowest case seems to be i = 2p)!

Conclusion: module-valued functors over Alg_A and abelian group valued functors on mod-A somehow ends up giving the same ext groups for the functors associated to A-modules described above.

During the last few weeks I have been working on a way to described the category of quasi-coherent modules on a scheme X in terms of the big fppf site of X. I think I have succeeded to some extend, and I’d like to explain some of the results here. But first, let me say why this may be (somewhat) useful.

Let us denote O_X the structure sheaf on the scheme X and O_{big} the structure sheaf on the big fppf site of X. Similarly, given a morphism f : X —> Y of schemes we have the usual pushforward f_* and the pushforward f_{big, *} on sheaves on the big sites. My goal was to understand the following two phenomena:

1. The fully faithful embedding i_X : QCoh(X) —> Mod(O_{fppf}) isn’t exact in general.
2. When f : X —> Y is quasi-compact and quasi-separated, then f_{big, *}i_X(F) is not equal to i_Yf_*F in general.

Of course for schemes this isn’t a problem, but 1 and 2 also happen for algebraic stacks where we do not have the luxury of an underlying ringed space whose category of quasi-coherent modules is the “right one”. This means that in order to define pushforward for quasi-coherent modules (along quasi-compact and quasi-separated maps) one has to be a little bit careful (it isn’t hard — I’ll come back to this). Secondly, as was pointed out to me several times by Martin Olsson, the first problem means that D_{i_X(QCoh)}(O_{big}) isn’t a triangulated subcategory of D(O_{big}). This isn’t a problem for schemes because you can take D_{QCoh}(O_X) but again this doesn’t work for algebraic stacks and you have to do something. A solution for this second problem is to work with the lisse-etale site, but then you get embroiled in the nonfunctoriality of it…

OK, so I have a “solution” to these two problems. Let me say right away that the solution isn’t ideal, partly because it is rather complicated. But at the end of the story (after a certain amount of work) the picture that emerges is rather pleasing.

First assume that X is an affine scheme. It turns out that problem 1 for affine X was solved in a paper by Jaffe entitled Coherent Functors, with Application to Torsion in the Picard Group. In this wonderful paper he points out that if you just add kernels of maps of quasi-coherent O_{big}-modules, then you obtain an abelian subcategory of D(O_{big})! I’m going to call these adequate modules (these correspond to the “module-quasi-coherent A-functors” of Jaffe’s paper). On a general scheme X I am going to say an O_{big}-module F is adequate if there exists an affine Zariski open covering of X such that F restricts to adequate modules over the members of the covering. It moreover turns out that adequate modules are preserved under colimits and that they form a Serre subcategory of the abelian category all O_{big}-modules.

You can show quite easily that one has vanishing of cohomology of adequate modules over affines, so that they behave much like quasi-coherent modules. Moreover, any quasi-coherent module is adequate and (Zariski) locally any adequate module is a kernel of a map of quasi-coherent modules. Finally, if you have an adequate module on X and you restrict it to the small Zariski-site of X then you get a quasi-coherent module. This implies readily that

QCoh(X) = Adeq(X)/C(X)

where C(X) is the category of parasitic adequate modules. An adequate O_{big}-module is called parasitic if the restriction to S_{Zar} is zero (it was a suggestion by Martin Olsson that these should play an important role in the story).

So this solves problem 1 as Adeq(X) —> Mod(O_{big}) is exact by construction and in fact Adeq(X) is a Serre subcategory. What about 2? The answer is that R^if_{big, *}F is adequate for adequate modules F when f : X —> Y is quasi-compact and quasi-separated. Moreover, R^if_{big, *}F is in C(Y) if F is in C(X). Hence we see that R^if_{big, *} induces a functor

QCoh(X) = Adeq(X)/C(X) —-> Adeq(Y)/C(Y) = QCoh(Y)

and (you guessed it) this recovers our usual R^if_* for quasi-coherent sheaves! Thus a solution to 2.

Morally, this tells us that we should view QCoh(X) as a subquotient of Mod(O_{big}) and not as a subcategory. Taking a quotient of an abelian category by a Serre subcategory is achieved by Gabriel-Zisman localization. This suggests that we can do the same with derived categories. Indeed, it turns out that

D_{QCoh(X)}(O_X) = D_{Adeq(X)}(O_{big})/D_{C(X)}(O_{big})

(Verdier quotient) with no conditions on X whatsoever. I expect a description of the total direct image Rf_* on the left hand side as the functor induced by Rf_{big, *} on D_{Adeq}(O_{big}) in exactly the same way as above (details not yet written).