Thick subcategories

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):
Jason Starr
But who is this person here? It looks like this guy is both happy and crazy, always a dangerous combination.
Johan de Jong
These pictures are copyright C. J. Mozzochi, Princeton N.J. For more see this web site.

Instances of the Tate conjecture

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.

Shioda cycles, II

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.

Shioda cycles, I

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.