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.)

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.