Quasi-coherent modules

Let X be a scheme. Let F be a quasi-coherent O_X-module. Let G ⊂ F be an O_X-submodule, not necessarily quasi-coherent. Then there exists a quasi-coherent submodule G’ ⊂ G which is universal for maps of quasi-coherent modules into G. This is Lemma Tag 01QZ in the stacks project.

The condition that G is a submodule of a quasi-coherent module is necessary in order to construct G’ (I think; explicit counter examples welcome). This result has two funny looking applications

  1. Any morphism f : X —> Y of schemes has a scheme theoretic image (Lemma Tag 01R6), and
  2. i_* : QCoh(Z) —> QCoh(Y) has a right adjoint when i : Z —> X is a closed immersion of schemes (Lemma Tag 01R0).

At first sight it may seem that 1 is too strong. But I think it isn’t simply because there is no way you can deduce anything from the existence of a smallest closed subscheme of Y through which f factors. It is only when f is quasi-compact and quasi-separated that the scheme theoretic image commutes with restriction to open subschemes for example.

It may be that the existence in 2 of a right adjoint i^! of i_* : QCoh(Z) —> QCoh(Y) follows from general facts, but it is cute that one can explicitly write it down as in the proof of the lemma referenced above. Again, the construction of i^! is not local on X, except in case the sheaf of ideals defining the closed immersion i is of finite type (to see this use Lemma Tag 01PO).

Quotients of projective spaces

Consider the moduli stack M_1 parametrizing smooth (locally) projective genus one curves C. If C is a genus one curve over a field k, then there exists a minimal integer d > 0 such that C has an ample invertible sheaf of degree d. It turnsout there is no bound for the integer d, and it follows that M_1 does not have a presentation by a finite type scheme over Z.

Consider the moduli stack M_1(d) parametrizing pairs (C, L) where C is a smooth projective genus 1 curve, and L is an ample invertible sheaf of degree d. This does have a presentation by a finite type scheme over Z.  For example when d = 3 we see that the space U = P(\Gamma(P^2, O(3))) – \Delta maps smoothly and surjectively onto M_1(d). Moreover, we have M_1(3) = [U/GL_3] (edit Oct 14, 2011: changed PGL_3 into GL_3).

Now, what’s interesting is that U is an open subscheme of a projective space. You can do the “same thing” for M_1(5) by writing every degree 5 genus one curve in P^4 as the zeros of the 2×2 pfaffians of a skew symmetric 5×5 matrix of linear forms on P^4. You can also do this for M_1(4) by writing a degree 4 genus 1 curve in P^3 as the intersection of 2 quadrics. You can also do something similar for M_1(2).

These stacks came up in a conversation with Manjul Bhargava in my office last week, and so did the following question: Can the same be done for any d > 5?

On the level of algebraic stacks, a more general question would be: Can we find obstructions to being able to write an algebraic stack M as a quotient stack [U/G] where U is an open subspace of P(V) where V is a linear representation of G? Cohomological? Intersection theory? I am hoping there may be some things once can say that avoid appealing to a classification of representations of G’s with small dimensional orbit spaces.

Splitting chapters

So I was wondering if it would maybe help to split the chapters in the stacks project into shorter chapters. To do this I wrote a python script that can automatically fix tags and references that point to the wrong labels. Allthough this script has already turned out to be useful I have decided not to split up any chapter yet. In fact, playing around with it myself I do not find it confusing to have long chapters.

Really the challenge is to find classifications of the material that makes sense and then to divide the chapter in question into the corresponding pieces. The longest chapter by a long shot is the chapter on commutative algebra which has 100 sections and 250 pages (about). I personally do not see an easy way to divide this into meaningful chunks, but maybe you do? Let me know if so. Of course the initial part (which could be lengthened) about really basic stuff — elementary properties of tensor, localization, finite type, finite presentation, etc — i.e. stuff where you do not need prime ideals — could be split off. But what about the later material? Again, if you see a good way to make divisions leave a comment.

Note that I say classification of material. I do not intend the stacks project to be read from A — Z since that would presumably drive the reader insane. I do intend the material to be locally readable, and I want it to be easy to access the earlier lemmas, propositions, theorems that more advanced material relies on.

Of course whether splitting up chapters would be useful depends on who is reading and how they access the stacks project. Are you downloading the whole book at once, or do you browse chapters? Do you use acrobat reader? Do cross file links work? Etc, etc. It would be helpful and welcome to have some feedback or ideas on this (as a comment or just an email).

Push and pull

Here are some examples of morphisms f of topoi such that f^{-1}f_*F —> F is always surjective for any sheaf of sets:

  • If f : X —> Y is a continuous map of topological spaces which induces a homeomorphism of X with a subset of Y.
  • If f : Sets —> G-Sets is the morphism of topoi coming from mapping the point to the “classifying space” of the group G.
  • If C is a site and f : Sh(C) —> PSh(C) is the morphism of topoi with f^{-1} equal to sheafification and f_* the forgetful functor.

In the first case the map f^{-1}f_*F —> F is actually always an isomorphism but in the second case it isn’t if G is nontrivial. Bhargav pointed out the last one and he also pointed out that you can similarly produce lots of examples for exampl X_{etale} —> X_{Zar} by comparing topologies.

By the way I think it is true that if f : X —> Y is a continuous map of Kolmogorov topological spaces with the property that f^{-1}f_*F —> F is always surjective for any sheaf of sets, then f induces a homeomorphism of X with a subset of Y. (I haven’t written out all the details however.)

Here is an example: f : X —> Y and X = {p, q} with discrete topology and Y= {*}. Then for any sheaf of abelian groups F the map f^{-1}f_*F —> F is surjective, but this does not hold for every sheaf of sets. Namely, a sheaf of sets (resp abelian groups) on X corresponds to a pair of sets (resp abelian groups) F_p, F_q (namely the stalks of F at p and at q). Then f_*F corresponds to F_p \times F_q. Thus we see that f^{-1}f_*F —> F is surjective if and only if the projections F_p \times F_q —> F_p and F_p \times F_q —> F_q are surjective. This is the case if and only if either both F_p and F_q are nonempty or both are empty. But for sheaves of sets F_p not empty and F_q empty can occur!

Of course this is somehow incredibly trivial. But since I’m used to thinking mostly about sheaves of abelian groups it is also very confusing. Namely, any sheaf of abelian groups on {p, q} is globally generated but as seen above it is not the case that every sheaf of sets on {p, q} is “globally generated” (i.e., the target of an epimorphism from a constant sheaf).

Closed immersions and the fppf topology

Let i : Y —> X be a closed immersion of schemes. This gives rise to a morphism of topoi i_{big} : (Sch/Y)_{fppf} —> (Sch/X)_{fppf}. Question: Is the direct image functor i_{big, *} is exact on the category of abelian sheaves?

My guess is no. To find an example we can look for an Artinian local ring A with an ideal I and a finite flat local ring map A/I —> C such that there does not exist any finite flat ring map A —> B with the property that A/I —> B/IB factors through C. Namely, in this case the map of abelian sheaves

(Z/2Z)_{Spec(C)} —> Z/2Z

on Y = Spec(A/I) is fppf surjective because {Spec(C) —> Spec(A/I)} is an fppf covering. Here the first sheaf is the free Z/2Z-module on the fppf sheaf represented by Spec(C) over Y. But

i_*((Z/2Z)_{Spec(C)}) —-> i_*(Z/2Z)

is not surjective since the section 1 does not lift fppf locally on X = Spec(A) by our assumption on A/I —> C. To make an explicit example you probably can do something similar to Exercise Tag 02CV but I haven’t quite been able to make it work yet. Leave a comment if you have an example, or a reference, or if you think the answer to the question is yes.

Cocontinuous functors

In the stacks project a site is defined as in Artin’s notes on Grothendieck topologies, and not as in SGA4. Hence also our notion of a cocontinuous functor u : C —> D between sites C and D is a bit different (than Verdier’s original one). Namely, it means that, given any object U of C, and any covering {V_j —> u(U)}_j in D there should exist a covering {U_i —> U} in C such that the family of morphisms {u(U_i) —> u(U)}_i refines the given family {V_j —> u(U)}_j.

The reason this definition is convenient is twofold. On the one hand, it is easy to check that a functor is cocontinuous, and on the other hand, it is true that a cocontinuous functor u : C —> D gives rise to a morphism of topoi g : Sh(C) –> Sh(D). For example, for a sheaf G on D the sheaf g^{-1}(G) is the sheaf associated to the presheaf U |—> G(u(U)).

Here are two examples

  • Let f : X —> Y be an open continuous map of topological spaces. Then the functor u(U) = f(U) is a cocontinuous functor between the site of opens of X and the site of opens of Y. The induced morphism of topoi Sh(X) —> Sh(Y) is the usual one.
  • Let f : X —> Y be a morphism of schemes. The “forgetful” functor u : (Sch/X)_{fppf} —> (Sch/Y)_{fppf} is cocontinuous and the associated morphism of topoi is the usual morphism of big topoi f_{big} : Sh((Sch/X)_{fppf}) —> Sh((Sch/Y)_{fppf}).

A little less standard are the following examples, which are related to the discussion in the previous post. Suppose that i : X_0 —> X is a closed immersion of schemes defined by a sheaf of ideals of square zero. Consider the functor of sites u : X_{lisse-etale} —> (X_0)_{lisse-etale}, or u : (Sch/X)_{syntomic} —> (Sch/X_0)_{syntomic} given by the rule V |—> V_0 = X_0 \times_X V. Then you can check that u is cocontinuous (in both cases). Hence we obtain a morphisms of topoi

  • g_{lisse-etale} : Sh(X_{lisse-etale}) —> Sh((X_0)_{lisse-etale})
  • g_{syntomic} : Sh((Sch/X)_{syntomic}) —> Sh((Sch/X_0)_{syntomic})

These maps are somehow contracting the topos associated to X onto the topos associated to X_0. Now in the second case the functor u also gives rise to a morphism of topoi in the opposite direction, namely i_{big} (for the syntomic topology), but I think neither i_{big} nor g_{syntomic} is an equivalence of topoi. In the first case, even though u is continuous, it does not define a morphism of topoi in the other direction.

In any case, cocontinuous functors are very useful and often easier to deal with than the better known continuous ones. For more information see the chapter on Sites and Sheaves.

Deformations and topoi

Let k be a field and let D = k[epsilon] be the ring of dual numbers. Suppose V is some geometric object over k. A geometric object U over D is called a deformation of V if it  (1) is flat over D and (2) has special fibre U_k = U \otimes_D k isomorphic to V. (This is intentionally vague.)

When V is a scheme, then U can be conveniently thought of as a locally ringed space whose underlying topological space is identical with the underlying space of V. In other words, you just change the sheaf of rings, and not the actual space.

However, some types of deformations in the literature do cause the underlying space or rather topos to change! And this is just one of the reasons why deformations of algebraic stacks are just a little more confusing than the case of schemes.

Here is a silly example: Let’s look at the lisse-etale site of k, call it C_1, and the lisse-etale site of D, call it C_2. For simplicity (and because it doesn’t matter for the associated topoi), let’s assume we only look at affine schemes. So an object of C_1 is an smooth affine scheme V over k and an object of C_2 is a smooth affine scheme U over D. In fact the sets of isomorphism classes of objects of C_1 and C_2 are naturally bijective, via the rules V —> V \otimes_k D and U —> U_k (Hartshorne, Exercise II 8.7). Moreover, if U is such an object, then the categories of etale coverings of U and U_k are canonically identified (by topological invariance of etale morphisms, see Theorem Tag 039R). For every isomorphism class of objects pick a particular object U_i of C_2 and let V_i be the corresponding object of C_1. Then we can try to match a sheaf F on C_1 with a sheaf G on C_2 by the rule F(V_i) = G(U_i). Does this work?

It doesn’t! Given two objects U_i, U_j of C_2 the collection of morphisms in C_2 between U_i and U_j is drastically different from the collection of morphisms between V_i and V_j in C_1. For example the value G(Spec(k[epsilon, x])) is acted upon by all automorphisms of k[epsilon, x] not just the automorphisms of k[x]. And in fact there is no way of identifying the categories of sheaves on C_1 and C_2 in any reasonable way. (I have several ways of saying this precisely, but none that is completely satisfactory. If you have one, please leave a comment. In fact, I would love a direct argument showing that Sh(C_1) and Sh(C_2) are not isomorphic as abstract topoi.)

Maybe this is just another reason for thinking that the lisse-etale site was a bad idea in the first place?