Let T be an algebraic space. A first order thickening of T is a closed immersion T —> T’ of algebraic spaces which is defined by an ideal of square zero. If T is over an algebraic space Y then we can talk about first order thickenings over Y. These form a category with an obvious notion of morphism.

Let X —> Y be a morphism of algebraic space. Consider first order thickenings T —> T’ over Y together with a morphism T —> X over Y. This gives a category of diagrams (T’ <— T —> X) over Y. Claim: If X —> Y is formally unramified then this category has a final object. Moreover, the universal object is actually a first order thickening X —> X’ of X over Y (i.e., T = X for the universal object). Let’s call this the universal first order thickening of X over Y.

Now, given X —> Y formally unramified we define the conormal sheaf of X over Y as the conormal sheaf of X in its universal first order thickening of X over Y. Notation C_{X/Y}. This construction is suitably functorial. For example if you have a morphism of arrows (f, g) : (X —> Y) —> (X’ —> Y’), and both arrows are formally unramified then you get a map f^*C_{X’/Y’} —> C_{X/Y}.

Why is this interesting? Well, I wanted to use this to clarify the notion of the module of differentials of a morphism of algebraic spaces. Namely, if f : X —> Y is an arbitrary morphism of algebraic spaces, then Δ : X —> X \times_Y X is not an immersion, just a monomorphism. Thus we need a slightly more general notion of a conormal sheaf in order to compare \Omega_{X/Y} to the conormal sheaf of Δ.

Note that a very natural definition of \Omega_{X/Y} is to define it as the module of differentials of the map of sheaves of rings f^{-1}O_Y —> O_X on the small etale site of X. (This is the one currently in the stacks project.) The result is that it is canonically isomorphic to the conormal sheaf of Δ. This can then be used to link with infinitesimal deformations of maps into X with \Omega_{X/Y}.

Moreover, as Jarod Alper pointed out, the material in the paragraph above should continue to work for morphisms of Deligne-Mumford stacks (as defined in the stacks project).

What is a skyscraper sheaf? Even for just sheaves on topological spaces there seem to be various definitions that one can use and that are used in the literature. Here are a few:

1. It is an abelian sheaf F whose stalks F_x are nonzero for finitely many points x if X.
2. It is an abelian sheaf F whose stalks F_x are nonzero for finitely many closed points x of X.
3. It is an abelian sheaf of the form i_{x,*}A where i_x : {x} —> X is the inclusion of a point into X and A is an abelian group.
4. It is an abelian sheaf of the form i_{x,*}A where i_x : {x} —> X is the inclusion of a closed point into X and A is an abelian group.

Of course the exact definition of any mathematical object is subject to minor variations. In each paper you have to look carefully what the definition really is. Moreover, sometimes authors have “hidden” assumptions (for example when an algebraic geometry paper says “everything is over C” they probably mean everything is at least locally of finite type over C and maybe even that everything is a variety over C).

Coming back to skyscraper sheaves, I think that for the stacks project the most natural choice is the one where a skyscraper sheaf is a sheaf of the form i_{x, *}A for any point x of X. An advantage of this choice is that we can also define skyscraper sheaves in the category of sheaves of sets.

As an aside we note that a sheaf satisfying 1 is not necessarily a finite direct sum of abelian skyscraper sheaves (think of the combinatorial circle and a nontrivial local system).

For a topos Sh(C) we define a skyscraper sheaf to be any sheaf of the form p_*A where p is a point of the site C.

This has the perhaps unfortunate consequence that if X is a topological space then the topos Sh(X) may have skyscraper sheaves which are not skyscraper sheaves on X. Namely, the points of the topos Sh(X) correspond 1-1 with irreducible closed subsets of X, and hence there may be more points of the topos than points of the space. This does not happen for schemes, which have underlying sober topological spaces.

It also means that the logical choice, in the stacks project, for a skyscraper sheaf on the small etale site S_{etale} of a scheme S is to require it to be a sheaf of the form s_*A where s : Spec(k) —> S is a geometric point of S. The reason is that points of the small etale site of S do indeed correspond to geometric points of S.

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).

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.

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).

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).

Edit: For some reason I keep making mistakes related to this material. I have had to edit the above several times to correct errors. and in the end it became such a mess that I decided to remove all of it…

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.

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.

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?