Since the last update we have worked on examples of stacks, stack of torsors, quotients stacks [X/G], Picard stack, epimorphisms of rings, change of partial universe for algebraic spaces, Gabriel-Zisman localization, pushforward of stacks, pullback of stacks, change of partial universe for algebraic stacks, change of base scheme for algebraic stack, and finally we started working on a new chapter entitled Properties of Algebraic Stacks.

As you can see from the list, we worked through a lot of very formal material. Some of this is a bit rough as written, although almost all of it is “obviously correct”. This is OK as at least some of it is just meant to explain set theoretical issues.

But now we’ve finally caught up with this material and we can start working on algebraic stacks! Moreover, I have decided to introduce the customary abuse of language in the first section of the new chapter linked to above. The idea is that any confusion which is caused by this abuse of language should be explained by pointing to a lemma in Algebraic Stacks or earlier. For example, we will say that an algebraic stack is an algebraic space if it is representable by an algebraic space (which is equivalent to have trivial inertia stack, see Proposition Tag 04SZ). Then suppose X, Y, Z, W are algebraic stacks and we are given a diagram

and X, Y, Z, W “are” algebraic spaces. What does it mean to say “the diagram is commutative” or “the diagram is cartesian”? Well, it could either mean that the diagram is a commutative, resp. cartesian in the category of algebraic spaces, or that the diagram is 2-commutative, resp. a 2-fibre product diagram in the 2-category of algebraic stacks. The abuse of language is not confusing in this case since these conditions agree. More precisely: the diagram is 2-commutative if and only if the corresponding diagram of sheaves of isomorphism classes of objects of fibre categories is commutative, and the diagram is a 2-fibre product if and only if the corresponding diagram of sheaves of isomorphism classes of objects of fibre categories is cartesian.

Well actually 2006 pages at this very moment. Also

• 193614 lines of tex,
• 6267 tags, and
• 1326 commits since I started using git on May 20, 2008.

Enjoy!

We added a section on epimorphisms of rings to the algebra chapter. Everything is completely straightforward except for the following fact: If A —> B is an epimorphism of rings then |B| ≤ |A|. Since an epimorphism of rings is not necessarily surjective this is not a triviality. We learned this from the exposee by Mazet in the Seminaire Samuel.

You can use this to show that if X —>Y is a monomorphism of schemes then size(X) ≤ size(Y), which is a technical condition on the cardinalities of some sets associated to X and Y. See the chapter on sets.

Here is a consequence: Given a scheme Y there is a set worth of isomorphism classes of monomorphisms X —> Y. I don’t think this is formal since I vaguely remember reading somewhere about a category (maybe spaces up to homotopy?) where such a thing is not true. Leave a comment if you know the correct statement.

Take a look at the following commit. Observe how the author of this commit is Hendrik! But if you look at the summary of all commits, then you find that most commits have been authored by me.

What happened is that Hendrik sent me a patch formatted in such a way that git was able to understand who authored this patch. Maybe he used the instructions here, or he figured it out himself. It doesn’t matter.

But, if you want your name to show as an author in the logs of the stacks project, then you can do this too!

A morphism of finite presentation X —> S is a morphism which is (a) locally of finite presentation, (b) quasi-separated, and (c) quasi-compact.

Let κ be an infinite cardinal. What should be a morphism of κ-presentation? By analogy with the above I think it should be a morphism f : X —> S such that

1. for any affine opens U, V of X, S with f(U) ⊂ V the algebra O(U) is of the form O(V)[x_i; i ∈ I]/(f_j; j ∈ J) with |I|, |J| ≤ κ,
2. for any U, U’ affine open in X over an affine V of S the intersection U ∩ U’ can be covered by κ affine opens, and
3. for any affine V in S the inverse image f^{-1}(V) can be covered by κ affine opens.

It is my guess that all the usual things we prove for morphisms of finite presentation also hold for morphisms of κ-presentation. Namely, it should be enough to check the conditions over the members of an affine open covering of Y, the base change of a morphism of κ-presentation is a morphism of κ-presentation, etc. In particular, if should also be true that if {S_i —> S} is an fpqc covering and X_i —> S_i is the base change of f : X —> S, then

X —> S is of κ-presentation ⇔ each X_i —> S_i is of κ-presentation

Of course this is completely orthogonal to most of algebraic geometry and I hope you’ve already stopped reading several lines above (maybe when I used the key word “cardinal”). For those of you still reading let me indicate what prompted me to write this post. Namely, suppose that X, Y are schemes over a base S which are fpqc locally isomorphic. Then the above says that X and Y have roughly the same “size” (this is defined precisely in the chapter on sets in the stacks project).

As an application this tells us for example that given a group scheme G over S there is a set worth of isomorphism classes of principal homogeneous G-spaces over S! A principal homogeneous G-space is defined in the stacks project, as in SGA3, to be a pseudo G-torsor which is fpqc locally trivial — and note that the collection of fpqc coverings of S forms a proper class, which does not contain a cofinal subset!

Another potential application, internal to the stacks project and with notation and assumptions as in the stacks project, is that, given a group algebraic space G over S, it guarantees that the stack of principal homogeneous G-spaces form a stack in groupoids over (Sch/S)_{fppf}. Instead of working this out in detail in the stacks project I will for now put in a link to this blog post.

Let U be a scheme. Let us define a family of d dimensions proper algebraic spaces over U to be a morphism X —> U from an algebraic space X to U which is flat, proper, locally of finite presentation, such that all geometric fibres are equidimensional of dimension d. Let Fam_d denote that full subcategory of the stack Spaces whose objects X/U are families of d dimensions proper algebraic spaces. Then as discussed in the preceding post we conclude that Fam_d is a stack over (Sch).

In this post I want to point out that for this to work out it is absolutely necessary that we work inside the category of algebraic spaces, and not with schemes. Let me start discussing the low dimensions.

[d = 0] It is a fact that any family X —> U of 0-dimensional proper algebraic spaces over a scheme U is automatically represented by a scheme. This follows from Proposition Tag 03XX.

[d = 1] Let X —> U be a family of 1-dimensional proper algebraic spaces over a scheme U. Then etale locally on U the space X is projective over U (in particular a scheme). But, even if you assume the fibres of X —> U are geometrically integral it is not the case that Zariski locally on U the space X is a scheme. An explicit example is the example of non-effective descent in Bosch-Lutkebohmert-Raynaud, Neron Models, Section 6.7 (since after all in Fam_1 we do have effective descent).

[d = 2] Here there are even examples of X —> U where all fibres are smooth projective surfaces, and U is a smooth curve, but the total space is an algebraic space and not a scheme. The examples comes from degenerating a general degree 513* surface in P^3 to a surface with a single node and doing a small resolution of the node on the total space (after performing a 2:1 base change). Moreover, there is no finite type, faithfully flat base change after which X becomes a scheme.

So you see that in order to do moduli of geometrically very interesting objects it is really convenient to work with algebraic spaces! In fact, if you don’t then you will not see all of the families that you want to see…

*Footnote: Degree 514 works also, and degree 21 too, and…

Consider the fibred category p : Spaces —> (Sch) where an object of Spaces over the scheme U is an algebraic space X over U. A morphism (f, g) : X/U \to Y/V is given by morphisms f : X —> Y and g : U —> V fitting into an obvious commutative diagram.

Theorem: This is a stack over (Sch)_{fppf}.

In essence the thing you have to prove here is that any descent data for spaces relative to an fppf covering of a scheme is effective. This follows immediately from the results discussed in this post, see Lemma Tag 04SK. You can find a detailed discussion in the chapter Examples of Stacks of the stacks project (in the stacks project we have only formulated this exact statement for the full subcategory of pairs X/U whose structure morphism X —> U is of finite type; this is due to our insistence to be honest about set theoretical issues).

Note how absurdly general this is! There are no assumptions on the morphisms X —> U at all. Now we can use this to show that suitable full subcategories of Spaces form stacks. For example, if we want to construct the stack parametrizing flat families of d-dimensional proper algebraic spaces, all we have to do is show that given an fppf covering {U_i —> U} of schemes and an algebraic space X —> U over U such that for each i the base change U_i \times_U X —> U_i is flat, proper with d-dimensional fibres, then also the morphism X —> U is flat, proper and has d-dimensional fibres. This is peanuts (compared to what goes into the theorem above).

Of course, to show that (under additional hypotheses on the families) we sometimes obtain an algebraic stack is quite a bit more work! For example you likely will have to add the hypothesis that X —> U is locally of finite presentation, which I intentionally omitted above, to make this work.

A site in the stacks project is different from what is called a site in SGA4. What we call a site is what is called a category endowed with a pretopology (see Exposee II, Definition 1.3 of SGA4). In other words a site is category C endowed with a set Cov of families of morphisms with fixed target called coverings such that

1. If V —> U is an isomorphism then {V —> U} is a covering,
2. if {U_i —> U} is a covering and {V_{ij} —> U_i} is a covering for each i, then {V_{ij} —> U} is a covering,
3. if {U_i \to U} is a covering and V —> U is a morphism of C then U_i \times_U V exists and {U_i \times_U V —> V} is a covering.

A sheaf on C is then a presheaf which satisfies the sheaf axiom for all the coverings. Note that in general there are many choices of Cov which give rise to the same category of sheaves. For example on (Sch), see previous post for notation, the etale coverings and the smooth coverings give rise to the same category of sheaves. For this reason you will sometimes hear people say that the etale and smooth topology are the same. But for us the etale site and the smooth site are different.

In this post I wanted to mention that working with sites as above is useful in that the types of coverings you allow can be used to express properties of the site which cannot be expressed in terms of the topology alone. For example, we can say that a property P of objects of C is local on the site if given a covering {U_i —> U} we have P(U) <=> P(U_i) for all i. Then it is clear that the property P(X) =”dim(X) < 17" is local on the etale site (Sch)_{etale} but not local on the smooth site (Sch)_{smooth}. Similarly for properties of morphisms, e.g., P(f)="f is locally quasi-finite" is local on the target on the etale site, but not local on the target on the smooth site. For a previous discussion of what it means for a property of morphisms to be "etale local on source and target", see this post.

Today I was able to add the result that the quotient stack [U/R] associated to a smooth groupoid in algebraic spaces is an algebraic stack. See Theorem Tag 04TK. Very satisfying!

There are several things that have to be done next:

1. Work on the chapter Examples of Stacks, and start a parallel chapter Examples of Algebraic Stacks, where we discuss in detail some very basic examples of algebraic stacks.
2. Add some foundational material on changing the base scheme and the underlying big site. This could then be used to define a 2-category of algebraic stacks which lumps all algebraic stacks together regardless of big site that was used to define them (but I’m not sure this would be useful, so I may not add this).
3. Write a Chapter on properties of algebraic stacks
4. Write a Chapter on morphisms of algebraic stacks
5. Write about separation axioms for morphisms of algebraic stacks
6. And so on and so forth.

Next week I will not have time to work on the stacks project, so you can start working on the topics above yourself!

This post is a bit of a rant.

One subgoal of the stacks project is to work through the beginnings of etale cohomology and algebraic stacks without making use of universes. Most of this is completely straightforward (and already done), and there is only one point at which you have to make an argument.

First I would like to point out that there is a completely well established (axiomatic) theory of sets, and that is ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Thus, virtually any mathematician who uses a set means the type of object that is described by the axioms of ZFC. Paradoxically it is the set theorists themselves who enjoy thinking about other kinds of sets. They like to add and substract axioms from ZFC and see what happens. It is probably for that reason that you cannot find a book that simply takes the axioms of ZFC and develops the theory of sets (if you do know such a book or lecture notes, please email me or leave a comment). So whenever you take up a book on set theory to learn something about the sets you work with every day, you have to be very careful to see whether the author has added some bizarre additional hypotheses to the theory, or works with a different axiom system. To me it seems a bit of a crime that some of the undergraduate level books do not use ZFC.

Of course, some of the most interesting results in set theory (that I know) are those having to do with consistency, etc. As an example I want to mention the result that if ZFC is consistent, then you cannot prove the existence of a strongly inaccessible cardinal in ZFC.

What is a universe? Roughly, a universe is a set X such that all the axioms of ZFC hold for the elements of X. It turns out that the existence of a universe is equivalent to the existence of a strongly inaccessible cardinal, hence cannot be proved inside ZFC (and neither can the nonexistence, actually). Thus I argue that most mathematicians use a set theory which does not have universes.

Grothendieck added the existence of universes U as an axiom so he could say “let (Sch) = the category of schemes which are elements of U”. This is somewhat convenient. For example it means that if I is an element of U and X_i is an element of U which is a scheme for all i, then \coprod X_i (suitably constructed) is an element of U. On the other hand (Sch) is not an element of U and considering it takes you outside of U. Moreover, since U is just some set, (Sch) is just some set of schemes, and there are schemes not in U. Furthermore, the topos of sheaves on (Sch) is of course a proper class and not a set at all. Finally, changing the universe gives you a different topos.

The axioms of ZFC provide many techniques for constructing large sets. How close can we come to constructing a universe U inside of ZFC? I don’t know; I’ve looked around, but I haven’t found somebody addressing this directly (likely because for a set theorist this is utterly trivial). But here is what is true; I’ll formulate this in terms of (Sch) = the category of schemes which are elements of U, because if you are reading this then you probably do not care about “large sets”. Given any set of schemes (Sch_0) you can construct a U such that

1. (Sch_0) is contained in (sch),
2. (Sch) has fibre products, and fibre products agree with fibre products in the category of all schemes,
3. more generally you can construct U such that limits and colimits over at most countable diagrams in (Sch) exist whenever they exist in the category of all schemes and are the same,
4. you can make (Sch) be closed under immersions, morphisms of finite type, morphisms which are locally of finite type such that the inverse image of an affine can be covered by countably many affines, and
5. if X is in (Sch) and Y is a scheme whose “size” is at most the 2^(size of X), then Y is isomorphic to an element of (Sch).

The key is to construct U such that (3) and (5) hold; the size of a scheme X is defined in terms of the cardinality of the set of sections of O_X over affine opens and the cardinality of the set of affine opens. But now you cannot _also_ require that (Sch) is closed under disjoint unions of objects of (Sch) indexed by elements I of U, and, if you think about it for a bit, you will see that this is the only difference from the case of a universe. So, although it is clear that properties 1 — 4 imply that in many cases the disjoint union does exist in (Sch), it is just not always true!

This is the approach we chose in the stacks project. Another one might have been to construct U’s such that the disjoint unions always exist, but then you need to weaken condition (5) by quite a bit; I’m not exactly sure how much.