Since the last update about a week ago we have started adding some very basic material on algebraic stacks. It should be possible to lay out the basics of theory of algebraic stacks in the rest of the summer. Mainly I mean material on types of morphisms of algebraic stacks (how to define them, how do you check for a property, how are they related, etc), perhaps a little bit on quasi-coherent sheaves on algebraic stacks, more examples of algebraic stacks, etc. It will probably turn out that during this process the existing material on algebraic spaces will have to be expanded upon and improved.

What are some more exciting topics that I hope to get to in the not too distant future? Here are some possibilities:

1. Artin’s axioms, starting with a chapter on Schlessinger’s paper.
2. Olsson’s paper on the equivalence of the different definitions on proper.
3. Translating some of the more interesting results on algebraic spaces (regarding points, decent spaces, etc) into results on algebraic stacks.
4. A proof of Artin’s trick for algebraic stacks to prove that [U/R] is an algebraic stack if s, t : R —> U are flat and locally of finite presentation.
5. Diagonal unramified <=> DM
6. Prove the correct version of the Keel-Mori theorem in our setting.
7. Discuss the lisse-etale site (I may somehow want to avoid this — perhaps you can always avoid working with this beast?)
8. Treatment of simplicial schemes (as an example of simplicial objects) and relation to algebraic stacks. Formulate possible extension to higher algebraic stacks (e.g., hands on version of algebraic 2-stacks with example moduli of stacky curves)
9. Gabber’s lemma and more on finding nice maps from schemes to algebraic stacks.
10. Tricks with limits. Quasi-coherent sheaves are directed colimits of finite type quasi-coherent sheaves if X is…
11. Gerbs and relation to H^2.
12. Stacky curves. What are 1-dimensional stacks like?
13. General Neron desingularization (needed for 1).

Of course it will take a long time to add all of these topics. But it probably is a good idea to take one or two of these topics and keep them in mind while developing the general theory, so as to have a good reason for building theory. Moreover, many of these topics require additional material on schemes and algebraic spaces. For example, 4 requires a bit about Hilbert spaces (only relative dimension 0 however).

Finally two questions:

1. A while back David Rydh suggested we give a name to algebraic stacks with locally quasi-finite diagonal. Should we? If so any suggestions?
2. And what to call an algebraic stack with finite diagonal? (Maybe these should have been called separated and the ones with only proper diagonal something else. Too late to change now since it has been used like this for a while now in the literature.)

Maybe we could call an algebraic stack with locally quasi-finite diagonal “space-like”? And then the algebraic stacks with finite diagonal are space-like separated algebraic stacks. Hmm… not sure.

Today I learned a new basic fact on morphisms of schemes, namely the result mentioned in the title of this post. I started wondering about this question as I was thinking about separation conditions for algebraic stacks. Namely: it appears that the standard definition for a separated algebraic stack is one whose diagonal is proper, and I was wondering if we could get away with just requiring the diagonal to be universally closed and separated. It turns out we can due to the result of the title. After trying to think about it for a bit I decided to look for it on the web, and I quickly found a mathoverflow question asking exactly whether universally closed implies quasi-compact for morphisms of schemes, as well as the proof provided by Bjorn Poonen!

Note that all posts on mathoverflow are under CC-BY-SA, which is (unfortunately) not compatible with GFDL which is the license that the stacks project is under. Moreover, they ask to link back to their site, see here; and actually I think they are really stretching the meaning of the license since I think no linking should be required (IANAL). Anyway, I asked Bjorn if he agreed to relicense his material, and he said “Yes, that’s fine”. This means I need not link to their site if I do not want to (I did anyway).

It seems that CC-BY-SA is winning over GFDL in some respects, so I may switch the stacks project over to it in the future (there are still not too many authors so it shouldn’t be difficult to do). If I do this then I imagine I am allowed to take any latex code submitted to Mathoverflow by mathematicians and add it to the stacks project as long as I make sure to attribute it to the author of the comment. But for the moment, contacting the author of the comment and asking for permission directly makes more sense. Of course this is a bit difficult to do since it often isn’t clear who the author is especially for some very prolific contributors on Mathoverflow such as BCnrd…

Anyway, on a completely different note: I finally figured out how to set up the Makefile so that I can run the latex compiles in parallel. You will know why this is a problem if you’ve ever tried to write such a Makefile. If not it probably makes sense to stop reading this now. It is really quite simple (and I’m sure it is an often used trick). Instead of running latex stem.tex you execute a bash script which

1. creates a temporary directory using mktemp
2. copies all aux files and temp.toc to the temporary directory
3. creates symbolic links in the temporary directory to stem.tex and stem.bbl (and maybe some style files, etc)
4. changes directory into the temporary directory
5. runs latex stem.tex
6. moves stem.dvi, stem.aux, stem.toc back to the main directory
7. removes temporary directory

Anyway, using this I was able to cut overall compilation time in half. On the server at work the times were

• make dvis -j1 takes 1m42s
• make dvis -j2 takes 50s
• make dvis -j3 takes 33s
• make dvis -j4 takes 25s

Not too shabby.

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.