Googling for Grothendieck’s lemma turns up a whole slew of different lemmas. For some reason I started thinking of Grothedieck’s lemma as the following result, of which there are two versions:

• If A –> B is a flat local ring map of Noetherian local rings and f in B is a nonzero divisor in the fiber ring B/m_AB, then B/fB is flat over A.
• If A –> B is a flat local ring map of local rings, B is essentially of finite presentation over A and f in B is a nonzero divisor in the fiber ring B/m_AB, then B/fB is flat over A.

Leave a comment if you have an opinion about how to refer to this lemma. This result is (very) related to the local criterion for flatness which says instead

• If A –> B is a local ring map of Noetherian local rings and I is an ideal of A such that B/IB is flat over A/I and moreover Tor_1^A(B, A/I) is zero, then B is flat over A.
• If A –> B is a local ring map of local rings, B is essentially if finite presentation over A and I is an ideal of A such that B/IB is flat over A/I and moreover Tor_1^A(B, A/I) is zero, then B is flat over A.

This is particularly useful when I = m_A because B/m_A B is automatically flat over the field A/m_A. In the Algebra chapter of the stacks project we prove both of these independently although it might have been better/quicker to deduce the first from the second. Finally, there is another very related result which I think is usually called the critère de platitude par fibre which says roughly

• If X –> Y is a morphism of locally Noetherian schemes flat over a locally Noetherian base S and if f induces flat morphisms between fibers, then f is flat.
• If X –> Y is a morphism of schemes of flat and locally of finite presentation over a base S and if f induces flat morphisms between fibers, then f is flat.

You can in fact weaken the assumptions a bit. Of course this is a completely algebraic fact which can be reformulated in terms of maps of local rings as above. There are also versions for modules which are potentially much more useful; for these and some other results see Algebra, Section Tag 00MD, Algebra, Section Tag 00R3, and More on Morphisms, Section Tag 039A.

In this post I mentioned a theorem usually attributed to Michael Artin which basically says that an fppf sheaf which has a flat, finitely presented cover by a scheme is an algebraic space. I am still working on adding this to the stacks project, and more or less all the preliminary work is done.

But what I wanted to say here is that to prove this one does not have to use “Artin’s trick”. What I mean is the argument in Artin’s versal deformations paper that rests on the following fact: Given a morphism f : X –> Y which is flat and of finite presentation then the space H_n(X/Y) of length n complete intersections in fibers of f is smooth over Y, and moreover \coprod_n H_n(X/Y) –> Y is surjective.

Instead one can use a slicing argument to go down to relative dimension zero (see Lemma Tag 0461) and etale localization of groupoids (see Lemma Tag 03FM) to get an etale covering by a scheme (by dividing out by the P-part of the groupoid scheme). Note that the last lemma is a version of Keel-Mori, Proposition 4.2 and that they in their proof use some form of Hilbert schemes also… but they needn’t have and standard etale locailzation techniques would have sufficed.

Morally speaking it is clear that Hilbert schemes needn’t be considered when proving this result since the original flat finitely presented covering X –> Y might have had relative dimension zero with connected fibres, and then only one H_n(X/Y) is nonempty (locally on Y), namely that one where n is the relative degree and H_n(X/Y) = Y. In other words you are just directly proving that Y is an algebraic space!

On the other hand, as Jarod Alper pointed out, when we try to prove the analogous result for algebraic stacks, then we have to construct a smooth cover which will have in general a positive relative dimension over the stack and the remark in the preceding paragraph doesn’t apply. Of course this was the point of Artin’s trick and this is how he used it in his paper.

If you’ve downloaded the whole stacks project as book.pdf and opened it in any reasonable pdf viewer, then you’ve been able to click on internal references to go straight to lemmas referred to in this or that proof. This is kind of essential as the document is too long to manually go back and forth between references. As far as I know this works in adobe reader (acroread), xpdf , okular and even evince.

But if you go to the “browse chapters page“and open a pdf file in an embedded pdf viewer, then the cross-file-hyperlinks do not work unless you use adobe reader as far as I know. For example if you do this on an apple machine using “preview” then this does not work (at least it did not last time I tried). Here I am talking not just about opening the correct file, but also opening it at the correct spot (named destination).

If you download the tar file with all the pdf files and untar and view them using xpdf, okular or acroread in the resulting directory, then cross-file-hyperlinks work.

Of course this is not a big problem since the whole book version should work for everybody. But as an online document it doesn’t work that well since it is kind of a large download. On the other hand, since not everybody has acrobat reader installed (or the browser plugin enabled) having smaller chapters with cross file links doesn’t work either…

In the discussion of groupoid stacks [U/R] it turns out that given objects x, y of [U/R] over some scheme T, then Isom(x, y) is fppf locally on T an algebraic space. Thus it makes sense to go back to algebraic spaces and prove a result characterizing algebraic spaces. Namely, an fppf sheaf of sets F for which there exists an algebraic space X and a map f : X –> F which is

• representable by algebraic spaces, and
• surjective, flat and locally of finite presentation

is an algebraic space. The only ingredient missing for the proof is an analogue of Keel-Mori, Lemma 3.3. Hopefully we will have some time to write this in the near future.

At the moment I am writing a chapter on groupoid spaces. In this chapter I introduce the notion of a “group space” and the notion of a “groupoid space”. Of course, most of the theory is exactly the same as for groupoid schemes, so in fact I am simply editing a copy of the chapter on groupoid schemes. What I am wondering is whether it is OK to use “groupoid space”‘ and “group space”, or if I should use the longer and perhaps more correct “algebraic groupoid space” and “algebraic group space”? Or, is it better to use “groupoid in algebraic spaces” and “group algebraic space”?

For now I’ll stick to my first choice, but if you object please leave a comment.

Here is a question somebody asked today which used to be answered in an older version of the stacks project, but which got excised a while ago.

The question is: How different are the notions of a stack in groupoids and a sheaf of groupoids?

The answer is that there are 2 differences. The first is a minor one: Although every stack in groupoids is equivalent to a split category fibred in groupoids, it is not always isomorphic to one. Here a split category fibred in groupoids over a category is the category associated to a contravariant functor from the category into the category of groupoids. Of course such a functor is nothing else than a presheaf F of groupoids on the site.

The second difference is more serious. Namely, when you say that F is a sheaf, then apart from the requirement that morphisms descend you are only requiring that descent data for objects are effective for a somewhat restrictive class of descent data. In fact you are only requiring that if x_i are objects of the split fibred category over the members U_i of the covering, and if the restrictions x_i|_{U_i \times_U U_j} and x_j|_{U_i \times_U U_j} are equal then this should be effective. Clearly this is different from the requirement that all descent data are effective.

The “explanation” of this in the earlier version of the stacks project is that the category F(U) should be the homotopy limit of the diagram

\prod F(U_i) ==> \prod F(U_i \times_U U_j) ==> \prod F(U_i \times_U U_j \times U_k)  …

and not the usual limit. And of course this is a nice way of saying it since it leads to possible generalizations such as higher stacks.

The semester is starting today, so I will have less time to work on the stacks project unfortunately. In any case, after some revisions to “Morphisms of Spaces” and adding the necessary lemmas to “Descent on Spaces”, I have now started to edit the chapter “Algebraic Stacks”. This chapter is only going to have the stuff comparing the different ways of thinking about algebraic stacks (including presentations), and some examples. My idea is that we do not use the customary abuse of language (such as a scheme is a stack) in that chapter but in the chapters following it we do. So the first chapter on algebraic stacks is awkward, and the later chapters less so. In fact, I think I won’t even mention the separation axioms in that chapter since it works better in a chapter on morphisms of algebraic stacks.

If you take a look at the current version of the stacks project you will see that essentially the last chapter is the one on algebraic stacks, and moreover that it is practically empty. Why?

One of the decisions made early on was to build up the material along historical lines in the usual manner. Namely:

1. We develop some commutative algebra and some theory of sheaves on topological spaces.
2. We define schemes as locally ringed spaces which locally look like the spectrum of a ring.
3. We develop the theory of sheaves on a site and we discuss schemes and morphisms of schemes. We define the big fppf/etale site of a scheme. We also discuss in some detail the notion of descent for schemes and descent of properties of morphisms of schemes.
4. We define algebraic spaces as fppf sheaves which etale locally look like a scheme.
5. We study the notion of stacks fibred in groupoids over a site. We study properties of algebraic spaces and morphisms of algebraic spaces. We study descent for algebraic spaces.
6. Finally we use this material to define algebraic stacks as stacks fibred in groupoids on the category of schemes in the fppf topology whose diagonal is representable by algebraic spaces and which have a smooth covering by a scheme.

This seems like a bit of overkill at first. Why can’t we do a little bit of commutative algebra, and then go straight to the definition of algebraic stacks as certain stacks fibred in groupoids over the opposite category of the category of rings endowed with the fppf topology?

The answer is of course that in principle you can do this. One advantage of doing this is that you might not have the kind of repetition that the progression 1,2,3,4,5,6 above shows. Moreover, every geometric object would be an algebraic stack and you would not have to use the customary abuse of notation (such as statements of the form “a scheme is an algebraic stack”) which one finds in papers on algebraic stacks. In fact, feel free to clone the stacks project and to rewrite it in this way.

On the other hand, any full discussion of the theory of algebraic stacks is going to mention affine schemes, schemes, and algebraic spaces. It will still be the case that the most interesting objects of study are algebraic varieties, and their moduli spaces. For example curves, abelian varieties, and Jacobians of curves, moduli spaces of such, Shimura varieties, K3-surfaces, etc, etc. Proofs of foundational theorems such as “coherence of proper pushforward”, or Zariski’s Main Theorem will likely still be proved by proceeding via arguments through the case of schemes.

Currently, the arguments dealing with schemes and morphisms of schemes are of a different nature than those for algebraic spaces and the arguments dealing with algebraic spaces will be I am sure of a different nature again. This means that there is actually less repetition in the sequence 1,2,3,4,5,6 as one expects at first. Also, when working on a new result for say algebraic spaces, by first proving the needed algebra lemmas, then proving the results for schemes, and finally the result for algebraic spaces, we automatically organize the material, and we prove each result in its natural setting.

A related observation is that the reader need only know about schemes when reading any of the results in the theory of schemes. (Of course eventually we will prove results which can be formulated in the language of schemes but whose proof uses algebraic stacks.) Similarly for the algebra results and the results on algebraic spaces.

There is also a psychological component. Sure you can define algebraic stacks without first defining any intermediate geometric objects. However, once this is done, there you are, and there is nothing that you can hold onto and relate the objects to… it seems  a bit similar to introducing quantum physics without first talking about classical mechanics. Sure it is fundamentally more important, but what is it really telling us about the real world?

This is a blog about algebraic geometry and more specifically algebraic stacks and the stacks project. So let’s start with some basic information on the stacks project.

The stacks project is an open source text book on algebraic stacks and the algebraic geometry that is needed to define them. The current version of the complete book can always be found here.

How can you contribute? You can leave comments on this blog if you want to give feedback in an informal manner. Another simple way is to download the tex file and the pdf file of a chapter, and while you are reading correct any typos, or mathematical errors you see. At the end simply email the new tex file to stacks.project@gmail.com. You can take a look at the todo-list to see what needs to be done.