My student Yanhong Yang today noticed this typo in the definition of a geometric quotient in GIT (Definition 0.6).It states that the morphism \phi should be submersive and then has a blank line, after which it says

U’ \subset Y’ is open if and only if \phi’^{-1}(U’) is open in X’.

without any further explanation. This is present in all the editions that I have been able to find, including the latest one. Kollar, Keel and Mori, and Rydh have concluded from this that what is meant is that \phi should be universally submersive. (Just look at their papers. Maybe they even called up Mumford to ask him — another possibility is that they looked at the proof of Remark (4) of Paragraph 2 of Chapter 0 where Mumford seems to be using the fact that it is universally submersive.) There are also plenty of places in the literature where authors do not use universally submersive, only submersive.

Not only that, there is also a huge variance in the literature as to what a geometric quotient is. In the two papers of Kollar dealing with quotients (on by actions of group schemes, the other by finite equivalence relations) his definitions are adapted to the problem at hand, and in particular include the condition that the quotient map is locally of finite type, or finite — either of which seems like the wrong thing to require when considering the problem in general. In Rydh’s paper he includes the condition that the quotient map is universally submersive in the constructible topology (although this is in almost all cases implied by the other conditions). In a paper of Sheshadri he requires the quotient morphism to be affine. And so on. (I even found an article where they use Remark (4) of Paragraph 2 in Chapter 0 but do not require universally submersive…)

Let (U, R, s, t, c) be a groupoid in algebraic spaces (so R = G \times U if we are talking about an action of a group algebraic space). Let \phi : U —> X be an R-invariant map of algebraic spaces. It seems to me, but I may be wrong, that in each of these references, except for those where the author misread the typo, everybody always at least requires the following:

• X is an orbit space, i.e., the maps U —> X and R —> U \times_X U are surjective,
• \phi is universally submersive, and
• the structure sheaf O_X of X is the sheaf of R-invariant functions.

(The last condition is thrown in to attempt to make the quotient unique, but that only holds if you work in the category of schemes.) I think that in analogy with the introduction of algebraic spaces, we should use these three conditions to define the notion of a geometric quotient in the stacks project. Then we can have fun and add adjectives to describe additional properties of geometric quotients. For example, I particularly like the condition, introduced in the paper of Rydh, that a quotient is strong if it has the property that R —> U \times_X U is universally submersive.

By the way, I really really dislike the numbering scheme in GIT. Don’t you?

In the last couple of days I have added a few results on group schemes over fields to the stacks project. I mainly wanted to add the result that group schemes locally of finite type over a characteristic zero field are smooth which I hope to use later in an idea I have relating to finite groupoids in characteristic zero.

The sheaf of differentials of a group scheme over a field  is free (this holds in any characteristic). But actually I am not sure that a scheme over a field of characteristic zero whose sheaf of differentials is free is even necessarily reduced. In fact, in a paper entitled “Algebraic group schemes in characteristic zero are reduced” (1966) Frans Oort asks: Is every group scheme over a field of characteristic zero reduced? I googled and tried mathscinet but this question seems to be still open.

Another question I have is: Does any group scheme over a field have an open subgroup scheme which is quasi-compact? It seems that this could be true… but maybe I simply do not know any of the truly enormous group schemes that exist out there?

Leave a comment if you have an idea about either of these questions.

In this comment David Rydh formulates the conjecture that for a finite affine groupoid (U, R, s, t, c) the spectrum of the ring of invariants may be a geometric quotient for the stack [U/R]. In fact, the same question came up in a recent conversation with Jarod Alper here in the department.

I have an idea for generating invariant functions, which sounds so familiar to me that I am sure it is in the literature (let me know if you have a reference), or maybe I have already tried using it in the past. First, recall that if s,t are finite locally free flat so B is finite locally free over A then for any element x of A gives rise to an invariant element y by taking y = Nm_s(t(x)). In words y is the norm of t(x) with respect to the finite locally free ring map s : A —> B. Thus, in the general case where s, t are finite we try to find an element y in A which behaves like the norm of t(x) with respect to s. Maybe a falsifiable version of the conjecture above would be to conjecture the existence of a y in A such that for every prime p of A the value of y in k(p) is a power of the Nm of t(x) restricted to B \otimes_{s, A} k(p)?

My idea is to try to do the following. Take a finite free extension phi : A —> B’ and a surjection pi : B —> B’ such that pi o phi = s. (It may be convenient for later arguments to allow only certain types of ring maps A —> B, such as my personal favorite: finite flat relative complete intersections.) Now for any element x of A we can let y = Nm_phi(x’) where x’ in B’ is any element with pi(x’) = t(x). It is clear that y will NOT be R-invariant in general, simply because we have put too little restrictions on B’. But on the other hand, I am pretty confident that the ideal generated by all y of the form Nm_phi(x’) will be R-invariant. Namely, it should just cut out the set of points which are R-equivalent to a zero of the function x.

However, if A is an Artinian ring, then we can choose B’ so that B’ and B have the same maximal ideals. In this case if A has positive residue characteristics then it is quite easy to show that y^{p^n} for large n is independent of the choice of x’ and presumably is an invariant element of A (I haven’t checked this completely). This could then be the start of a kind of induction argument in the Noetherian case. But in characteristic zero I do not even know how to produce enough invariant functions in the Artinian case.

Well, my claims in this post were a little premature and may not be correct after all. It turns out that an argument in the lemma on etale localization of groupoids was wrong and has to be fixed. It doesn’t matter much for the overall picture since I can certainly use the arguments in the paper of Keel and Mori to fix it, but I wanted to prove it using only “etale localization” which may be hard.

The problem is the following: Suppose we have a groupoid scheme (U, R, s, t, c) with s, t flat, locally of finite presentation, and locally quasi-finite. Then we want to find many etale morphisms U’ –> U such that the restriction R’ of R to U’ is a disjoint union R’ = P \coprod Rest, with P a subgroupoid which is finite over U’ under both s’ and t’. The technique that I used (wrongly) in the current version of proof of the lemma mentioned above does prove that when we take U’ to be the spectrum of the henselization of the local ring of U at a point of U. But this doesn’t give you an etale morphism U’ —> U! What I am trying to see is if there is a kind of limit argument to descend P from the henselization to a smaller ring…

Let me just state here for the record that I think this means we can use this version of the lemma (with the henselization I mean) to define the (strictly) henselian local ring of the coarse moduli space without knowing that the coarse moduli space exists! Namely, since s, t : P —> Spec(O_{U,u}^h) are finite, flat and locally of finite presentation, we obtain that P is affine, and the “usual” arguments show that O_{U,u}^h is integral over the subring C of P-invariant elements of O_{U, u}^h. Presumably C (or its strict henselization) is what we are looking for. I haven’t thought this through completely, however.

I’ll post more here when I figure out how to repair the lemma.

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.