On my desktop at work I switched to the pretest version of Texlive 2010. This was probably a bad idea, and I may have to switch back if things don’t work out. But for the moment it looks like everything works fine. As an added bonus pdflatex generates pdf 1.5 with more compression which means that the pdf files are a bit smaller now than they were before. In fact, now book.pdf is a smaller download than book.dvi! (This is also true for the algebra chapter but not for the smaller chapters.)

Anyway, let me know if your pdf viewer doesn’t handle the new versions, or if you find something else wrong with the new setup.

Today I wrote a bit about the finite part of a morphism. The goal is to show: If f : X —> Y is locally of finite type and separated then the functor (X/Y)_{fin} which associates to a scheme T the set

{(a, Z) where a : T —> Y is a map and Z ⊂ T x_Y X is open and finite over T}

is representable by an algebraic space. It is easy to prove that it is a sheaf for the fppf topology. What is very cute is that it is trivial to show that (X/Y)_{fin} has representable diagonal. Hence now the only thing left to prove is that it has a surjective etale covering by a scheme which I think I know how to do.

As I expected this is quite a bit easier than proving representability theorems for Hilbert functors, which is the other method to approach the current short term goal: etale splitting of groupoids.

My next goal is to work out the material of this post. To do this I am going prove that, given a separated morphisms f : X —> Y of algebraic spaces which is separated and locally of finite type, the functor which associates to T/Y the set of open subspaces Z \subset T \times_Y X which are finite over T is representable by an algebraic space. As a very first trivial step we will prove that the functor remains the same if we replace X by the open part of X where f has relative dimension 0…

But as happens frequently, we don’t have the prerequisites available. Namely, we have not yet discussed the relative dimension of morphisms of algebraic spaces in the stacks project. Thus the recent work on the stacks project is all about dimension of schemes, local rings, algebraic spaces, fibers of morphisms of algebraic spaces, etc. Everything seems to work exactly as expected — although we are not entirely done writing it all out — and given a morphism f : X —> Y of algebraic spaces which is locally of finite type, and an integer d there exists an open U_d of X which is exactly the set of points where f has relative dimension <= d.

PS: You can figure out what was added to the stacks project recently by clicking on the links under the heading “Development Logs” on the right hand side. The material in green is what was added and the material in red is the deleted lines. The commit messages sometimes give a brief indication of what is happening in the commit.

Barring more embarrassing mistakes the slicing lemma (Lemma Tag 0461) is now fixed, as well as the only application of it in the stacks project, namely Lemma Tag 0489. Roughly speaking this last lemma states that given an equivalence relation j : R —> U x U of schemes such that both morphisms R —> U are flat and locally of finite presentation, there exists another equivalence relation j’ : R’ —> U’ x U’ such that the quotient sheaves are isomorphic: U/R = U’/R’ and such that the two morphisms R’ —> U’ are flat, locally of finite presentation and locally quasi-finite.

This is one step towards the goal of proving that R/U is an algebraic space if j : R —> U x U satisfies the assumptions above. The final steps are to fix the etale localization lemma (as discussed before on this blog), and apply it to U’/R’.

I think there are two interesting aspects of the fix we just implemented.

The first is that we used the notion of a point of finite type of a scheme, see Definition Tag 02J1. Basically a point of finite type of a scheme S is a closed point of an open affine of S. If you like working with very general schemes (non-Noetherian or non-quasi-separated, etc, etc) then using the points of finite type can be useful since (a) there are always enough of them: they are dense in any locally closed subset of S, and (b) they behave pretty much like closed points do. Take a look at the section on points of finite type in the chapter on Morphisms of Schemes.

The second is the digression on groupoids on fields we added. Its main goal was to prove Lemma Tag 04MQ which states that dim(R) = dim(G) for a locally finite type groupoid on a field. It is a bit subtle to explain precisely what this means, but the underlying result that makes it work is not hard to understand: It says simply that if we have a scheme X and two morphisms X —> Spec(k_1) and X —> Spec(k_2) both of which turn X into a geometrically integral variety over k_i, then actually k_1 = k_2 and the maps are identified too, see Proposition Tag 04MK.

The semester is completely over here at Columbia University, so I have more time to work on the stacks project. Since the last update (May 14) we have made the following changes to the stacks project

1. Moved the more technical and advanced material on groupoid schemes to its own chapter, with the unimagitive title “More on Groupoid Schemes”.
2. Rewrote some lemmas on local properties of groupoids for greater clarity, and to make them more widely applicable.
3. Small reorganization of the material on quotient stacks.
4. Added the following result to the chapter on varieties: If X is a variety over an algebraically closed field k then O(X)^*/k^* is a finitely generated abelian group. Somehow we will need this result in the near future.
5. This forced us to rethink some of the material on geometrically irreducible/reduced/connected schemes over fields. Leading to a bunch of small improvements.
6. Fixed a circular reasoning in the algebra chapter.
7. Finally, we added some stuff on groupoids on fields which we discuss below.

Of course this is a bit boring but I wanted to show that in the course of working towards a new result in the stacks project there is a kind of tendency to explain earlier material better and more precisely. In fact, in this manner we go over most of the material in the stacks project multiple times, and the material that gets used more is looked at more often — hopefully leading to an ever improved version of the most used algebraic geometry results in the stacks project.

A “groupoid on a field” means a groupoid scheme (U, R, s, t, c) where U is the spectrum of a field. These are quite interesting objects to work with, somehow analogous to group schemes over fields. For example here are some results we have added to the stacks project so far:

1. c : R \times_{s, U, t} R \to R is open,
2. R is a separated scheme,
3. there is a unique irreducible component Z of R which passes through the identity e,
4. Z is geometrically irreducible via both s and t,
5. if s, t are locally of finite type, then R is equidimensional.

We intend to add a few more soon. The main target is to show that if s,t are locally of finite type, then dim(R) = dim(G) where G is the stabilizer group scheme, and the material above goes into the details of the approach to this I have in mind. But I wonder if there is some completely general theorem saying that a groupoid on a field is somehow an extension of a group acting on a field by a group scheme over the field (how to formulate this precisely is not completely clear to me). Ideas?

Finally, we are done proving the assertion in this post. In fact the proof of the result is completely mechanical once you know the result for morphisms of schemes (see this post), and once you have developed enough machinery regarding localization of topoi, and ringed topoi. In fact, entirely the same argument is I think going to prove the result for morphisms of DM stacks mentioned briefly in this post but as usual there is the disclaimer that I haven’t worked out the details yet.

I am going to postpone the application of this result to deformations of maps till later, since I first want to start building theory for algebraic stacks. I will start with fixing the two errors in the chapter on groupoids. In fact I know how to fix the errors due to conversations I had, on this blog and by email, with David Rydh and Jarod Alper.

My prediction at the end of the last post was complete nonsense! Here are some examples of actions where the stabilizer jumps in codimension 1:

1. The action G_a^n x P^n —> P^n given by (a_1, …, a_n), (x_0: …: x_n) maps to (x_0: x_1 + a_1x_0: … : x_n + a_nx_0). The generic stabilizer is trivial and over the divisor x_0 = 0 the stabilizer is G_a^n. So the dimension of the stabilizer can jump up arbitrarily high in codimension 1.
2. A special case of the example above is the case n = 1 which Jarod Alper pointed out. If y = x_0/x_1 then the action looks like y maps to y/(1 + ty) where t is the coordinate on G_a.
3. Note that there are many formal actions \hat{G_a} x \hat{A^1} —> \hat{A^1}, because if theta is the derivation ty^k(d/dy) acting on C[[y]] then if k > 1 we can exponentiate and get automorphisms phi_t = e^theta : C[[y]] —> C[[y]] which satisfy phi_t \circ phi_s = phi_{s + t}.
4. Another example due to Jarod is the action G_a x A^2 —> A^2 given by t, (x, y) maps to (x + ty, y). The locus of points where the stabilizer is G_a is y = 0. This action seems very different from the action in case 2, allthough it may not be so easy to prove.
5. Take the product P^1 x P^1 with the action of G_a which is trivial on the first component and as in example 2 on the second. Then we may blow up (several times) in invariant points. If you do this in a suitable manner you will find an exceptional curve E consisting of fixed points where the local ring of the blow up at the generic point of E looks like C[x, y]_{(y)} and where the action is given by (x, y) maps to (x/(1 + txy^n), y). This gives infinitely many actions which cannot be etale locally isomorphic since the action is trivial modulo y^n and not y^{n + 1}. Note that y is the uniformizer of the local ring in question.

The conclusion is that if you allow the jump of the inertia group to be non-reductive, then many examples exist (there may even be moduli in the examples).

In this post I want to continue the discussion of the previous post by asking: How do space and automorphisms get mixed up in codimension 1.

Everybody’s favorite example of this phenomenon is the algebraic stack [A^1/mu_n] over a field. Namely this is a smooth separated stack of dimension 1 with generically trivial stabilizer and special stabilizer the group scheme mu_n of nth roots of 1. Consider the morphism

[A^1/mu_n] —> A^1

given by z maps to z^n on the covering A^1 of the stack. This is an isomorphism everywhere except over 0 where we get as stack theoretic fiber the algebraic stack [Spec(k[z]/(z^n)/mu_n]. One of the many cute things about this example is that if you look at the canonical morphism

[Spec(k[z]/(z^n)/mu_n] —> [Spec(k)/mu_n]

then the push forward of the structure sheaf corresponds to the regular representation of mu_n. I suggest we compare this with the fact that the push forward of the structure sheaf via the morphism Spec(k) —> [Spec(k)/mu_n] corresponds to the regular representation as well. For me this signifies that [Spec(k[z]/(z^n)/mu_n] is really a “single point”. Another fact is that if you consider the inverse of the morphism above, namely

A^1 – {0} —> [A^1/mu_n]

then the corresponding mu_n-torsor over A^1 – {0} is a generator of H^1_{fppf}(A^1 – {0}, mu_n). There are more canonical and coordinate independent ways of formulating these properties, which we leave to the reader…

Now I think that for any smooth separated algebraic stack over a field of characteristic zero having generically trivial stabilizer this is the only kind of jump that happens in codimension 1. (Haven’t proved it. If you add the condition that the stack is Deligne-Mumford then this is easier to prove.) In characteristic p > 0 there are many other finite groups that can occur as jumps in codimension 1. This is true for example because large finite p-groups act faithfully on k[[t]] if k is a field of characteristic p; the simplest action being perhaps the action of Z/pZ given by t maps to t/(1 + t). Note: Z/pZ is very different from mu_p in characteristic p.

If we look at still smooth but not necessarily separated algebraic stacks (back in characteristic zero) then many other jumps of automorphism groups happen in codimension 1. Here are some examples:

1. The stack [A^1/G_m] gives an example where G_1 = {1} and G_0 = G_m.
2. The stack [symmetric bilinar forms/GL_n] gives an example where G_1 = O(n) and G_0 is an extension of G_m x O(n – 1) by an n-1 dimensional additive group.
3. The stack [skew symmetric bilinear forms/GL_{2n}] gives an example where G_1 = Sp(2n) and G_0 is an extension of GL_2 x Sp(2n-2) by an 2(2n – 2) dimensional additive group.
4. The stack M_1 of genus 1 curves gives an example where G_1 is an elliptic curve semidirect Z/2Z and G_0 is an elliptic curve semi-direct Z/6Z.
5. The stack \bar M_1 of generalized genus 1 curves gives an example where G_1 is an elliptic curve semidirect Z/2Z and G_0 is G_m semidirect Z/2.

There is much worse that can happen (namely, nonseparated group schemes) if you allow the diagonal morphism to be nonseparated itself. But somehow if the stack is smooth, the characteristic is zero, and the diagonal morphism is separated, then I think (this is nonsense see below) that the picture should always be that in codimension 1 the stack fibers over [A^1/G_m] or [A^1/mu_n] with “fibre” B(H) where H is a flat group scheme. The proof should be that one takes H the closure of the generic stabilzer and then one divides it out.

[Edit: Jarod Alper pointed out that the last paragraph I also have to allow for [P^1/G_a] action via translation locally around infinity as a possibility. Maybe there are even others? Answer: yes, many. Will explain in next post.]

Let X be an algebraic stack. Let x_1, x_0 be points of X such that x_1 specializes to x_0 (here point means equivalence class of morphisms from spectra of fields). What can we say about the automorphism group schemes G_1 and G_0 of x_1 and x_0?

I wanted to write a bit about this question, and lead up to some related questions on higher algebraic stacks. But now I realize that (a) in the general case there is not a lot I can say, and (b) I haven’t though enough about this. Maybe you can help me out.

Let (U, R, s, t, c) is a groupoid in algebraic spaces such that X = [U/R]. Then there may exist a specialization of points u_1 -> u_0 of U such that u_i maps to x_i, but this is not always the case as examples of algebraic spaces show (for the unsuspecting reader we point out that in the stacks project an algebraic stack/space is defined with no separation conditions whatsoever). If this holds, then we see that the stabilizer group algebraic space G —> U has fibres G_{u_1} and G_{u_0} which are geometrically isomorphic to G_1 and G_0. This implies that dim(G_0) >= dim(G_1) for example.

Can we say anything more  if the generic stabilizer G_1 is trivial? In other words, given G_1 = \{1\} are there some G_0 which are “forbidden”?

Let’s reformulate the question in a slightly different form: Suppose that R is a valuation ring and G is a group algebraic space locally of finite type over R. Does there exist an algebraic stack X and a morphism Spec(R) —> X whose automorphism group scheme is G?

General remark: If G is locally of finite presentation and flat then the answer is yes, since in that case the quotient stack [Spec(R)/G] is algebraic.

Consider the case where R = k[[t]], char(k) = p > 0 and G = Spec(R[x]/(x^p, tx)) with group law given by addition. I.e., G is the group scheme whose special fibre is \alpha_p and whose general fibre is the trivial group scheme at the special fibre. Does G occur? The answer is yes. Namely, let \alpha_p act on affine 2-space over k by letting x act as the matrix
(1 x)
(0 1)
and let Spec(R) —> A^2 be given by t maps to (1, t). If you compute the automorphism scheme of this you get G.

Does such a construction work for every complete discrete valuation ring R and finite group scheme H over the residue field of R? If R is equicharacteristic p then a similar construction works, but if R has mixed characteristic I’m not so sure how to do this. Namely, if the group scheme has a flat deformation over R, then I think you can make it work, but if not, then I do not know how to construct a suitable algebraic stack. Do you?

There are noncommutative finite group schemes over fields of characteristic p which do not lift to characteristic zero. There are group schemes of order p^2 which do not lift, see paper by Oort and Mumford from 1968. I also think the kernel of frobenius on GL_n if p is not too small relative to n should not lift, but I do not know why I think so… So these may be good examples to try.

Suppose that f : Y —> X is a morphism of schemes with f locally of finite type and Y affine. Then there exists an immersion Y —> A^n_X of Y into affine n-space over X. See the slightly more general Lemma Tag 04II.

Now suppose that f : Y —> X is a morphism of algebraic spaces with f locally of finite type and Y an affine scheme. Then it is not true in general that we can find an immersion of Y into affine n-space over X.

A first (nasty) counter example is Y = Spec(k) and X = [A^1_k/Z] where k is a field of caracteristic zero and Z acts on A^1_k by translation (n, t) —> t + n. Namely, for any morphism Y —> A^n_X over X we can pullback to the covering A^1_k of X and we get an infinite disjoint union of A^1_k’s mapping into A^{n + 1}_k which is not an immersion.

A second counter example is Y = A^1_k —> X = A^1_k/R with R = {(t, t)} \coprod {(t, -t), t not 0}. Namely, in this case the morphism Y —> A^n_X would be given by some regular functions f_1, …, f_n on Y and hence the fibre product of Y with the covering A^{n + 1}_k —> A^n_X would be the scheme

{(f_1(t), …, f_n(t), t)} \coprod {(f_1(t), …, f_n(t), -t), t not 0}

with obvious morphism to A^{n + 1} which is not an immersion. Note that this gives a counter example with X quasi-separated.

I think the statement does hold if X is locally separated, but I haven’t written out the details. Maybe it is somehow equivalent to X being locally separated?

Perhaps the correct weakening of the lemma that holds in general is that given Y —> X with Y affine and f locally of finite type, there exists a morphism Y —> A^n_X which is “etale locally on X and then Zariski locally on Y” an immersion? (This does not seem to be a very useful statement however… although you never know.)