Let me discuss a bit the possible separation conditions to impose on algebraic stacks.

Before we talk about stacks, let’s review the conditions we have for algebraic spaces X. Here is a list:

1. Decent. This means that every point of X can be represented by a quasi-compact monomorphism from the spectrum of a field into X.
2. Reasonable: This means that for an affine scheme U any etale morphism U —> X has universally bounded fibres.
3. Very reasonable: This means that there exist schemes U_i and an etale surjective morphism \coprod U_i —> X such that each U_i —> X is quasi-compact onto its image.
4. Quasi-separated: This means that the diagonal morphism X —> XxX is quasi-compact.
5. Locally separated: This means that the diagonal morphism X —> XxX is an immersion.
6. Separated: This means that the diagonal morphism X —> XxX is a closed immersion.

Most algebraic geometers will work with either quasi-separated or locally separated spaces (note that in the stacks project a locally separated algebraic space is not required to be quasi-separated, e.g., any scheme is a locally separated algebraic space). On the other end of the spectrum requiring a space to be “decent” is a very mild condition that implies the points on a space behave like points on a scheme. All of the other conditions imply that X is decent (the hardest one to prove is 5 => 1 which is due to David Rydh and not yet in the stacks project). It seems that the class of all decent spaces, singled out by David Rydh, is a very nice class of algebraic spaces to work with.

Now for algebraic stacks there are going to be many, many different flavors of separation conditions. The reason is that if X is an algebraic space over S, then we can impose conditions on the diagonal Δ : X —> X x_S X but we may also impose conditions on the diagonal of the diagonal

Δ_2 : X —> X x_{Δ , X x_S X, Δ} X

Note that this is just the identity section of the inertia stack of X. So for example requiring this second diagonal to be quasi-compact is equivalent to the condition that Aut(x) —> T is quasi-separated for any object x of X over affine schemes T. Then by a standard trick (Lemmas Tag 02YI and Tag 0455) this implies that Isom(x, y) —> T is quasi-separated for any pair of objects x, y of X over T.

What David Rydh suggested to me in an email (if I understood correctly) is that we number diagonals as follows:

1. The structure morphism X —> S is the zeroth diagonal Δ_0 of X.
2. The usual diagonal Δ : X —> X x_S X is the first diagonal Δ_1 of X.
3. The second diagonal is Δ_2 : X —> X x_{Δ , X x_S X, Δ} X as above.

Presumably higher diagonals will not be needed since we work with stacks, and not higher stacks. Using this terminology we can define “X is of finite presentation over S” as “X is locally of finite presentation and Δ_0, Δ_1, and Δ_2 are quasi-compact”.

Moreover, in an ancient email (Mar 6, 2006) of Martin Olsson about the definitions of stacks in the stacks project he suggested that it might be a good idea to look at those stacks which are “locally separated on the diagonal”. In the language above I think translates into saying that Δ_2 is an immersion. This means that given a, b : x —> y morphisms of objects of X over an affine scheme T the locus “a = b” is represented by an open sub scheme of T. I think Martin’s point was that this is a natural condition which is often satisfied in moduli problems.

A groupoid on a field is just a groupoid scheme (U, R, s, t, c) where U = Spec(k) with k a field.

Groupoids on fields are very similar to groupschemes. Many results on group schemes have analogues for groupoids on fields. I recently added a bunch of new results on these types of objects to the stacks project. The goal was to see to whether I could now prove: If s, t are finite type, does there exist a finite index subfield k_0 of k such that s, t agree as morphisms to Spec(k_0).

Now, it turns out that this is a somewhat tricky thing to prove from the material we have in the stacks project so far. But looking into the future this is really quite straightforward. Let me explain.

[Hypothetical Argument] Suppose more generally that (U, R, s, t, c) is a groupoid scheme with s, t flat, of finite presentation whose stabilizer group scheme G is flat and locally of finite presentation over U. Then G acts freely on R and the quotient R’ = R/G is an algebraic space (future result). Then R’ is an equivalence relation on U with s’, t’ : R’ —> U flat and locally of finite presentation. So U/R’ is an algebraic space (future result). In fact, U/R’ is the coarse moduli space of [U/R] and actually [U/R] —> U/R’ is a gerbe (banded by G somehow).

Going back to U = Spec(k), s, t finite type, the spectrum of the field k_0 is going to be the coarse moduli space in the previous paragraph. The index of k_0 in k will be finite as a gerbe acquires a point over a finite extension (+ small argument).

The hypothetical argument above (hypothetical in that I haven’t written out all the details, and some hypotheses might need to be added) can also be used as the basis for decomposing any algebraic stack with suitable finiteness assumptions into gerbes over algebraic spaces. In the language of groupoid schemes: given (U, R, s, t, c) and say everybody finite type over a Noetherian base find a stratification U = \bigcup U_i such that the restriction of the stabilizer group scheme G to U_i is flat.

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.]