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.)

Here is something introduced today:

1. If G is an abstract group, then G-Sets denotes the category of sets endowed with left G-action
2. If G is a topological group, then G-Sets denotes the category of sets X endowed with a continuous G-action where X is given the discrete topology.
3. If G is an abstract group, then Mod_G denotes the abelian group objects in the category G-Sets.
4. If G is a topological group, then Mod_G denotes the abelian group objects in the category G-Sets.

This works well in the sense that if G is the absolute Galois group of a field K, then G-Sets is equivalent to the category of sheaves of sets on the small etale site of Spec(K). Similarly, Mod_G is equivalent to the category of abelian sheaves on the small etale site of Spec(K).

I suppose that if G is a topological group, then one may also want to consider the category of topological spaces endowed with continuous G-action. For this category we could use Top_G, or G-Tops, or G-Spaces (although Spaces/S has been used for the category of algebraic spaces over S already… only a few times and it may be better to give that a really expensive name). Then there is the category of abelian group objects, usually called topological G-modules, in G-Tops/Top_G/G-Spaces, sigh! These already come up briefly in the chapter on etale cohomology when introducing group homology on the category of compact topological G-modules (Warning: This part is still very rough and not yet cleaned up).

I also changed the definition of a geometric point in the chapter on etale cohomology to require the field in question to be algebraically closed (from just requiring it to be separably closed). Sure, for discussing stalk functors of sheaves on the small etale site you only need separably algebraically closed, but it is just more convenient to have the same definition everywhere.

Suppose that (Sh(C), O) is a locally ringed topos. When is this the small etale topos of a DM stack? I think the condition is just that it is “locally isomorphic to the small etale topos of a scheme”. Here is why: (again I haven’t worked out all the details — so some of this may not work exactly as stated)

The condition means there exists a sheaf F in Sh(C) such that the localization (Sh(C)/F, O_F) is isomorphic to (Sh(U_{etale}), O_U) as a locally ringed topos. Consider the product sheaf F \times F and think of it as a sheaf over F via one of the projections. Via the isomorphism Sh(C)/F = Sh(U_{etale}) we can think of F \times F as an etale sheaf on U. Since every sheaf  on U_{etale} is representable by an algebraic space over U we conclude that (Sh(C)/F \times F, O_{F \times F}) is isomorphic to (Sh(R_{etale}), O_R) for some algebraic space R. By the fully faithfulness discussed in previous posts we obtain two morphisms s, t : R —> U. Moreover, we can do the same trick with F \times F \times F and obtain a composition morphism R \times_U R —> R (this will require a bit of work relating fibre products of etale morphisms of algebraic spaces to what happens on the side of small etale topoi, but I’m not worried). Hence (U, R, s, t, c) will be an etale groupoid algebraic space. The final step is to show that the DM-stack X = [U/R] has an associated locally ringed small etale topos (X_{etale}, O_X) which is equivalent to the locally ringed topos we started out with.

Note that [U/R] is a DM-stack since in the stacks project we work with algebraic stacks having no separation conditions whatsoever.

To characterize algebraic spaces among these will require a discussion of the inertia in terms of the language of locally ringed topoi as in the post dicussing the difference between Spec(R) and [Spec{C)/{+1, -1}].

Actually this sounds extremely familiar to me and I wouldn’t be surprised if I attended a talk or read an article/book which contained exactly this argument. There is after all an enormous literature on topoi, ringed topoi, etc (starting for example with Hakim, Topos anneles et schemas relatifs). It is also possible that somebody explained this to me in a conversation. Please think of it as being part of algebraic geometry already. But one of the things that is really fun about doing mathematics for me is the discovery process: As I work through material it feels as if I’m discovering it, even if I am reading a 100 year old text.

Certainly this argument will not become part of the stacks project until much later (if at all). My goal for the summer is to really start hacking away at basic properties of algebraic stacks.

PS: For experts on topoi: I keep writing Sh(C) whenever I mean topos since in the stacks project I define a topos to be Sh(C) where C is a site (a la Artin’s notes on Grothendieck topologies).

Theorem Tag 04I7 now has a complete proof. It is the case of schemes for the result I mentioned in this post. It says that given two schemes X, Y any morphism of locally ringed topoi

(Sh(X_{etale}), O_X) —> (Sh(Y_{etale}), O_Y)

comes from a morphism of schemes X —> Y. To prove it you use that an affine scheme V etale over Y can be embedded into A^n_Y for some n (and that it is cut out by polynomial equations in there).

Of course, it would perhaps be quicker to try and directly prove the corresponding result for algebraic spaces or Deligne-Mumford stacks (haven’t worked out the details yet), but I want mostly to stick with the philosophy that each result is proved in various levels of generality: commutative algebra, schemes, algebraic spaces, algebraic stacks, higher topos theory, etc, etc.

In a related discussion Brian Conrad pointed me to Theorem A.4.1 of the preprint by Conrad-Lieblich-Olsson entitled “Nagata compactification for algebraic spaces”. This theorem states that the category of all first order thickenings of algebraic spaces is equivalent to the category of pairs (X, A —> O_X) where X is an algebraic space and A –> O_X is a surjection of sheaves of rings on X_{etale} with quasi-coherent square zero kernel.

It seems to me that it is useful to think about the locally ringed small etale topos of an algebraic space in order to formulate and prove such results, even though it will not necessarily simplify, or shorten the proofs. Namely, in that language Theorem A.4.1 can be reformulated as follows:

• if X —> X’ is a first order thickening of algebraic spaces, then X_{etale} = X’_{etale}, i.e., the topos doesn’t change,
• define a locally ringed topos (Sh(C), A) to be an algebraic space if it is equivalent to (Sh(X_{etale}), O_X) for some algebraic space X, and
• if (Sh(C), A) is an algebraic space and A’ –> A is a surjection of rings with square zero quasi-coherent kernel then (Sh(C), A’) is an algebraic space.

The functoriality takes care of itself by the result discussed higher up.

This post is a response to Brian Conrad asking the following question: “How come the stacks project includes a base scheme S in the definition of algebraic spaces? Namely, we could think of an algebraic space over S as just an algebraic space over Spec(Z) equipped with a morphism to S.”

The short answer is that everywhere in the stacks project you can just think of X as an algebraic space over Z endowed with a morphism to S whenever you see the statement “let X be an algebraic space over S”. If you do this, then in many statements mentioning S is indeed completely superfluous.

A longer answer is that it is related to the setup in the stacks project, including our choices regarding set-theory.

When you see “Let S be a scheme” at the beginning of a lemma/proposition/theorem about algebraic spaces then this really means “Choose a partial universe of schemes to work with which contains S”. I can quantify exactly what I mean with “partial universe” and we prove using ZFC that partial universes exist containing any given set of schemes. (See Lemma Tag 000J.)

For the stacks project an algebraic space is a functor defined on the comma category C/S where C is this partial universe. So an algebraic space is a functor F : (C/S)^{opp} —> Sets. If you want to get an algebraic space over Spec(Z) you have to apply “Change of base scheme” (Section Tag 03I3 of the chapter “Algebraic Spaces”). Of course this is a completely trivial operation, but to get all the details right this is what you have to do.

A consequence is that an algebraic space over Spec(Z) doesn’t (a priori) have a value on all schemes, only on the schemes in the partial universe C. But you can apply “Change of big site” (Section Tag 03FO of the chapter “Algebraic Spaces”) to enlarge your partial universe to contain any given set of schemes.

A similar story goes for algebraic stacks. But… what we’ve done for algebraic stacks in Properties of Stacks, Section Tag 04XA is introduce the customary abuse of language which forgets about all of this set-theoretical nonsense. This language is also less precise.

We could (and maybe should) do the same thing for algebraic spaces. On the other hand, it mostly doesn’t hurt; it just looks a bit funny here and there.

[Post edited on May 30, 2012.]