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.]

I think that what I was saying in this post also works for Deligne-Mumford stacks. If so, then this is kind of fun, since it make us think about how the topos distinguishes between group actions and Galois actions. It doesn’t!

Let R, C denote the real and complex numbers. Consider the scheme Spec(R), and the Deligne-Mumford stack [Spec(C)/{+1, -1}] where the group acts trivially. The these have isomorphic small etale topoi, since in both cases an etale sheaf is just given by a set with an action of the group {+1, -1}.

How can we distinguish these? Well, we should look at the ringed small etale topoi, and the structure sheaves are different. Namely, in both cases the structure sheaf corresponds to C with an action of {+1, -1} and in the first case the element -1 acts via complex conjugation and in the second case it acts trivially.

By the way, is there an example of two non isomorphic varieties over C whose small etale topoi are isomorphic? Maybe not since you can recover Galois groups from the topos and the absolute Galois group of a function field determines the function field up to isomorphism IIRC. In other words, perhaps you can sometimes get away with only looking at the topos.

[Edit 5/19/2010: Bhargav mentioned that this cannot be true since the elliptic curves y^2 = x(x-1)(x-e) and y^2 = x(x-1)(x-pi) have isomorphic small etale topoi. Here e = 2.71828… and pi = 3.14159… I googled the function field thing and found a note by Florian Pop where the result is stated over the algebraic closure of Q or the algebraic closure of a finite field (which is what Bhargav was saying), but only for function fields of transcendence degree > 1. Anyway, the above suggestion that sometimes the topos is enough is true. In fact is related to Grothendieck’s anabelian conjecture — for all I know Grothendieck formulated his conjecture in terms of topoi. Of course there is an enormous literature on that subject, and all of those results can be thought of as results towards the “etale topos determines variety” thing if you like.]

As you may have noticed the numbering of lemmas, propositions, theorems, sections, chapters etc is not constant over time in the stacks project. Hence if you want to reference a result in the stacks project then you cannot refer to a number in the text, because what works now is almost surely not going to work in the future. Still, I want the stacks project to have stable references.

The reason for not requiring a stable numbering is that I want to be free to edit the latex files in the project, move results around, reorganize chapters, add chapters, and even move results to different chapters. Maybe later I want to change the format altogether and dispense with having chapters, and so on.

We cannot use the latex labels to reference results for various reasons: I want the latex labels to be “human readable”, and we may want to edit them. A random example is the label lemma-stalk-exact which refers to a lemma in etale-cohomology.tex on exactness of the stalk functor. At some point we may want to rename this lemma-stalk-functor-exact. Or perhaps future versions of the stacks project will be written in XML and labels and references work entirely differently.

To provide for stable references Cathy O’Neil came up with the tags system. I implemented it about a year ago, and some people have started using it already. The idea is very simple. Each mathematical result gets a tag, which is just a four character string made out of capitals and digits. Once a tag has been assigned to a mathematical result or definition, it will always point to that exact mathematical result or definition.

Let’s see how this works in practice. For example the lemma mentioned above has tag 03PT. Then if you want to refer to this lemma you refer to the tag instead. You would put something like

\cite[Lemma 03PT]{stacks-project}

in your latex file. How can the reader of your manuscript, or blog post, or email figure out what result you are referring too? They would search for a tag on this page. Entering the tag into the search bar on that page returns a page telling you exactly the location of the corresponding mathematical result in the current version of the stacks project.

Conversely, how do you find the tag of a result in the stacks project? As you may have noticed the lemmas, proposition, theorems in the online version of the stacks project are hyperlinks. If you click on them you will be brought to a page telling you the tag of the result you clicked on (or just hover your mouse over the hyperlink to discover the tag).

There is more we can do with this system. For example the pdf files contain hyperanchors whose anchor name is the tag. Here are two small tests. Currently whether this works still depends on your browser and pdf viewer setup.

No matter. The basic functionality of the tags system, which is to provide stable references, works fine and has been working fine for a while now. You could call it a stable feature of the stacks project!

OK, so currently I am writing a bit of material on etale sites, points of etale sites and (upcoming) morphisms of etale sites. The reason for doing this is that I want to verify that my claim in the preceding post is correct.

Also, I finished rewriting the material on formally unramified, etale and smooth morphisms of schemes. I introduced the notion of a thickening of schemes (which is a closed immersion whose ideal is locally nilpotent), and a first order thickening. Then I introduced the notion of a universal first order thickening of a scheme formally unramified over another scheme. Using this a formally etale morphism is one which is formally unramified such that the universal first order thickening is trivial. I also rewrote some of the material on formally smooth morphisms, splitting the main result into separate lemmas (and slightly generalizing the result). I think this has substantially improved the exposition.

I want to use the same ideas to discuss formally unramified, etale and smooth morphisms of algebraic spaces, as well as modules of differentials and conormal sheaves for morphisms of algebraic spaces (see my previous post). In order to do this it is going to be really helpful to have the claim of the preceding post, so I am trying to finish this up first.

Meanwhile, work on the ideas mentioned in this post has been delayed a bit, but I hope to return to it soon. The reason for the delay is that I decided that the construction of the (badly named) f_!X mentioned there should go in the chapter “More on Morphisms of Spaces”. As a result I started thinking about some of the material that should go into this chapter, and so on and so forth.

PS: I am in the market for a good symbol to use as a replacement for f_!X…

Given an algebraic space X we obtain a ringed topos (Sh(X_{etale}), O_X) of sheaves on the small etale site of X endowed with the structure sheaf. This is a locally ringed topos (as in SGA4, Expose IV, Exercise 13.9). Moreover, a morphism X —> Y of algebraic spaces induces a morphism of ringed topoi in the same direction. In fact it is a morphism of locally ringed topoi (see reference above for definition). In fact I think that

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

is a bijection, i.e., the category of algebraic spaces is a full subcategory of the category of locally ringed topoi. This is sooooo cool!

Allow me to get excited even if you already knew this ages ago. Namely, it means we can describe algebraic spaces as certain locally ringed topoi. This could be helpful for example when we think about thickenings of algebraic spaces: it will allow us to use the same underlying topos of sheaves and just change the structure sheaf (as we do in the case of schemes).

But it goes farther than that. It also means that we can forget about an algebraic space as just a functor on the category of schemes, and consider it as a geometric object it in its own right. Moreover, one of the things that is currently bogging down the stacks project a bit is writing the interface between schemes theory and the theory of algebraic spaces, where in the schemes language we often use points and locality and on the side of algebraic spaces we constantly worry about all scheme valued points of X. It is conceivable that this can be clarified a bit by using the idea above.

Of course we are not going to rewrite the whole thing from scratch, but I hope to add the observation above to the stacks project and then use it whenever I can!