The following results now have a complete proof in the stacks project:

1. If F = U/R where R is an equivalence relation on U such that R —> U are flat and locally of finite presentation then F is an algebraic space.
2. If F is a sheaf such that there exists U —> F which is representable by algebraic spaces, surjective, flat, and locally of finite presentation, then F is an algebraic space.

See Theorem Tag 04S6. This is the culmination of a lot of hard work and I am very happy that it is finally done!

The original reason for adding this to the stacks project was that I wanted to start writing about presentations of algebraic stacks. This immediately leads to the following two questions:

1. Suppose that X is an algebraic stack with trivial inertia. Why is X an algebraic space?
2. Suppose that (U, R, s, t, c) is a groupoid in algebraic spaces with s, t smooth. Why is the associated stack in groupoids [U/R] on (Sch/S)_{fppf} is an algebraic stack?

I would like to stress that both questions are nontrivial. Let me discuss why.

Part (a) is a bit easier if X is a Deligne-Mumford stack, see Lemma Tag 045H although it already uses a bootstrap argument for the diagonal. In the general case, besides bootstrapping the diagonal, you have to show that starting with a smooth equivalence relation you can get an etale equivalence relation with the same quotient sheaf. You can try to prove this by carefully slicing, which probably works, although it isn’t that easy (one problem is that you don’t know a priori which points to slice at even if everything is of finite type over a Noetherian base). Our approach is to see (a) as a direct consequence of (1) since after all a smooth morphism is flat and locally of finite presentation. Thus our proof of (a) completely takes one outside the realm of smooth presentations.

Part (b), besides a lot of general nonsense which already is documented in the stacks project, requires proving that the Isom sheaves of [U/R] are representable by algebraic spaces. This is relatively straightforward if you take the associated stack in the etale topology: you have to show a sheaf over a base scheme S which etale locally on S becomes an algebraic space is an algebraic space. But in the stacks project we use the fppf topology and it is not so straightforward: you have to show a sheaf over S which fppf locally on S becomes an algebraic space is an algebraic space. Although I haven’t written out all the details, I think this is a simple consequence of (2) above.

In the future we will need to discuss another theorem similar to the results above. Namely, Artin’s result that if (U, R, s, t, c) is a groupoid in algebraic spaces and s, t are flat and locally of finite presentation then the associated stack [U/R] is algebraic. The results above tell us that the only thing we need to do is show there exists a scheme and a surjective smooth morphism from that scheme onto [U/R], i.e., all the other properties have already been taken care of. To do this we will use Artin’s trick of looking at complete intersections in fibres of U —> [U/R]. But that will be another day!

If you use gvim and xdvi to edit and view your latex and corresponding dvi files then you can set it up so that you can jump effortlessly back and forth between the location in the dvi viewer and the location in the editor. This is explained on this webpage, but I wanted to rephrase it here.

Firstly, in order for inverse search to work you have to use
latex -src filename.tex
when compiling your latex files. The make command in the stacks project does this automatically for you. The default is that control + left button in the xdvi window brings you to the corresponding place in the editor. If this isn’t working you need to check if a line such as
xdvi.editor: gvim --servername xdvi --remote +%l %f
occurs in the file .xdvirc (which is probably somewhere in your home directory). If not then try to see if your xdvi lets you set it (in the options/preferences), else you can edit .xdvirc directly.

Secondly, to use forward search place the following line
map <F3> :execute "!xdvi -sourceposition " . line(".") . expand("%") . " " . expand("%:r") . ".dvi"<CR><CR>
in your .gvimrc file (in your home directory). Having done this then when you press F3 in gvim you will jump to the corresponding place in xdvi.

There is still a minor issue with this if you are editing multiple files concurrently (in tabs of gvim). Namely, in that case case a new instance of xdvi pops up if you use F3 after you switch to editing another file in gvim. (If you switch files using ctrl-click in the corresponding xdvi window this doesn’t happen.) This is a small price to pay for the convenience.

Finally! The method of proof was discussed in this post. Actually, the procedure for finding the sub groupoid is better than what I wrote in that blog post, and follows more closely the material in Keel-Mori.

We will soon add another lemma with more hypotheses where the output is a scheme, and not an algebraic space (namely when s, t are separated, locally of finite presentation, and flat). This avoids using Hilbert schemes at the cost of leaving the category of schemes temporarily.

This is a splendid example of an application of the theory of algebraic spaces: Namely, you define some functor, show it is an algebraic space, and then a posteriori you prove it is a scheme by some additional arguments.

What does it mean for a property P of morphisms of schemes to be etale local on the source and target? In Deligne-Mumford they use the following definition (page 100): for any family of commutative squares

where {h_i : X_i —> X}, {g_i : Y_i —> Y} are etale coverings we have P(f) <=> P(f_i) for all i. And of course this is exactly the minimum needed to be able to define what it means for a morphism of Deligne-Mumford stacks to have a certain property…

However, here are some very confusing points

1. the condition does NOT imply that P is preserved under post-composing with open immersions,
2. if P is etale local on the source and P is etale local on the target, then P does not necessarily satisfy Deligne and Mumford’s condition.

Now it turns out that this NEVER leads to any confusion, since if P is preserved under post-composing with open immersions, which is a condition always satisfied in practice, then all three conceivable notions agree. Moreover, in that case the property is preserved under post-composing with etale morphisms. To see all the gory details, see the section entitled “Properties of morphisms local on source-and-target” in Descent.pdf.

PS: This may be good material to read if you are having trouble falling asleep.

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.