Let U be a scheme. Let us define a family of d dimensions proper algebraic spaces over U to be a morphism X —> U from an algebraic space X to U which is flat, proper, locally of finite presentation, such that all geometric fibres are equidimensional of dimension d. Let Fam_d denote that full subcategory of the stack Spaces whose objects X/U are families of d dimensions proper algebraic spaces. Then as discussed in the preceding post we conclude that Fam_d is a stack over (Sch).

In this post I want to point out that for this to work out it is absolutely necessary that we work inside the category of algebraic spaces, and not with schemes. Let me start discussing the low dimensions.

[d = 0] It is a fact that any family X —> U of 0-dimensional proper algebraic spaces over a scheme U is automatically represented by a scheme. This follows from Proposition Tag 03XX.

[d = 1] Let X —> U be a family of 1-dimensional proper algebraic spaces over a scheme U. Then etale locally on U the space X is projective over U (in particular a scheme). But, even if you assume the fibres of X —> U are geometrically integral it is not the case that Zariski locally on U the space X is a scheme. An explicit example is the example of non-effective descent in Bosch-Lutkebohmert-Raynaud, Neron Models, Section 6.7 (since after all in Fam_1 we do have effective descent).

[d = 2] Here there are even examples of X —> U where all fibres are smooth projective surfaces, and U is a smooth curve, but the total space is an algebraic space and not a scheme. The examples comes from degenerating a general degree 513* surface in P^3 to a surface with a single node and doing a small resolution of the node on the total space (after performing a 2:1 base change). Moreover, there is no finite type, faithfully flat base change after which X becomes a scheme.

So you see that in order to do moduli of geometrically very interesting objects it is really convenient to work with algebraic spaces! In fact, if you don’t then you will not see all of the families that you want to see…

*Footnote: Degree 514 works also, and degree 21 too, and…

Consider the fibred category p : Spaces —> (Sch) where an object of Spaces over the scheme U is an algebraic space X over U. A morphism (f, g) : X/U \to Y/V is given by morphisms f : X —> Y and g : U —> V fitting into an obvious commutative diagram.

Theorem: This is a stack over (Sch)_{fppf}.

In essence the thing you have to prove here is that any descent data for spaces relative to an fppf covering of a scheme is effective. This follows immediately from the results discussed in this post, see Lemma Tag 04SK. You can find a detailed discussion in the chapter Examples of Stacks of the stacks project (in the stacks project we have only formulated this exact statement for the full subcategory of pairs X/U whose structure morphism X —> U is of finite type; this is due to our insistence to be honest about set theoretical issues).

Note how absurdly general this is! There are no assumptions on the morphisms X —> U at all. Now we can use this to show that suitable full subcategories of Spaces form stacks. For example, if we want to construct the stack parametrizing flat families of d-dimensional proper algebraic spaces, all we have to do is show that given an fppf covering {U_i —> U} of schemes and an algebraic space X —> U over U such that for each i the base change U_i \times_U X —> U_i is flat, proper with d-dimensional fibres, then also the morphism X —> U is flat, proper and has d-dimensional fibres. This is peanuts (compared to what goes into the theorem above).

Of course, to show that (under additional hypotheses on the families) we sometimes obtain an algebraic stack is quite a bit more work! For example you likely will have to add the hypothesis that X —> U is locally of finite presentation, which I intentionally omitted above, to make this work.

A site in the stacks project is different from what is called a site in SGA4. What we call a site is what is called a category endowed with a pretopology (see Exposee II, Definition 1.3 of SGA4). In other words a site is category C endowed with a set Cov of families of morphisms with fixed target called coverings such that

1. If V —> U is an isomorphism then {V —> U} is a covering,
2. if {U_i —> U} is a covering and {V_{ij} —> U_i} is a covering for each i, then {V_{ij} —> U} is a covering,
3. if {U_i \to U} is a covering and V —> U is a morphism of C then U_i \times_U V exists and {U_i \times_U V —> V} is a covering.

A sheaf on C is then a presheaf which satisfies the sheaf axiom for all the coverings. Note that in general there are many choices of Cov which give rise to the same category of sheaves. For example on (Sch), see previous post for notation, the etale coverings and the smooth coverings give rise to the same category of sheaves. For this reason you will sometimes hear people say that the etale and smooth topology are the same. But for us the etale site and the smooth site are different.

In this post I wanted to mention that working with sites as above is useful in that the types of coverings you allow can be used to express properties of the site which cannot be expressed in terms of the topology alone. For example, we can say that a property P of objects of C is local on the site if given a covering {U_i —> U} we have P(U) <=> P(U_i) for all i. Then it is clear that the property P(X) =”dim(X) < 17" is local on the etale site (Sch)_{etale} but not local on the smooth site (Sch)_{smooth}. Similarly for properties of morphisms, e.g., P(f)="f is locally quasi-finite" is local on the target on the etale site, but not local on the target on the smooth site. For a previous discussion of what it means for a property of morphisms to be "etale local on source and target", see this post.

Today I was able to add the result that the quotient stack [U/R] associated to a smooth groupoid in algebraic spaces is an algebraic stack. See Theorem Tag 04TK. Very satisfying!

There are several things that have to be done next:

1. Work on the chapter Examples of Stacks, and start a parallel chapter Examples of Algebraic Stacks, where we discuss in detail some very basic examples of algebraic stacks.
2. Add some foundational material on changing the base scheme and the underlying big site. This could then be used to define a 2-category of algebraic stacks which lumps all algebraic stacks together regardless of big site that was used to define them (but I’m not sure this would be useful, so I may not add this).
3. Write a Chapter on properties of algebraic stacks
4. Write a Chapter on morphisms of algebraic stacks
5. Write about separation axioms for morphisms of algebraic stacks
6. And so on and so forth.

Next week I will not have time to work on the stacks project, so you can start working on the topics above yourself!

This post is a bit of a rant.

One subgoal of the stacks project is to work through the beginnings of etale cohomology and algebraic stacks without making use of universes. Most of this is completely straightforward (and already done), and there is only one point at which you have to make an argument.

First I would like to point out that there is a completely well established (axiomatic) theory of sets, and that is ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Thus, virtually any mathematician who uses a set means the type of object that is described by the axioms of ZFC. Paradoxically it is the set theorists themselves who enjoy thinking about other kinds of sets. They like to add and substract axioms from ZFC and see what happens. It is probably for that reason that you cannot find a book that simply takes the axioms of ZFC and develops the theory of sets (if you do know such a book or lecture notes, please email me or leave a comment). So whenever you take up a book on set theory to learn something about the sets you work with every day, you have to be very careful to see whether the author has added some bizarre additional hypotheses to the theory, or works with a different axiom system. To me it seems a bit of a crime that some of the undergraduate level books do not use ZFC.

Of course, some of the most interesting results in set theory (that I know) are those having to do with consistency, etc. As an example I want to mention the result that if ZFC is consistent, then you cannot prove the existence of a strongly inaccessible cardinal in ZFC.

What is a universe? Roughly, a universe is a set X such that all the axioms of ZFC hold for the elements of X. It turns out that the existence of a universe is equivalent to the existence of a strongly inaccessible cardinal, hence cannot be proved inside ZFC (and neither can the nonexistence, actually). Thus I argue that most mathematicians use a set theory which does not have universes.

Grothendieck added the existence of universes U as an axiom so he could say “let (Sch) = the category of schemes which are elements of U”. This is somewhat convenient. For example it means that if I is an element of U and X_i is an element of U which is a scheme for all i, then \coprod X_i (suitably constructed) is an element of U. On the other hand (Sch) is not an element of U and considering it takes you outside of U. Moreover, since U is just some set, (Sch) is just some set of schemes, and there are schemes not in U. Furthermore, the topos of sheaves on (Sch) is of course a proper class and not a set at all. Finally, changing the universe gives you a different topos.

The axioms of ZFC provide many techniques for constructing large sets. How close can we come to constructing a universe U inside of ZFC? I don’t know; I’ve looked around, but I haven’t found somebody addressing this directly (likely because for a set theorist this is utterly trivial). But here is what is true; I’ll formulate this in terms of (Sch) = the category of schemes which are elements of U, because if you are reading this then you probably do not care about “large sets”. Given any set of schemes (Sch_0) you can construct a U such that

1. (Sch_0) is contained in (sch),
2. (Sch) has fibre products, and fibre products agree with fibre products in the category of all schemes,
3. more generally you can construct U such that limits and colimits over at most countable diagrams in (Sch) exist whenever they exist in the category of all schemes and are the same,
4. you can make (Sch) be closed under immersions, morphisms of finite type, morphisms which are locally of finite type such that the inverse image of an affine can be covered by countably many affines, and
5. if X is in (Sch) and Y is a scheme whose “size” is at most the 2^(size of X), then Y is isomorphic to an element of (Sch).

The key is to construct U such that (3) and (5) hold; the size of a scheme X is defined in terms of the cardinality of the set of sections of O_X over affine opens and the cardinality of the set of affine opens. But now you cannot _also_ require that (Sch) is closed under disjoint unions of objects of (Sch) indexed by elements I of U, and, if you think about it for a bit, you will see that this is the only difference from the case of a universe. So, although it is clear that properties 1 — 4 imply that in many cases the disjoint union does exist in (Sch), it is just not always true!

This is the approach we chose in the stacks project. Another one might have been to construct U’s such that the disjoint unions always exist, but then you need to weaken condition (5) by quite a bit; I’m not exactly sure how much.

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.