This is a continuation of previous post on flattening stratifications. The experts reading this blog could probably tell that I hadn’t really understood what is going on at all. I still haven’t mastered the subject but I think I know a little bit more now.

Let f : X —> S be a morphism of schemes. Consider the functor F : (Sch)^{opp} —> (Sets) which to a scheme T associates the set of morphisms T —> S such that the base change X_T is flat over T. Clearly the map F —> S is a monomorphism. We propose to introduce the following

Definition: We say the flattening stratification of f exists if F is an algebraic space.

What I added to the stacks project last Friday is the following: Assume S is the spectrum of a Noetherian complete local ring and f is of finite type. Then there exists a biggest closed subscheme Z of S such that X_Z —> Z is flat at all the points of the closed fibre. Moreover, Z satisfies a universal property which is formulated in terms of local morphisms of local schemes and flatness at points of the special fibre. If in addition X —> S is closed, then it follows that X_Z —> Z is flat as the set of points where X_Z —> Z is flat is an open set.

Assume S Noetherian and f of finite type and proper. In terms of Artin’s axioms for F the result in the previous paragraph takes care of the existence of a formal versal deformation. I think there is a straightforward little argument which takes care of openness of versality (but I did not write this out completely). Since f is of finite presentation, it follows that F is of finite presentation by the usual arguments on limits and flatness. Relative representability is OK too. Hence, if S is excellent then F is an algebraic space by Artin’s theorem. But of course we can descend X —> S to a situation of finite type over Z and hence we get the result in general (with same hypotheses). In fact, using limit arguments we may be able to prove the same thing when S is arbitrary and f proper and of finite presentation.

Still, my answer to Jason’s question here was a bit premature. Some of the above may work exactly as stated in the generality of Jason’s question. But I was trying to prove flattening stratifications exist without using Artin’s theorem. In particular, it should be possible to avoid using general N\’eron desingularization.

The reason I started looking at flattening stratifications was to construct Quot and Hilbert schemes/spaces/stacks. And the reason to discuss those was that Artin’s trick uses Hilbert spaces. However, it only uses the Hilbert space parametrizing closed subschemes of length n on a space. Of course I could take the easy way out and just use one of the explicit constructions of Hilb^n. But once I started looking at the problem of constructing flattening stratifications (which is related to descent of flat modules) I just couldn’t stop myself.

Consider the functor F which to a scheme T associates the set of closed subschemes Z of T \times A^1 such that the projection Z —> T is an open immersion. In other words F is the functor of flat families of closed sub schemes of degree <= 1 on A^1, whence the title of this post. We note that F is a sheaf for the fppf topology.

What is fun about this functor  is that it is a natural candidate for a 1-point compactification of A^1, as the following discussion shows.

Namely, consider for each integer n >= 1 the scheme P_n which is P^1 but crimped at infinity to order n. What I mean is this: If y = x^{-1} denotes the usual coordinate on the standard affine of P^1 which contains infinity, then the local ring of P_n at infinity is the Z-algebra generated by y^n, y^{n + 1}, y^{n + 2}, … Note that there are morphisms

P_1 —> P_2 —> P_3 —> …

and that for each n there is a natural map P_n —> F compatible with the transition maps of the system. Hence we obtain a transformation of sheaves

colim P_n —> F.

It seems likely that this map is an isomorphism (we take the colimit in the category of fppf sheaves), but I did not write out all the details.

Does anybody have a reference? What about the same thing for A^2?

[Edit 18:57 July 31 2010: Original definition of F omitted the condition that the fibres are a point or empty. Replaced by the open immersion condition. This make sense because a morphism Z —> T which is flat and locally of finite presentation whose fibres Z_t are either empty or Z_t = t (scheme theoretically) is an open immersion.]

Let f : X —> S be a morphism of schemes. A flattening stratification for f is a disjoint union decomposition of S into locally closed subschemes S_i such that for a morphism of schemes T —> S with T connected we have that T times_S X —> T is flat if and only if T —> S factors through S_i for some i.

There is also the notion of a flattening stratification for F where F is a quasi-coherent sheaf on X. In the case that X and S are affine this leads to the notion of a flattening stratification of Spec(A) for a module M over a ring B relative to a ring map A —> B.

Flattening stratifications do not always exist, but here are some examples where it does:

1. If A = B and M is a finitely presented A-module, then the flattening stratification corresponds to the stratification of A given by the fitting ideals of M.
2. If (A, m) is a complete local Noetherian ring, A—> B arbitrary, and M is m-adically complete, then the closed stratum of the flattening stratification for M in Spec(A) exists. (Intentionally vague statement; haven’t worked it out precisely.)

What you should keep in mind is that the flattening stratification does exist whenever the module is finite or formal locally in general.

Here is an example where the flattening stratification does not exist. Namely, take the ring map C[x, y] —> C[s, 1/(s + 1)] given by x |—> s – s^3 and y |—> 1 – s^2. Let f : X —> S be the associated morphism of affine schemes. Note that the image of f is contained in the curve D : x^2 – y^2 + y^3 = 0. Note that D has an ordinary double point at (0, 0). The problem is the stratum which contains the point (0, 0) of S. Namely, working infinitesimally around (0, 0) this is going to give you one of the two branches of the curve D at (0, 0), namely the one with slope 1. But globally, there is no locally closed sub scheme which gives you just that one branch!

The example above is not so bad yet, because there is a stratification of S by monomorpisms which does the job. Here is a simpler, somehow worse example. Namely, let S = Spec(C[x, y]) = A^2 be affine two space. Let X = A^2 ∪ G_m be the disjoint union of a copy of S and a line minus a point. The map f : X —> S is the identity on A^2 and the inclusion of G_m into the line y = 0 with the origin the “missing” point of G_m. Then looking infinitesimally around the origin in A^2 we are led to think that the stratum containing 0 should have complete local ring equal to C[[x, y]]. But looking at the overall picture we see that f(G_m) has to be removed, i.e., we have to take V(y) – V(x, y) out of Spec(C[[x, y]]). This shows that a flattening stratification cannot exist in this case (not even by monomorphisms).

Of course, somehow the main result on flattening stratifications is that it exists if f is a projective morphism and S is Noetherian. You can prove it by applying result 1 above to the direct images of high twists of the structure sheaf of X. The examples above show that it is unlikely that there exists a proof of this fact which uses the flattening stratifications for affine morphisms, as these do not always exist.

Certain results have a variant for generic points, and a variant which works over a dense open. As an example let’s discuss “generically finite morphisms” of schemes.

The first variant is Lemma Tag 02NW: If f : X —> Y is of finite type and quasi-separated, η is a generic point of an irreducible component of Y with f^{-1}(η) finite, then there exists an affine open V of Y containing η such that f^{-1}(V) —> V is finite.

The second variant is Lemma Tag 03I1: If f : X —> Y is a quasi-finite morphism, then there exists a dense open V of Y such that f^{-1}(V) —> V is finite.

Comments: (a) In the second variant it isn’t necessarily the case that every generic point of every irreducible component of Y is contained in the open V, although this would follow from the first variant if we assumed f quasi-separated. (b) The proof of the first variant in the stacks project is basically elementary; the proof of the second variant currently uses (a technical version of) Zariski’s main theorem.

The point I am trying to make (badly) is that you can often get around making any separation assumptions by trying to prove a variant “over a dense open”. Maybe the archetype is the following result (Lemma Tag 03J1): Every quasi-compact scheme has a dense open subscheme which is separated.

I just added some generic flatness results to the stacks project (only for morphisms of schemes so far). There are two interesting features of the presentation in the stacks project:

1. Assuming only the morphism is of finite type the conclusion is that the morphism is flat and finite presentation over a dense open of the base, and
2. it suffices to assume the base is reduced.

Using these results we can discuss “flattening stratifications”. But I want to discuss this in a maximally general setting. Reader beware!

Let f : X —> S be a morphism of schemes of finite type. I want to find a stratification of S by reduced locally closed subschemes S_i such that X_{S_i} —> S_i is flat. If f is of finite presentation we can reduce to S Noetherian and there is (locally) a finite stratification that does the job; so what I am interested in here is the case where S is not Noetherian.

Step 0: Find the open stratum. Just replace S by its reduction S_{red} and let S_0 be the open dense U ⊂ S you get from generic flatness. Step 1: Let S_1 be the dense open of (S – S_0)_{red} you get from generic flatness. Step 2: Let S_2 be the dense open of (S – S_0 – S_1)_{red} you get from generic flatness. Etc.

Now we get S_0, S_1, … but it may not be the case that S = \bigcup S_i. For example the last post contains an example. So then you start all over again. Namely, note that the complement of S_0 ∪ S_1 ∪ S_2 ∪… is closed in S hence a scheme. So we restrict our family to this closed subset and we continue. Doesn’t it feel like we can just continue forever using transfinite induction? And moreover, the process does really have to stop as S has an underlying topological space which has a finite cofinality. Thus we do get our desired stratification of S.

But this is madness! Surely there are at most countably many strata…!?!

Here is a fun example. Take U = Spec(k[x_0, x_1, x_2, …]) and let G_m act by t(x_0, x_1, x_2, …) = (tx_0, t^px_1, t^{p^2}x_2, …) where p is a prime number. Let X = [U/G_m]. This is an algebraic stack. There is a stratification of X by strata

• X_0 is where x_0 is not zero,
• X_1 is where x_0 is zero but x_1 is not zero,
• X_2 is where x_0, x_1 are zero, but x_2 is not zero,
• and so on…
• X_{infty} is where all the x_i are zero

Each stratum is a gerb over a scheme with group \mu_{p^i} for X_i and G_m for X_{infty}. The strata are reduced locally closed substacks. There is no coarser stratification with the same properties.

So clearly, in order to prove a very general result as in the title of this post then we need to allow infinite stratifications…

Since the last update about a week ago we have started adding some very basic material on algebraic stacks. It should be possible to lay out the basics of theory of algebraic stacks in the rest of the summer. Mainly I mean material on types of morphisms of algebraic stacks (how to define them, how do you check for a property, how are they related, etc), perhaps a little bit on quasi-coherent sheaves on algebraic stacks, more examples of algebraic stacks, etc. It will probably turn out that during this process the existing material on algebraic spaces will have to be expanded upon and improved.

What are some more exciting topics that I hope to get to in the not too distant future? Here are some possibilities:

1. Artin’s axioms, starting with a chapter on Schlessinger’s paper.
2. Olsson’s paper on the equivalence of the different definitions on proper.
3. Translating some of the more interesting results on algebraic spaces (regarding points, decent spaces, etc) into results on algebraic stacks.
4. A proof of Artin’s trick for algebraic stacks to prove that [U/R] is an algebraic stack if s, t : R —> U are flat and locally of finite presentation.
5. Diagonal unramified <=> DM
6. Prove the correct version of the Keel-Mori theorem in our setting.
7. Discuss the lisse-etale site (I may somehow want to avoid this — perhaps you can always avoid working with this beast?)
8. Treatment of simplicial schemes (as an example of simplicial objects) and relation to algebraic stacks. Formulate possible extension to higher algebraic stacks (e.g., hands on version of algebraic 2-stacks with example moduli of stacky curves)
9. Gabber’s lemma and more on finding nice maps from schemes to algebraic stacks.
10. Tricks with limits. Quasi-coherent sheaves are directed colimits of finite type quasi-coherent sheaves if X is…
11. Gerbs and relation to H^2.
12. Stacky curves. What are 1-dimensional stacks like?
13. General Neron desingularization (needed for 1).

Of course it will take a long time to add all of these topics. But it probably is a good idea to take one or two of these topics and keep them in mind while developing the general theory, so as to have a good reason for building theory. Moreover, many of these topics require additional material on schemes and algebraic spaces. For example, 4 requires a bit about Hilbert spaces (only relative dimension 0 however).

Finally two questions:

1. A while back David Rydh suggested we give a name to algebraic stacks with locally quasi-finite diagonal. Should we? If so any suggestions?
2. And what to call an algebraic stack with finite diagonal? (Maybe these should have been called separated and the ones with only proper diagonal something else. Too late to change now since it has been used like this for a while now in the literature.)

Maybe we could call an algebraic stack with locally quasi-finite diagonal “space-like”? And then the algebraic stacks with finite diagonal are space-like separated algebraic stacks. Hmm… not sure.

Today I learned a new basic fact on morphisms of schemes, namely the result mentioned in the title of this post. I started wondering about this question as I was thinking about separation conditions for algebraic stacks. Namely: it appears that the standard definition for a separated algebraic stack is one whose diagonal is proper, and I was wondering if we could get away with just requiring the diagonal to be universally closed and separated. It turns out we can due to the result of the title. After trying to think about it for a bit I decided to look for it on the web, and I quickly found a mathoverflow question asking exactly whether universally closed implies quasi-compact for morphisms of schemes, as well as the proof provided by Bjorn Poonen!

Note that all posts on mathoverflow are under CC-BY-SA, which is (unfortunately) not compatible with GFDL which is the license that the stacks project is under. Moreover, they ask to link back to their site, see here; and actually I think they are really stretching the meaning of the license since I think no linking should be required (IANAL). Anyway, I asked Bjorn if he agreed to relicense his material, and he said “Yes, that’s fine”. This means I need not link to their site if I do not want to (I did anyway).

It seems that CC-BY-SA is winning over GFDL in some respects, so I may switch the stacks project over to it in the future (there are still not too many authors so it shouldn’t be difficult to do). If I do this then I imagine I am allowed to take any latex code submitted to Mathoverflow by mathematicians and add it to the stacks project as long as I make sure to attribute it to the author of the comment. But for the moment, contacting the author of the comment and asking for permission directly makes more sense. Of course this is a bit difficult to do since it often isn’t clear who the author is especially for some very prolific contributors on Mathoverflow such as BCnrd…

Anyway, on a completely different note: I finally figured out how to set up the Makefile so that I can run the latex compiles in parallel. You will know why this is a problem if you’ve ever tried to write such a Makefile. If not it probably makes sense to stop reading this now. It is really quite simple (and I’m sure it is an often used trick). Instead of running latex stem.tex you execute a bash script which

1. creates a temporary directory using mktemp
2. copies all aux files and temp.toc to the temporary directory
3. creates symbolic links in the temporary directory to stem.tex and stem.bbl (and maybe some style files, etc)
4. changes directory into the temporary directory
5. runs latex stem.tex
6. moves stem.dvi, stem.aux, stem.toc back to the main directory
7. removes temporary directory

Anyway, using this I was able to cut overall compilation time in half. On the server at work the times were

• make dvis -j1 takes 1m42s
• make dvis -j2 takes 50s
• make dvis -j3 takes 33s
• make dvis -j4 takes 25s

Not too shabby.

Since the last update we have worked on examples of stacks, stack of torsors, quotients stacks [X/G], Picard stack, epimorphisms of rings, change of partial universe for algebraic spaces, Gabriel-Zisman localization, pushforward of stacks, pullback of stacks, change of partial universe for algebraic stacks, change of base scheme for algebraic stack, and finally we started working on a new chapter entitled Properties of Algebraic Stacks.

As you can see from the list, we worked through a lot of very formal material. Some of this is a bit rough as written, although almost all of it is “obviously correct”. This is OK as at least some of it is just meant to explain set theoretical issues.

But now we’ve finally caught up with this material and we can start working on algebraic stacks! Moreover, I have decided to introduce the customary abuse of language in the first section of the new chapter linked to above. The idea is that any confusion which is caused by this abuse of language should be explained by pointing to a lemma in Algebraic Stacks or earlier. For example, we will say that an algebraic stack is an algebraic space if it is representable by an algebraic space (which is equivalent to have trivial inertia stack, see Proposition Tag 04SZ). Then suppose X, Y, Z, W are algebraic stacks and we are given a diagram

and X, Y, Z, W “are” algebraic spaces. What does it mean to say “the diagram is commutative” or “the diagram is cartesian”? Well, it could either mean that the diagram is a commutative, resp. cartesian in the category of algebraic spaces, or that the diagram is 2-commutative, resp. a 2-fibre product diagram in the 2-category of algebraic stacks. The abuse of language is not confusing in this case since these conditions agree. More precisely: the diagram is 2-commutative if and only if the corresponding diagram of sheaves of isomorphism classes of objects of fibre categories is commutative, and the diagram is a 2-fibre product if and only if the corresponding diagram of sheaves of isomorphism classes of objects of fibre categories is cartesian.

Well actually 2006 pages at this very moment. Also

• 193614 lines of tex,
• 6267 tags, and
• 1326 commits since I started using git on May 20, 2008.

Enjoy!