# Pull requests

OK, just a quick update. Somehow I have been kind of swamped with work this semester and I haven’t gotten to do a lot of writing on the stacks project recently. I intend to get back to it at the end of the semester and perhaps write about semi-stable reduction.

On the other hand, others have continued to contribute: We have had a new section added on getting triangulated categories out of differential graded categories, written by John Yu and Yifei Zhao. Also, some of the readers of the stacks project have chimed in by leaving comments and found mistakes as well as typos. Finally, there has been a few pull requests which is particularly nice because all I have to do to incorporate those is click a couple of buttons.

# Chow stability of curves

This post is part of my effort to understand what is currently the cheapest way to prove semi-stable reduction of curves. One possible approach, championed by Gieseker, is to prove it using GIT stability. To explain the strategy, fix a genus g > 1. Fix an integer d >> g. Set P(t) = dt + 1 – g ∈ Z[t]. Suppose we can prove

1. the Hilbert point of any smooth projective curve C of genus g embedded by the full linear system of a degree d line bundle is GIT stable, and
2. every closed subscheme of P^{d – g} with Hilbert polynomial P whose Hilbert point is GIT semi-stable, is a nodal curve.

Then semi-stable reduction pops out. Namely, it is a basic fact of GIT that a specialization of a stable point can be replaced, modulo the group action, to a specialization whose limit is semi-stable (one may in addition assume the limit point has a closed orbit in the semi-stable points). It always seemed to me that proving 1 is more difficult than proving 2, so we’ll focus on that in this post.

I’d like to observe here that if one is only interested in proving semi-stable reduction, then one can play around with the quantifiers above in a fun way. For example, to use the argument above it suffices to prove for a given curve C that there exists some very ample line bundle of large degree such that the corresponding Hilbert point is stable. Next, it suffices to prove this after specializing the curve, hence we may assume that the ground field is finite. In that case there are only finitely many flags to consider in the Hilbert-Mumford criterion, hence this makes some of the uniformity questions that arise trivial. I find this a psychologically useful thing to observe, but I am not sure it actually make things easier to prove.

In a paper by Ross and Thomas it is mentioned that one can get a geometric proof of Chow stability of curves. In particular they claim that one can avoid doing some of the combinatorics that often arise when proving stability. I think I have now understood a simple way to do this (which is, if not equal, very close to what Ross and Thomas say in that article), which I am going to explain here. I haven’t worked out all the details, so caveat emptor.

Let C be a smooth projective curve sitting in P^n. We are going to check the Hilbert-Mumford criterion for stability of its Chow point. To do this we consider closed subschemes X of P^n x A^1, flat over A^1, and invariant under the action of G_m acting by a 1-parameter subgroup of SL_{n + 1} on P^n and by the standard action on A^1. We normalize weights of the action of G_m such that the coordinate t on A^1 has weight 1 (this is, I think, the opposite of the normalization in the paper of Ross-Thomas). Let w_l be the weight of the G_m-action on H^0(X, O_X(l))/tH^0(X, O_X(l)). It is shown in Mumford, Stability of projective varieties, that

w_l = a_0 l^2 + a_1 l + a_2

for l >> 0 and some rational numbers a_i. Mumford also shows that a_0 (up to a factor or 2, 1, or 1/2) is the weight of G_m on the ample line bundle of the Chow variety corresponding to the Chow point of the G_m fixed point corresponding to the special fibre X_0. Thus the Hilbert-Mumford criterion tells us we have to show

for every X as above we have a_0 < 0

Now, we are going to apply Lemma 5.1 of the paper of Ross-Thomas. It says that we can compute a_0 on any modification of X. I claim that with some simple algebraic geometry we can find (after possibly taking an nth root of t and replacing G_m accordingly) a G_m-equivariant map

f : Y —> X

where Y is flat over A^1, where f induces an isomorphism of generic fibres, and where the special fibre Y_0 is a nodal curve. Because the generic fibre is constant with value C, this implies that Y_0 is equal to C union a bunch of chains of P^1′s. Moreover, the G_m-action on these chains is easy to describe: for t -> 0 it pushes points towards the component C. The pullback of O(1) to Y_0 is an invertible sheaf L, equipped with a G_m-action, with the following properties

1. L is globally generated, and
2. the weight on H^0(Y_0, L) – H^1(Y, L) is nonpositive.

It takes a bit of work to prove this. Now I claim that this, together with the description of how G_m acts on the chains of P^1′s I mentioned above, implies that the weights on H^0(Y_0, L^l) grows asymptotically as (neg) l^2 unless the scheme X is the trivial degeneration of C. To prove this involves a little bit of combinatorics of the chains of P^1′s but not much. (I’ve gone over this computation twice, and both times it came out correctly. If you’re interested, stop by my office and I’ll explain it to you.)

Although beautiful, this method of proving semi-stable reduction for curves takes too many twists and turns to make it suitable for inclusion into the Stacks project, at least for the moment…

PS: I strongly encourage anybody trying to understand Kempf’s result on uniqueness of destabilizing flags, to look at Sections 2 and 3 of Burt Totaro’s paper on tensor products in p-adic Hodge theory. It is a marvelous piece of mathematical exposition if I ever saw one.

A graded (preadditive) category is a preadditive category such that the hom groups have a Z-grading compatible with composition. In Heller’s paper of 1958 he talks about direct sums in graded categories: one requires the projections and the coprojections to be homogeneous (I would also require them to have degree 0 but Heller doesn’t require this).

Today seems to be the day for silly questions, because I was wondering if a graded category which has direct sums as an additive category (i.e., ignoring the grading) necessarily has direct sums as a graded category.

Today, I was on and off wondering about idempotents in Z-graded associative algebras with a unit (which is assumed homogeneous). In my googling of this, I have found the terminology graded idempotents which refers to idempotents which are homogeneous of degree 1. This suggests that there exist others. And indeed, it is easy to make examples of non-homogeneous idempotents by conjugation with units. But we can ask for more.

1. Is there an example of an idempotent which is not conjugate to a graded idempotent?
2. Is there an example of a Z-graded associative algebra with a nontrivial idempotent but no nontrivial graded idempotents?

Hmm…?

Some more searching and google finally turned up the paper Idempotents in ring extensions by Kanwar, Leroy, and Matczuk which provides the answer to 1. There’s probably tons of papers that make this observation. Namely, suppose that R is a (commutative) domain such that R[x, x^{-1}] and R don’t have the same Picard group. For example R = k[t^2, t^3] with k a field (details omitted). Let L be an invertible module over R[x, x^{-1}] which is not isomorphic to the pullback of an invertible module from R. Pick a surjection

R[x, x^{-1}]⊕ n —> L

As L is a projective R[x, x^{-1}]-module we obtain an idempotent e in the Z-graded ring M_n(R[x, x^{-1}]) = M_n(R)[x, x^{-1}]. And this idempotent is not conjugate to an element of M_n(R) as that would mean L does come from R.

So this answers 1. I do not know the answer to 2.

# 4000 pages

Well, actually 4001 pages at this very moment. Also

• 381203 lines of tex,
• 12104 tags, and
• 3268 commits since we started using git on May 20, 2008.

Enjoy!

# DQCoh(X) = D(E)

Over the last week I added the final arguments to show the result of the title of this post when X is a quasi-compact and quasi-separated scheme or algebraic space X.

Let X be a quasi-compact and quasi-separated algebraic space. There exists a differential graded algebra E with only a finite number of nonzero cohomologies, such that DQCoh(X) = D(E).

For schemes this result is due to Bondal and van den Bergh and can be found in their wonderful paper Generators and representability of functors in commutative and noncommutative geometry. The proof for algebraic spaces is exactly the same (we claim no originality, as always). You can find more precise statements and proofs in the Stacks project here:

1. the case of schemes is in Section Tag 09M2
2. the case of algebraic spaces is in Section Tag 09M9

The statement involves the derived category of differential graded modules over a differential graded ring. Moreover, the proof of the result as given in the paper by Bondal and van den Bergh invokes a general result a la Gabriel-Popescu of Keller which can be found in Keller’s paper Deriving DG categories, in Section 4.3 to be precise. A very general (perhaps the most general available) version of Keller’s result is in a more recent paper by Marco Porta, entitled “The Popescu-Gabriel theorem for triangulated categories”.

So, this got me a bit worried as I am not an expert in differential graded categories, Frobenius categories, etc. But it turns out, as Michel van den Bergh hinted at in an email, that one needs only the most basic material on differential graded algebras, differential graded modules, and a tiny bit about differential graded categories. This material can be found in the new chapter on differential graded algebra. The key construction needed for the proof of the theorem of the title can be found in Section Tag 09LU.

Enjoy!

PS: In my graduate student seminar I will lecture on this and related material during the semester and I will go over the new material a second time, adding a few more details and corrections. But as always, I’d be most happy if you can find mathematical errors (this will earn you a Stacks project mug) or have suggestions for improvements of exposition. Thanks!

# The Sagemath Cloud

Please read William Stein‘s blog post about the online LaTeX editor he created for the Sagemath Cloud. I’ve tried it out and it is pretty incredible, especially considering the fact that when I wrote my first post about the Sagemath Cloud two weeks ago, there wasn’t anything like this.

Of course, since the online latex editor is new there will be some bugs, etc. For me (and maybe because of me or my local setup) not all features work perfectly for some of the files in the Stacks project. But all the basic stuff works. So I would use it if I needed to do some edits and was using a laptop for example. (I am just too used to my own setup on my desktop at home/work to consider using an online editor when I am in my office.)

I strongly recommend you get an account on the Sagemath Cloud and play with it.

# Generating the derived category

A paper by Bondal and ven den Bergh shows that one can generate D_{QCoh}(X) by a single perfect complex if X is a quasi-compact and quasi-separated scheme.

There are two key ingredients to the proof.

The first concerns the orthogonal to a Koszul complex. Let I = (f_1, …, f_n) be a finitely generated ideal of a ring A. Let K be the Koszul complex on f_1, …, f_n over A. Set X = Spec(A) and let U = D(f_1) ∪ … ∪ D(f_n). Let j : U —> X be the inclusion morphism. Let E be a complex of A-modules, in other words an object of D_{QCoh}(X). What can we say about E if Hom(K, E[n]) = 0 for all integers n? Well, if I’m not mistaken, this means exactly that E is Rj_* of an object of D_{QCoh}(U).

Namely, one can build the Koszul complexes K^e = K(f_1^e, …, f_n^e) from the complex K(f_1, …, f_n) by taking cones. Hence the assumed vanishing gives vanishing of Hom(K^e, E[n]) also. Then we consider the distinguished triangles

I^e —> A —> K^e —> I^e[1]

where A —> K is the obvious map. Any element of H^n(U, E) comes from a map I^e —> E for some e (see for example Tag 08DD). Thus our assumption implies that RΓ(U, E) = RΓ(X, E) which is what we wanted to show.

The second ingredient is Thomason-Throghbaugh’s result that any perfect object on a quasi-compact open of a qc + qs scheme X is a direct summand of the restriction of a perfect object on X. In Bondal – van den Bergh this is proved using some abstract machinery (due to Neeman IIRC; this machinery is interesting by itself), but in essence it is elementary from the induction princple and the case of affine schemes (this is how we do it in the Stacks project).

Finally, how do we get our perfect object generating D_{QCoh}(X)? Let X = U_1 ∪ … ∪ U_n be a finite covering by affine opens with n minimal. We will find our perfect object by induction on n. Let Z be the complement of U_2 ∪ … ∪ U_n in X. Then Z is a closed subset of U_1 whose complement is quasi-compact. Writing U_1 = Spec(A_1) we can find a Koszul complex K for a finitely generated ideal I_1 in A_1 cutting out Z. This complex has no cohomology outside of Z hence is a perfect complex K on X. By the discussion above, for E in D_{QCoh}(X) being orthogonal to all shifts of K implies that E comes from U_2 ∪ … ∪ U_n. By induction we can find a perfect complex P on U_2 ∪ … ∪ U_n which generates D_{QCoh}(U_2 ∪ … ∪ U_n). By TT (see above) we can find a perfect complex Q on X whose restriction to U_2 ∪ … ∪ U_n contains P as a direct summand. Then the reader sees that K ⊕ Q is the desired generator of D_{QCoh}(X).

These results can now be found in the stacks project

1. For schemes look at Section Tag 09IP, and
2. for algebraic spaces look at Section Tag 09IU.

Enjoy!

# Semi-stable reduction

There are many proofs of the stable reduction theorem for curves. A good overview is given in Abbes : Réduction semi-stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters, …. I personally found the introduction of Temkin’s paper Stable modification of relative curves quite helpful. Yet another proof (not discussed in the references just given) can be found in a preprint by Kai Arzdorf and Stefan Wewers entitled “Another proof of the Semistable Reduction Theorem”. In this blog post I’d like to discuss their argument (up to a point).

General remark: The goal on this blog is not new or original research, but rather the goal is to understand material in a way that is easy to explain with what is currently available in the Stacks project.

Let’s start with a complete discrete valuation ring R with fraction field K whose residue field k is algebraically closed. Assume we have a smooth projective geometrically irreducible curve C over K. Then we can choose a flat projective sheme X over R whose generic fibre is C. We may normalize X and assume X is a normal scheme. Denote X_0 the special fibre. This is a projective connected scheme over k which satisfies (S_1).

During the proof we will finitely many times

1. replace R by the integral closure R’ of R in a finite K’/K and X by the normalized base change X’, and
2. replace X by a normalized blowup of X.

The assertion of the stable reduction theorem is that in doing so we can get to a situation where X_0 has at worst nodes as singularities.

Step 1. A theorem of Epp guarantees that we can do an operation of type 1 to get reduced special fibre X_0. See for example Tag 09IJ. We observe that any further base change of X (by an extension of dvrs) is normal, so that in particular the isomorphism type of the special fibre is preserved under this operation.

Step 2. If X_0 is reduced, then for a closed point p \in X_0 we have two local invariants: the number of formal branches m_p of X_0 at p and the δ-invariant δp.

Step 3. Let p be a singular point of X_0 with δp > 1. Choose a normalized blow up Y —> X at p such that each of the formal branches B_i at p lifts to a nonsingular branch at some point q_i of Y lying over p and all the q_i are distinct. (Of course you have to show such a blow up exists, but I think this is completely standard. Moreover we have results like this available in the Stacks project; for example Tag 080P can be used to separate the branches and the proof of Tag 00P8 gives a sequence of blowups that normalize the branches.) Of course now Y no longer needs to have reduced special fibre. Thus we replace Y and X by normalized base changes so that both X and Y have reduce special fibre.

Step 4. Consider the map Y —> X we produced in the previous step. To finish the proof it suffices to show that the maximal local δ-invariant of Y at points mapping to p is < δp.

To do this my first guess was to try and prove following criterion (this is probably wrong, although I have no counter example):

If R1f*OY0 has nonzero stalk at p, then Step 4 works.

It is easy to see that the vanishing of R1f*OY0 has all kinds of pleasurable consequences (such as the rationality of the irreducible components of Y0 lying over p — kind of like having a rational singularity at p), so there is hope we can prove that this vanishing isn’t possible if δp > 1.

To prove the statement quoted in italics above, we try to use the relationship between local invariants and the genus of the curve plus the fact that X_0 and Y_0 have the same (arithmetic) genus. However it won’t work. Let’s take the example suggested by Anand where X_0 is a cuspidal rational curve. Then what might happen is that Y_0 is a smooth rational curve (the normalization of X_0) attached transversally to a cuspidal rational curve (the exceptional fibre). Here we could end up doing an infinite sequence of normalized blow-ups over and over again. Moreover, somehow at each step there is a unique point where you blow up.

This kind of problem is solved in the paper by a (more) careful choice of the ideal to blow up in (I haven’t yet looked enough at the paper to understand how). In resolution of singularities of surfaces, a similar problem comes up. For example, look at the discussion of resolution of rational double points in Artin’s paper “Lipman’s proof of ….”. The problem is dealt with by showing that an infinite sequence of infinitely near but not satellite points gives a nonsingular arc passing through the singular point and showing that along a nonsingular arc the procedure of normalized blowups always works. I think the references mentioned at the beginning of this post, tell us a similar method will work here, but the question is how difficult it will be.