blegs


I’ve been trying to write up a bunch of notes on Zariski’s Main Theorem and its applications, which is the main reason I haven’t blogged in a while. They should be done (as in, in a rough but complete state) soon, after which I will post them. In writing them up, I ran into the following doubt. Can anyone clarify? I don’t really know whether this is MO level, so I’ll just ask it here.

Is an unramified, radicial morphism an immersion?

I know this is true for etale morphisms (SGA I, expose 1.5); in that case the map is even open.

Edit: I know this is also true if the morphism is unramified and proper (it follows easily from the fact that a proper and quasi-finite morphism is finite). If the properness hypothesis is unnecessary, then we have a very nice functorial description of what an immersion is.

Edit 2: Never mind, this is false.

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Adeel Khan has set up a git server for the CRing project. In particular, you can follow how open source commutative algebra evolves in real time. More practically, you can download source files from there; they’re also on the main website, of course, but the ones there are likely to be slightly newer: I can’t update the website on my college account instantly. Plus, you can see who contributed what in what is really an intuitive and transparent manner. You can also use the git server to submit contributions, though for that you’ll need the password. For this, you can write to cring.project(at)gmail.  Again, if you don’t want to use git, we’re happy to receive contributions by email.

So why and how should you contribute? Johan deJong has explained it here (for the Stacks Project); the same applies to the CRing project. After all, the source code to your old homework sets or class notes isn’t doing anything on your hard drive. We’d be thrilled to receive it and to list you as a contributor. And we’ll work out how to edit it in (unless you want to, which you are welcome to).

So I signed up to give a talk at HMMT some time back, which will be this Saturday. As expected, I procrastinated preparation for it until now. The problem is, I’m not sure what to talk about. In high school, I wasn’t really into math contests such as HMMT — my mind was never able to find creative solutions with the necessary speed, and I’d consistently turn in abysmal performances. So as a result, I was never exposed to much of the culture of high school math contests (the existence of which I found out not that long ago). Anyway, as a result I’m not completely sure how to prep this talk, or even what to talk about. Some topics that I consider talk-worthy and interesting are:

  1. Lecture one of algebraic geometry class. Define varieties and algebraic sets, and state (or even prove) the Nullstellensatz. But I suspect this will use too much commutative algebra than I should assume. I understand that plenty of extremely accomplished HMMTers may not know what a ring is.
  2. The p-adic numbers.  This has the benefit of my being able to recycle an old talk.  But I might have to re-tool it.
  3. Quadratic reciprocity. Perhaps the proof via Gauss sums, for instance. But this is something that people will tend to know, right?
  4. A brief intro to computability theory (as in — Turing machines, unsolvability of the halting problem, complexity classes, maybe say something about Kolmogorov complexity)

The basic problem is that such topics essentially amount to picking your favorite textbook on subject X, choosing five or six pages, and reading them aloud to the students — in short, a normal class. Which is probably not what they’re looking for.

But some of you readers have better ideas than I.  So, any thoughts? Pretend, or not, that you were in high school. What would you wish to know that I could cover in an hour?

(If I end up using your topic, I’ll mention you in the talk!)

So, the blog stats show that semisimple Lie algebras haven’t exactly been popular.  Traffic has actually been unusually high, but people have been reading about the heat equation or Ricci curvature rather than Verma modules.  Which is interesting, since I thought there was a dearth of analysts in the mathosphere.  At MathOverflow, for instance, there have been a few complaints that everyone there is an algebraic geometer.     Anyway, there wasn’t going to be that much more I would say about semisimple Lie algebras in the near future, so for the next few weeks I plan random and totally disconnected posts at varying levels (but loosely related to algebra or algebraic geometry, in general).

I learned a while back that there is a classification of the simple modules over the semidirect product between a group and an commutative algebra which works the same way as the (more specific case) between a group and an abelian group.  The result for abelian groups rather than commutative rings appears in a lot of places, e.g. Serre’s Linear Representations of Finite Groups or Pavel Etingof’s notes.  I couldn’t find a source for the more general result though.  I wanted to work that out here, though I got a bit confused near the end, at which point I’ll toss out a bleg.

Let {G} be a finite group acting on a finite-dimensional commutative algebra {A}over an algebraically closed field {k} of characteristic prime to the order of {G}. Then, the irreducible representations of {A} correspond to maximal ideals in {A}, or equivalently (by Hilbert’s Nullstellensatz!) homomorphisms {\chi: A \rightarrow k}, called characters. In other words, {A} acts on a 1-dimensional space via the character {\chi}. (more…)

Today’s main goal is the Leray theorem (though at the end I have to ask a question):

Theorem 1 Let {\mathcal{F}} be a sheaf on {X}, and {\mathfrak{U} = \{ U_i, i \in I\}} an open cover of {X}. Suppose\displaystyle H^n( U_{i_1} \cap \dots \cap U_{i_k}, \mathcal{F}|_{ U_{i_1} \cap \dots \cap U_{i_k}}) = 0

for all {k}-tuples {i_1, \dots , i_k \in I}, and all {n>0}. Then the canonical morphism\displaystyle H^n( \mathfrak{U}, \mathcal{F}) \rightarrow H^n( X, \mathcal{F})

is an isomorphism for all {n}

 

This seems rather useless, because the theorem presupposes the vanishing of (regular) cohomology on the covering. However, in many cases it turns out to be helpful. If {X} is a separated scheme, {U_i} an open affine cover of {X}, and {\mathcal{F}} quasi-coherent, it applies. The reason is that each of the intersections { U_{i_1} \cap \dots \cap U_{i_k}} are all affine by separatedness, so {\mathcal{F}} has no cohomology on them by a basic property of quasi-coherent sheaves. This gives a practical way of computing sheaf cohomology in algebraic geometry. Hartshorne uses it to compute the cohomology of line bundles on projective space.

Another instance arises when {\mathcal{O}} is the sheaf of holomorphic functions over some Riemann surface {X}. In this case {\{U_i\}} is a covering of charts. It is a theorem (which I will eventually prove) that for any open subset of {\mathbb{C}} (which any intersection of the {U_i}‘s is isomorphic to), the sheaf {\mathcal{O}} has trivial cohomology. (more…)