May 2008


Sorry that this one is late, things are going to continue to be hit-or-miss through June, though hopefully hit more than miss. Anyway, we’ve got a cohomology theory for sheaves. Anyone who has worked with cohomology of manifolds will know just how useful Poincare Duality is. So what we want is some sort of duality theory for the cohomology of sheaves that will help us to reduce the amount of computation necessary. Here, we must appeal to the wisdom of Serre (and of course, it was generalized by Grothendieck, but I don’t quite get Grothendieck Duality yet).

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In order to get at surfaces, we don’t just need the scheme theory of the last two posts, but we also need to understand the cohomology of sheaves. We will not be proving all of the details, we will also not be doing things in full generality with derived functors, at least not until we absolutely must. Today, we’re going to talk about Čech Cohomology.

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Last time, we talked about \mathrm{Spec}, which takes rings and gives us affine schemes in a manner analogous to affine varieties. Now, though it is safe to say that projective varieties are schemes (well, with the extra points taken into account) because they are covered by affine varieties (and so affine schemes), they’re actually MUCH nicer than arbitrary schemes. So, we’re going to discuss \mathrm{Proj} which takes graded rings to special types of schemes.

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Sorry about the extended delay, should be back to regular Monday-Wednesday-Friday updates now. I hope. Anyway, we’re going to be heading into Intersection Theory soon, and in particular we’re going to be looking at surfaces. Sadly, at this point, abstract varieties aren’t good enough anymore. We’re going to need all sorts of “non-reduced” behavior, that is, nilpotent functions on our more generalized notion of varieties. This will, however, give us a rather nice way of interpreting Cartier Divisors, or at least the effective ones, so now we’ll just have to get started.

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A friend of mine from undergrad days has started up a math blog recently. He’s just getting started, but I, at least, expect him to do interesting things once he gets rolling (probably in another week or so, due to exams). So anyway, welcome Chris, and his blog Coffee and Mathematics, to the math blogosphere.

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