December 2007


So now we begin one of the great examples in algebraic geometry: algebraic groups. These are exceptionally nice, and we’ll talk about a couple of more general concepts before applying them to this case.

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Last time we brought in a bunch of the algebra necessary to do algebraic geometry, now we’ll talk a bit about topology, a bit about morphisms, and then note that we have a perfectly well-defined category (and we’ll even say what that means).

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Before moving on to morphisms of varieties, it would do us some good to talk about a bit of commutative algebra and to do a little bit more geometry of individual objects. The central notion of algebra that we’ll be needing is that of a commutative ring with identity.

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Yesterday, there was talk of affine varieties, and I pointed out the fact that intersections don’t work out nicely in this case. Today we’ll talk about an elegant solution to this problem: the notion of projective space.

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Ok, I know I haven’t posted in a bit, but it’s the end of my first semester of my first year, so hopefully I can be forgiven. Today, we’re going to start a (potentially doomed from the start) project: algebraic geometry without prerequisites, or at least, with minimal prerequisites. The goal is to get as much done as possible, with detours into many of the more interesting theorems, and today we’re going to start with affine varieties.

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