Algebraic Geometry Observation II: Sheaf Theory

Observation II. Sheaf theory is hard.

Per my earlier entry: Given a smooth complex algebraic variety X, I finally manage to track down a semi-manageable definition for the structure sheaf \mathcal{O}_X. But here’s the thing with super-abstract definitions of things:

There’s a big difference between “getting them” and understanding them. In this case, I read the definition a few dozen times, annotated the .pdf file by Elliott with what I found, and felt as if I truly “got it”. Then, I read an example regarding the topological space X=\mathbb{C}^n with the claim that \mathcal{O}_X=\mathbb{C}[x_1,x_2,\ldots,x_n]

…say what now?!

Finally, I reread a bunch of stuff and found a cool analogy between arbitrary varieties X and the case where X=M is a smooth k-times continuously differentiable manifold of dimension \dim X=n. This cleared things up for me.

It really is going to be a long summer. Heh.

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