# 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.