As a concrete example, consider the space $\mathbf P^2$ over some field $k$ and the embedded $\mathbf A^2\hookrightarrow\mathbf P^2, (x_1, x_2)\mapsto[1:x_1:x_2]$. I want to understand sections of $\mathcal O(1)$ and $\mathcal O(-1)$ on $\mathbf A^2$.
- First, the structure sheaf has sections $\Gamma(\mathcal O, \mathbf A^2)=k[x_0, x_1, x_2]_{(x_0)}=k[\frac{x_1}{x_0}, \frac{x_2}{x_0}]$
The tautological line bundle has sections $\Gamma(\mathcal O(-1), \mathbf A^2)=(k[x_0, x_1, x_2]_{(x_0)})_(-1)=\frac {1}{x_0} k[\frac{x_1}{x_0}, \frac{x_2}{x_0}]$ as $k[\frac{x_1}{x_0}, \frac{x_2}{x_0}]$-module.
Question: how do I see that this is the same as the more geometric approach, according to which $\mathcal O(-1)$ assigns to a point $[x_0:x_1:x_2]$ the one-dimensional space $k(x_0, x_1, x_2)\subseteq k^3$?
The twisting sheaf has sections $\Gamma(\mathcal O(1), \mathbf A^2)=(k[x_0, x_1, x_2]_{(x_0)})_1$, which is generated by $x_1, x_2$ as $k[\frac{x_1}{x_0}, \frac{x_2}{x_0}]$-module.
Question: How to I see that that the dual of $\Gamma(\mathcal O(1), \mathbf A^2)$ (as $\Gamma(\mathcal O, \mathbf A^2)$-modules) is $\Gamma(\mathcal O(-1), \mathbf A^2)$?
My attempt: For the last question, it somehow sounds reasonable that any monomial $\frac{x_i^n}{x_0^{n+1}}$ gives rise to a morphism $\langle x_1, x_2\rangle\to k[\frac{x_1}{x_0}, \frac{x_2}{x_0}], f\mapsto \frac{x_i^n f}{x_0^{n+1}}$ of $k[\frac{x_1}{x_0}, \frac{x_2}{x_0}]$-modules. Is it clear that this is an isomorphism?