Remark 111.52.2. Let $k$ be a field. Let $\mathbf{P}^2_ k = \text{Proj}(k[X_0, X_1, X_2])$. Any invertible sheaf on $\mathbf{P}^2_ k$ is isomorphic to $\mathcal{O}_{\mathbf{P}^2_ k}(n)$ for some $n \in \mathbf{Z}$. Recall that
\[ \Gamma (\mathbf{P}^2_ k, \mathcal{O}_{\mathbf{P}^2_ k}(n)) = k[X_0, X_1, X_2]_ n \]
is the degree $n$ part of the polynomial ring. For a quasi-coherent sheaf $\mathcal{F}$ on $\mathbf{P}^2_ k$ set $\mathcal{F}(n) = \mathcal{F} \otimes _{\mathcal{O}_{\mathbf{P}^2_ k}} \mathcal{O}_{\mathbf{P}^2_ k}(n)$ as usual.
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