Definition 17.23.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given an invertible sheaf $\mathcal{L}$ on $X$ we define the *associated graded ring* to be

\[ \Gamma _*(X, \mathcal{L}) = \bigoplus \nolimits _{n \geq 0} \Gamma (X, \mathcal{L}^{\otimes n}) \]

Given a sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we set

\[ \Gamma _*(X, \mathcal{L}, \mathcal{F}) = \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes n}) \]

which we think of as a graded $\Gamma _*(X, \mathcal{L})$-module.

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