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Definition 16.21.4. Let $(X, \mathcal{O}_X)$ be a ringed space. Assume that all stalks $\mathcal{O}_{X, x}$ are local rings. 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.
\begin{definition}
\label{definition-gamma-star}
Let $(X, \mathcal{O}_X)$ be a ringed space.
Assume that all stalks $\mathcal{O}_{X, x}$ are local rings.
Given an invertible sheaf $\mathcal{L}$ on $X$ we define
the {\it 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.
\end{definition}
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