# The Stacks Project

## Tag: 01CV

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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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Comment #11 by Pieter Belmans on July 22, 2012 at 9:17 am UTC

In the notes following this definition (there's no associated tag) there is the typo "allthough".

Comment #12 by Pieter Belmans on July 22, 2012 at 9:21 am UTC

I am not sure about your stance on using _* versus _\bullet, but given that complexes of cohomology in tag 07AS are written in the latter and less ambiguous way, it might be beneficial to do this here (and elsewhere) too?

Comment #13 by Pieter Belmans on July 22, 2012 at 9:23 am UTC

Missed this earlier on: there is the typo "compatibilties" in the last sentence of the remarks following this definition.

Comment #20 by Johan on July 22, 2012 at 10:58 pm UTC

@#11 and #13. Fixed. Thanks. @#12: Yes, hmm, I think of the notation $\Gamma_*$ as standard for the direct sum of global sections of the powers of an invertible sheaf. This isn't a complex!

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