The Stacks project

Definition 17.25.7. 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.

Comments (4)

Comment #11 by Pieter Belmans on

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

Comment #12 by Pieter Belmans on

I am not sure about your stance on using _* versus _\bullet, but given that complexes of cohomology in tag 104.3.1 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

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

Comment #20 by Johan on

@#11 and #13. Fixed. Thanks. @#12: Yes, hmm, I think of the notation 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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View Definition 17.25.7 as pdf