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!

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01CV. Beware of the difference between the letter 'O' and the digit '0'.