The Stacks project

Lemma 39.12.5. Let $A$ be a graded ring. Let $X = \mathop{\mathrm{Spec}}(A)$ with action $a : \mathbf{G}_ m \times X \to X$ as in Example 39.12.3. Let $\mathcal{F}$ be a $\mathbf{G}_ m$-equivariant quasi-coherent $\mathcal{O}_ X$-module. Then $M = \Gamma (X, \mathcal{F})$ has a canonical grading such that it is a graded $A$-module and such that the isomorphism $\widetilde{M} \to \mathcal{F}$ (Schemes, Lemma 26.7.4) is an isomorphism of $\mathbf{G}_ m$-equivariant modules where the $\mathbf{G}_ m$-equivariant structure on $\widetilde{M}$ is the one from Example 39.12.3.

Proof. You can either prove this by repeating the arguments of Lemma 39.12.4 for the module $M$. Alternatively, you can consider the scheme $(X', \mathcal{O}_{X'}) = (X, \mathcal{O}_ X \oplus \mathcal{F})$ where $\mathcal{F}$ is viewed as an ideal of square zero. There is a natural action $a' : \mathbf{G}_ m \times X' \to X'$ defined using the action on $X$ and on $\mathcal{F}$. Then apply Lemma 39.12.4 to $X'$ and conclude. (The nice thing about this argument is that it immediately shows that the grading on $A$ and $M$ are compatible, i.e., that $M$ is a graded $A$-module.) Details omitted. $\square$

Comments (2)

Comment #3775 by Herman Rohrbach on

Typo: in the last sentence of the statement of the lemma, "onde" should be "one".

There are also:

  • 12 comment(s) on Section 39.12: Equivariant quasi-coherent sheaves

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Your email address will not be published.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 0EKL. Beware of the difference between the letter 'O' and the digit '0'.