The Stacks project

Lemma 27.12.3. Let $S$ be a graded ring generated as an $S_0$-algebra by the elements of $S_1$. In this case the scheme $X = \text{Proj}(S)$ represents the functor which associates to a scheme $Y$ the set of pairs $(\mathcal{L}, \psi )$, where

  1. $\mathcal{L}$ is an invertible $\mathcal{O}_ Y$-module, and

  2. $\psi : S \to \Gamma _*(Y, \mathcal{L})$ is a graded ring homomorphism such that $\mathcal{L}$ is generated by the global sections $\psi (f)$, with $f \in S_1$

up to strict equivalence as above.

Proof. Under the assumptions of the lemma we have $X = U_1$ and the lemma is a reformulation of Lemma 27.12.2 above. $\square$

Comments (0)

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Your email address will not be published.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 01NA. Beware of the difference between the letter 'O' and the digit '0'.