The Stacks project

Lemma 71.11.4. In Situation 71.11.1. The relative Proj comes equipped with a quasi-coherent sheaf of $\mathbf{Z}$-graded algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n)$ and a canonical homomorphism of graded algebras

\[ \psi : \pi ^*\mathcal{A} \longrightarrow \bigoplus \nolimits _{n \geq 0} \mathcal{O}_{\underline{\text{Proj}}_ X(\mathcal{A})}(n) \]

whose base change to any scheme over $X$ agrees with Constructions, Lemma 27.15.5.

Proof. As in the discussion following Definition 71.11.3 choose a scheme $U$ and a surjective ├ętale morphism $U \to X$, set $R = U \times _ X U$ with projections $s, t : R \to U$, $\mathcal{A}_ U = \mathcal{A}|_ U$, $\mathcal{A}_ R = \mathcal{A}|_ R$, and $\pi : P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X$, $\pi _ U : P_ U = \underline{\text{Proj}}_ U(\mathcal{A}_ U)$ and $\pi _ R : P_ R = \underline{\text{Proj}}_ U(\mathcal{A}_ R)$. By the Constructions, Lemma 27.15.5 we have a quasi-coherent sheaf of $\mathbf{Z}$-graded $\mathcal{O}_{P_ U}$-algebras $\bigoplus _{n \in \mathbf{Z}} \mathcal{O}_{P_ U}(n)$ and a canonical map $\psi _ U : \pi _ U^*\mathcal{A}_ U \to \bigoplus _{n \geq 0} \mathcal{O}_{P_ U}(n)$ and similarly for $P_ R$. By Constructions, Lemma 27.16.10 the pullback of $\mathcal{O}_{P_ U}(n)$ and $\psi _ U$ by either projection $P_ R \to P_ U$ is equal to $\mathcal{O}_{P_ R}(n)$ and $\psi _ R$. By Properties of Spaces, Proposition 66.32.1 we obtain $\mathcal{O}_{P}(n)$ and $\psi $. We omit the verification of compatibility with pullback to arbitrary schemes over $X$. $\square$

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Your email address will not be published.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 084D. Beware of the difference between the letter 'O' and the digit '0'.