The Stacks project

Lemma 70.12.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\psi : \mathcal{A} \to \mathcal{B}$ be a map of quasi-coherent graded $\mathcal{O}_ X$-algebras. Set $P = \underline{\text{Proj}}_ X(\mathcal{A}) \to X$ and $Q = \underline{\text{Proj}}_ X(\mathcal{B}) \to X$. There is a canonical open subspace $U(\psi ) \subset Q$ and a canonical morphism of algebraic spaces

\[ r_\psi : U(\psi ) \longrightarrow P \]

over $X$ and a map of $\mathbf{Z}$-graded $\mathcal{O}_{U(\psi )}$-algebras

\[ \theta = \theta _\psi : r_\psi ^*\left( \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_ P(d) \right) \longrightarrow \bigoplus \nolimits _{d \in \mathbf{Z}} \mathcal{O}_{U(\psi )}(d). \]

The triple $(U(\psi ), r_\psi , \theta )$ is characterized by the property that for any scheme $W$ ├ętale over $X$ the triple

\[ (U(\psi ) \times _ X W,\quad r_\psi |_{U(\psi ) \times _ X W} : U(\psi ) \times _ X W \to P \times _ X W,\quad \theta |_{U(\psi ) \times _ X W}) \]

is equal to the triple associated to $\psi : \mathcal{A}|_ W \to \mathcal{B}|_ W$ of Constructions, Lemma 27.18.1.

Proof. This lemma follows from ├ętale localization and the case of schemes, see discussion following Definition 70.11.3. Details omitted. $\square$

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Your email address will not be published.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 085F. Beware of the difference between the letter 'O' and the digit '0'.