The Stacks project

Example 29.38.3. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$. In this case $\mathcal{O}_ X(1)$ is a very ample invertible sheaf on $X$. Namely, the morphism associated to the graded $\mathcal{O}_ S$-algebra map

\[ \text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A} \]

is a closed immersion $X \to \mathbf{P}(\mathcal{A}_1)$ which pulls back $\mathcal{O}_{\mathbf{P}(\mathcal{A}_1)}(1)$ to $\mathcal{O}_ X(1)$, see Constructions, Lemma 27.18.5.

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View Example 29.38.3 as pdf