Lemma 28.26.9 (01PZ)—The Stacks project

Lemma 28.26.9. Let $X$ be a scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Set $S = \Gamma _*(X, \mathcal{L})$ as a graded ring. If every point of $X$ is contained in one of the open subschemes $X_ s$, for some $s \in S_{+}$ homogeneous, then there is a canonical morphism of schemes

\[ f : X \longrightarrow Y = \text{Proj}(S), \]

to the homogeneous spectrum of $S$ (see Constructions, Section 27.8). This morphism has the following properties

$f^{-1}(D_{+}(s)) = X_ s$ for any $s \in S_{+}$ homogeneous,
there are $\mathcal{O}_ X$-module maps $f^*\mathcal{O}_ Y(n) \to \mathcal{L}^{\otimes n}$ compatible with multiplication maps, see Constructions, Equation (27.10.1.1),
the composition $S_ n \to \Gamma (Y, \mathcal{O}_ Y(n)) \to \Gamma (X, \mathcal{L}^{\otimes n})$ is the identity map, and
for every $x \in X$ there is an integer $d \geq 1$ and an open neighbourhood $U \subset X$ of $x$ such that $f^*\mathcal{O}_ Y(dn)|_ U \to \mathcal{L}^{\otimes dn}|_ U$ is an isomorphism for all $n \in \mathbf{Z}$.

Proof. Denote $\psi : S \to \Gamma _*(X, \mathcal{L})$ the identity map. We are going to use the triple $(U(\psi ), r_{\mathcal{L}, \psi }, \theta )$ of Constructions, Lemma 27.14.1. By assumption the open subscheme $U(\psi )$ of equals $X$. Hence $r_{\mathcal{L}, \psi } : U(\psi ) \to Y$ is defined on all of $X$. We set $f = r_{\mathcal{L}, \psi }$. The maps in part (2) are the components of $\theta $. Part (3) follows from condition (2) in the lemma cited above. Part (1) follows from (3) combined with condition (1) in the lemma cited above. Part (4) follows from the last statement in Constructions, Lemma 27.14.1 since the map $\alpha $ mentioned there is an isomorphism. $\square$

The Stacks project

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