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

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 $n \in \mathbf{Z}$.

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