Lemma 28.28.3. In Situation 28.28.1. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Set $M = \Gamma _*(X, \mathcal{L}, \mathcal{F})$ as a graded $S$-module. There are isomorphisms

\[ f^*\widetilde{M} \longrightarrow \mathcal{F} \]

functorial in $\mathcal{F}$ such that $M_0 \to \Gamma (\text{Proj}(S), \widetilde{M}) \to \Gamma (X, \mathcal{F})$ is the identity map.

**Proof.**
Let $s \in S_{+}$ be homogeneous such that $X_ s$ is affine open in $X$. Recall that $\widetilde{M}|_{D_{+}(s)}$ corresponds to the $S_{(s)}$-module $M_{(s)}$, see Constructions, Lemma 27.8.4. Recall that $f^{-1}(D_{+}(s)) = X_ s$. As $X$ carries an ample invertible sheaf it is quasi-compact and quasi-separated, see Section 28.26. By Lemma 28.17.2 there is a canonical isomorphism $M_{(s)} = \Gamma _*(X, \mathcal{L}, \mathcal{F})_{(s)} \to \Gamma (X_ s, \mathcal{F})$. Since $\mathcal{F}$ is quasi-coherent this leads to a canonical isomorphism

\[ f^*\widetilde{M}|_{X_ s} \to \mathcal{F}|_{X_ s} \]

Since $\mathcal{L}$ is ample on $X$ we know that $X$ is covered by the affine opens of the form $X_ s$. Hence it suffices to prove that the displayed maps glue on overlaps. Proof of this is omitted.
$\square$

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