Lemma 27.10.7 (0B5I)—The Stacks project

Lemma 27.10.7. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Set $M = \bigoplus _{n \in \mathbf{Z}} \Gamma (X, \mathcal{F}(n))$ as a graded $S$-module, using (27.10.1.4) and (27.10.1.3). Then there is a canonical $\mathcal{O}_ X$-module map

\[ \widetilde{M} \longrightarrow \mathcal{F} \]

functorial in $\mathcal{F}$ such that the induced map $M_0 \to \Gamma (X, \mathcal{F})$ is the identity.

Proof. Let $f \in S$ be homogeneous of degree $d > 0$. Recall that $\widetilde{M}|_{D_{+}(f)}$ corresponds to the $S_{(f)}$-module $M_{(f)}$ by Lemma 27.8.4. Thus we can define a canonical map

\[ M_{(f)} \longrightarrow \Gamma (D_+(f), \mathcal{F}),\quad m/f^ n \longmapsto m|_{D_+(f)} \otimes f|_{D_+(f)}^{-n} \]

which makes sense because $f|_{D_+(f)}$ is a trivializing section of the invertible sheaf $\mathcal{O}_ X(d)|_{D_+(f)}$, see Lemma 27.10.2 and its proof. Since $\widetilde{M}$ is quasi-coherent, this leads to a canonical map

\[ \widetilde{M}|_{D_+(f)} \longrightarrow \mathcal{F}|_{D_+(f)} \]

via Schemes, Lemma 26.7.1. We obtain a global map if we prove that the displayed maps glue on overlaps. Proof of this is omitted. We also omit the proof of the final statement. $\square$

Comments (2)

Comment #7541 by old friend on July 30, 2022 at 16:50

It would be helpful to cite equation (0AG2) when saying "the induced map $M_{0} \to \Gamma(X,\mathcal{F})$ is identity" which is confusing as written. Instead, replace it with "the induced map $M_{0} \to \Gamma(X, \mathcal{M}) \to \Gamma(X, \mathcal{F})$ (see tag(0AG2)) is identity"

Comment #7666 by Stacks Project on August 15, 2022 at 11:44

Going to leave as is.

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