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 ( and ( 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

It would be helpful to cite equation (0AG2) when saying "the induced map is identity" which is confusing as written. Instead, replace it with "the induced map (see tag(0AG2)) is identity"

There are also:

  • 2 comment(s) on Section 27.10: Invertible sheaves on Proj

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