Lemma 27.10.2 (01MS)—The Stacks project

Lemma 27.10.2. Let $S$ be a graded ring. Set $X = \text{Proj}(S)$. Let $f \in S$ be homogeneous of degree $d > 0$. The sheaves $\mathcal{O}_ X(nd)|_{D_{+}(f)}$ are invertible, and in fact trivial for all $n \in \mathbf{Z}$ (see Modules, Definition 17.25.1). The maps (27.10.1.1) restricted to $D_{+}(f)$

\[ \mathcal{O}_ X(nd)|_{D_{+}(f)} \otimes _{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_ X(m)|_{D_{+}(f)} \longrightarrow \mathcal{O}_ X(nd + m)|_{D_{+}(f)}, \]

the maps (27.10.1.2) restricted to $D_+(f)$

\[ \mathcal{O}_ X(nd)|_{D_{+}(f)} \otimes _{\mathcal{O}_{D_{+}(f)}} \mathcal{F}(m)|_{D_{+}(f)} \longrightarrow \mathcal{F}(nd + m)|_{D_{+}(f)}, \]

and the maps (27.10.1.5) restricted to $D_{+}(f)$

\[ \widetilde M(nd)|_{D_{+}(f)} = \widetilde M|_{D_{+}(f)} \otimes _{\mathcal{O}_{D_{+}(f)}} \mathcal{O}_ X(nd)|_{D_{+}(f)} \longrightarrow \widetilde{M(nd)}|_{D_{+}(f)} \]

are isomorphisms for all $n, m \in \mathbf{Z}$.

Proof. The (not graded) $S$-module maps $S \to S(nd)$, and $M \to M(nd)$, given by $x \mapsto f^ n x$ become isomorphisms after inverting $f$. The first shows that $S_{(f)} \cong S(nd)_{(f)}$ which gives an isomorphism $\mathcal{O}_{D_{+}(f)} \cong \mathcal{O}_ X(nd)|_{D_{+}(f)}$. The second shows that the map $S(nd)_{(f)} \otimes _{S_{(f)}} M_{(f)} \to M(nd)_{(f)}$ is an isomorphism. The case of the map (27.10.1.2) is a consequence of the case of the map (27.10.1.1). $\square$

The Stacks project

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