Lemma 27.10.3 (01MT)—The Stacks project

Lemma 27.10.3. Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Set $X = \text{Proj}(S)$. Assume $X$ is covered by the standard opens $D_+(f)$ with $f \in S_1$, e.g., if $S$ is generated by $S_1$ over $S_0$. Then the sheaves $\mathcal{O}_ X(n)$ are invertible and the maps (27.10.1.1), (27.10.1.2), and (27.10.1.5) are isomorphisms. In particular, these maps induce isomorphisms

\[ \mathcal{O}_ X(1)^{\otimes n} \cong \mathcal{O}_ X(n) \quad \text{and} \quad \widetilde{M} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(n) = \widetilde{M}(n) \cong \widetilde{M(n)} \]

Thus (27.9.0.2) becomes a map

27.10.3.1

\begin{equation} \label{constructions-equation-map-global-sections-degree-n-simplified} M_ n \longrightarrow \Gamma (X, \widetilde{M}(n)) \end{equation}

and (27.10.1.6) becomes a map

27.10.3.2

\begin{equation} \label{constructions-equation-global-sections-more-generally-simplified} M \longrightarrow \bigoplus \nolimits _{n \in \mathbf{Z}} \Gamma (X, \widetilde{M}(n)). \end{equation}

Proof. Under the assumptions of the lemma $X$ is covered by the open subsets $D_{+}(f)$ with $f \in S_1$ and the lemma is a consequence of Lemma 27.10.2 above. $\square$

The Stacks project

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