The Stacks project

Lemma 27.8.7. Let $S$ be a graded ring. The locally ringed space $\text{Proj}(S)$ is a scheme. The standard opens $D_{+}(f)$ are affine opens. For any graded $S$-module $M$ the sheaf $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules.

Proof. Consider a standard open $D_{+}(f) \subset \text{Proj}(S)$. By Lemmas 27.8.1 and 27.8.4 we have $\Gamma (D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}$, and we have a homeomorphism $\varphi : D_{+}(f) \to \mathop{\mathrm{Spec}}(S_{(f)})$. For any standard open $D(g) \subset \mathop{\mathrm{Spec}}(S_{(f)})$ we may pick an $h \in S_{+}$ as in Lemma 27.8.6. Then $\varphi ^{-1}(D(g)) = D_{+}(h)$, and by Lemmas 27.8.4 and 27.8.1 we see

\[ \Gamma (D_{+}(h), \mathcal{O}_{\text{Proj}(S)}) = S_{(h)} = (S_{(f)})_{h^{\deg (f)}/f^{\deg (h)}} = (S_{(f)})_ g = \Gamma (D(g), \mathcal{O}_{\mathop{\mathrm{Spec}}(S_{(f)})}). \]

Thus the restriction of $\mathcal{O}_{\text{Proj}(S)}$ to $D_{+}(f)$ corresponds via the homeomorphism $\varphi $ exactly to the sheaf $\mathcal{O}_{\mathop{\mathrm{Spec}}(S_{(f)})}$ as defined in Schemes, Section 26.5. We conclude that $D_{+}(f)$ is an affine scheme isomorphic to $\mathop{\mathrm{Spec}}(S_{(f)})$ via $\varphi $ and hence that $\text{Proj}(S)$ is a scheme.

In exactly the same way we show that $\widetilde M$ is a quasi-coherent sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules. Namely, the argument above will show that

\[ \widetilde M|_{D_{+}(f)} \cong \varphi ^*\left(\widetilde{M_{(f)}}\right) \]

which shows that $\widetilde M$ is quasi-coherent. $\square$

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