The Stacks project

Lemma 27.8.4. Let $S$ be a graded ring. Let $M$ be a graded $S$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\text{Proj}(S)}$-modules associated to $M$.

  1. For every $f \in S$ homogeneous of positive degree we have

    \[ \Gamma (D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}. \]
  2. For every $f\in S$ homogeneous of positive degree we have $\Gamma (D_{+}(f), \widetilde M) = M_{(f)}$ as an $S_{(f)}$-module.

  3. Whenever $D_{+}(g) \subset D_{+}(f)$ the restriction mappings on $\mathcal{O}_{\text{Proj}(S)}$ and $\widetilde M$ are the maps $S_{(f)} \to S_{(g)}$ and $M_{(f)} \to M_{(g)}$ from Lemma 27.8.1.

  4. Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$.

  5. Let $\mathfrak p$ be a homogeneous prime of $S$ not containing $S_{+}$, and let $x \in \text{Proj}(S)$ be the corresponding point. We have $\mathcal{F}_ x = M_{(\mathfrak p)}$ as an $S_{(\mathfrak p)}$-module.

  6. There is a canonical ring map $ S_0 \longrightarrow \Gamma (\text{Proj}(S), \widetilde S) $ and a canonical $S_0$-module map $ M_0 \longrightarrow \Gamma (\text{Proj}(S), \widetilde M) $ compatible with the descriptions of sections over standard opens and stalks above.

Moreover, all these identifications are functorial in the graded $S$-module $M$. In particular, the functor $M \mapsto \widetilde M$ is an exact functor from the category of graded $S$-modules to the category of $\mathcal{O}_{\text{Proj}(S)}$-modules.

Proof. Assertions (1) - (5) are clear from the discussion above. We see (6) since there are canonical maps $M_0 \to M_{(f)}$, $x \mapsto x/1$ compatible with the restriction maps described in (3). The exactness of the functor $M \mapsto \widetilde M$ follows from the fact that the functor $M \mapsto M_{(\mathfrak p)}$ is exact (see Algebra, Lemma 10.57.5) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 17.3.1. $\square$


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