## Tag `01M7`

Chapter 26: Constructions of Schemes > Section 26.8: Proj of a graded ring

Lemma 26.8.4. 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 $M$.

- For every $f \in S$ homogeneous of positive degree we have $$ \Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}. $$
- For every $f\in S$ homogeneous of positive degree we have $\Gamma(D_{+}(f), \widetilde M) = M_{(f)}$ as an $S_{(f)}$-module.
- 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 26.8.1.
- 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 $\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$.
- 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.
- 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 10.56.5) and the fact that exactness of short exact sequences may be checked on stalks, see Modules, Lemma 17.3.1. $\square$

The code snippet corresponding to this tag is a part of the file `constructions.tex` and is located in lines 1178–1221 (see updates for more information).

```
\begin{lemma}
\label{lemma-proj-sheaves}
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 $M$.
\begin{enumerate}
\item For every $f \in S$ homogeneous of positive degree we have
$$
\Gamma(D_{+}(f), \mathcal{O}_{\text{Proj}(S)}) = S_{(f)}.
$$
\item For every $f\in S$ homogeneous of positive degree
we have $\Gamma(D_{+}(f), \widetilde M) = M_{(f)}$
as an $S_{(f)}$-module.
\item 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
\ref{lemma-standard-open}.
\item 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
$\mathcal{O}_{\text{Proj}(S), x} = S_{(\mathfrak p)}$.
\item 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.
\item
\label{item-map}
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.
\end{enumerate}
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.
\end{lemma}
\begin{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 \ref{algebra-lemma-proj-prime})
and the fact that exactness of short exact sequences
may be checked on stalks, see
Modules, Lemma \ref{modules-lemma-abelian}.
\end{proof}
```

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