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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 $M$.

  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 26.8.1.
  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 10.56.5) 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}

    Comments (2)

    Comment #130 by Fred Rohrer (site) on February 15, 2013 a 1:36 pm UTC

    In 6, delete the first occurence of "above".

    Comment #136 by Johan (site) on February 16, 2013 a 12:59 am UTC

    Fixed, see here. Thanks!

    There are also 6 comments on Section 26.8: Constructions of Schemes.

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