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Tag 01HV

Chapter 25: Schemes > Section 25.5: Affine schemes

Lemma 25.5.4. Let $R$ be a ring. Let $M$ be an $R$-module. Let $\widetilde M$ be the sheaf of $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$-modules associated to $M$.

  1. We have $\Gamma(\mathop{\mathrm{Spec}}(R), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R$.
  2. We have $\Gamma(\mathop{\mathrm{Spec}}(R), \widetilde M) = M$ as an $R$-module.
  3. For every $f \in R$ we have $\Gamma(D(f), \mathcal{O}_{\mathop{\mathrm{Spec}}(R)}) = R_f$.
  4. For every $f\in R$ we have $\Gamma(D(f), \widetilde M) = M_f$ as an $R_f$-module.
  5. Whenever $D(g) \subset D(f)$ the restriction mappings on $\mathcal{O}_{\mathop{\mathrm{Spec}}(R)}$ and $\widetilde M$ are the maps $R_f \to R_g$ and $M_f \to M_g$ from Lemma 25.5.1.
  6. Let $\mathfrak p$ be a prime of $R$, and let $x \in \mathop{\mathrm{Spec}}(R)$ be the corresponding point. We have $\mathcal{O}_{\mathop{\mathrm{Spec}}(R), x} = R_{\mathfrak p}$.
  7. Let $\mathfrak p$ be a prime of $R$, and let $x \in \mathop{\mathrm{Spec}}(R)$ be the corresponding point. We have $\mathcal{F}_x = M_{\mathfrak p}$ as an $R_{\mathfrak p}$-module.

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

Proof. Assertions (1) - (7) are clear from the discussion above. The exactness of the functor $M \mapsto \widetilde M$ follows from the fact that the functor $M \mapsto M_{\mathfrak p}$ is exact 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 schemes.tex and is located in lines 691–719 (see updates for more information).

    \begin{lemma}
    \label{lemma-spec-sheaves}
    Let $R$ be a ring. Let $M$ be an $R$-module. Let $\widetilde M$
    be the sheaf of $\mathcal{O}_{\Spec(R)}$-modules
    associated to $M$.
    \begin{enumerate}
    \item We have $\Gamma(\Spec(R), \mathcal{O}_{\Spec(R)}) = R$.
    \item We have $\Gamma(\Spec(R), \widetilde M) = M$ as an $R$-module.
    \item For every $f \in R$ we have
    $\Gamma(D(f), \mathcal{O}_{\Spec(R)}) = R_f$.
    \item For every $f\in R$ we have $\Gamma(D(f), \widetilde M) = M_f$
    as an $R_f$-module.
    \item Whenever $D(g) \subset D(f)$ the restriction mappings
    on $\mathcal{O}_{\Spec(R)}$ and $\widetilde M$
    are the maps
    $R_f \to R_g$ and $M_f \to M_g$ from Lemma
    \ref{lemma-standard-open}.
    \item Let $\mathfrak p$ be a prime of $R$, and let $x \in \Spec(R)$
    be the corresponding point. We have
    $\mathcal{O}_{\Spec(R), x} = R_{\mathfrak p}$.
    \item Let $\mathfrak p$ be a prime of $R$, and let $x \in \Spec(R)$
    be the corresponding point. We have $\mathcal{F}_x = M_{\mathfrak p}$
    as an $R_{\mathfrak p}$-module.
    \end{enumerate}
    Moreover, all these identifications are functorial in the $R$
    module $M$. In particular, the functor $M \mapsto \widetilde M$
    is an exact functor from the category of $R$-modules
    to the category of $\mathcal{O}_{\Spec(R)}$-modules.
    \end{lemma}
    
    \begin{proof}
    Assertions (1) - (7) are clear from the discussion above.
    The exactness of the functor $M \mapsto \widetilde M$
    follows from the fact that the functor $M \mapsto M_{\mathfrak p}$
    is exact 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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