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Tag 03PU

Chapter 53: Étale Cohomology > Section 53.29: Neighborhoods, stalks and points

Theorem 53.29.10. Let $S$ be a scheme. A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective (resp. surjective) if and only if the map on stalks $a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is injective (resp. surjective) for all geometric points of $S$. A sequence of abelian sheaves on $S_{\acute{e}tale}$ is exact if and only if it is exact on all stalks at geometric points of $S$.

Proof. The necessity of exactness on stalks follows from Lemma 53.29.9. For the converse, it suffices to show that a map of sheaves is surjective (respectively injective) if and only if it is surjective (respectively injective) on all stalks. We prove this in the case of surjectivity, and omit the proof in the case of injectivity.

Let $\alpha : \mathcal{F} \to \mathcal{G}$ be a map of sheaves such that $\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$ is surjective for all geometric points. Fix $U \in \mathop{\mathrm{Ob}}\nolimits(S_{\acute{e}tale})$ and $s \in \mathcal{G}(U)$. For every $u \in U$ choose some $\overline{u} \to U$ lying over $u$ and an étale neighborhood $(V_u , \overline{v}_u) \to (U, \overline{u})$ such that $s|_{V_u} = \alpha(s_{V_u})$ for some $s_{V_u} \in \mathcal{F}(V_u)$. This is possible since $\alpha$ is surjective on stalks. Then $\{V_u \to U\}_{u \in U}$ is an étale covering on which the restrictions of $s$ are in the image of the map $\alpha$. Thus, $\alpha$ is surjective, see Sites, Section 7.11. $\square$

    The code snippet corresponding to this tag is a part of the file etale-cohomology.tex and is located in lines 3512–3521 (see updates for more information).

    \begin{theorem}
    \label{theorem-exactness-stalks}
    Let $S$ be a scheme.
    A map $a : \mathcal{F} \to \mathcal{G}$ of sheaves of sets is injective
    (resp.\ surjective) if and only if the map on stalks
    $a_{\overline{s}} : \mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$
    is injective (resp.\ surjective) for all geometric points of $S$.
    A sequence of abelian sheaves on $S_\etale$ is exact
    if and only if it is exact on all stalks at geometric points of $S$.
    \end{theorem}
    
    \begin{proof}
    The necessity of exactness on stalks follows from
    Lemma \ref{lemma-stalk-exact}.
    For the converse, it suffices to show that a map of sheaves is surjective
    (respectively injective) if and only if it is surjective (respectively
    injective) on all stalks. We prove this in the case of surjectivity, and omit
    the proof in the case of injectivity.
    
    \medskip\noindent
    Let $\alpha : \mathcal{F} \to \mathcal{G}$ be a map of sheaves such
    that $\mathcal{F}_{\overline{s}} \to \mathcal{G}_{\overline{s}}$
    is surjective for all geometric points. Fix
    $U \in \Ob(S_\etale)$
    and $s \in \mathcal{G}(U)$. For every $u \in U$ choose some
    $\overline{u} \to U$ lying over $u$ and an \'etale neighborhood
    $(V_u , \overline{v}_u) \to (U, \overline{u})$ such that
    $s|_{V_u} = \alpha(s_{V_u})$ for some
    $s_{V_u} \in \mathcal{F}(V_u)$.
    This is possible since $\alpha$ is surjective on
    stalks. Then $\{V_u \to U\}_{u \in U}$
    is an \'etale covering on which the restrictions of $s$
    are in the image of the map $\alpha$.
    Thus, $\alpha$ is surjective, see
    Sites, Section \ref{sites-section-sheaves-injective}.
    \end{proof}

    Comments (2)

    Comment #1707 by Yogesh More on November 30, 2015 a 12:24 pm UTC

    very minor remark: at the beginning of the second paragraph of the proof, "Let $\alpha:F \to G$ be a map of abelian sheaves", I don't think you need the word abelian; the statement of the theorem says it holds for sheaves of sets.

    Comment #1752 by Johan (site) on December 15, 2015 a 7:02 pm UTC

    THanks, fixed here.

    There are also 2 comments on Section 53.29: Étale Cohomology.

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