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Tag 09V3

Chapter 20: Cohomology of Sheaves > Section 20.17: Cohomology on Hausdorff quasi-compact spaces

Lemma 20.17.3. Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. For every abelian sheaf $\mathcal{F}$ on $X$ the canonical map $$ \mathop{\mathrm{colim}}\nolimits H^p(U, \mathcal{F}) \longrightarrow H^p(Z, \mathcal{F}|_Z) $$ where the colimit is over open neighbourhoods $U$ of $Z$ in $X$ is an isomorphism.

Proof. We first prove this for $p = 0$. Injectivity follows from the definition of $\mathcal{F}|_Z$ and holds in general (for any subset of any topological space $X$). Next, suppose that $s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$ such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$ comes from $s_i \in \mathcal{F}(U_i)$. It follows that there exist opens $W_{ij} \subset U_i \cap U_j$ with $W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that $s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying Topology, Lemma 5.13.7 we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and such that $V_i \cap V_j \subset W_{ij}$. Hence we see that $s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the open neighbourhood $\bigcup V_i$ of $Z$.

To finish the proof, it suffices to show that if $\mathcal{I}$ is an injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$ for $p > 0$. This follows using short exact sequences and dimension shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$. By Lemma 20.17.2 the element $\overline{\xi}$ comes from $\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$ for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$. Say $\overline{\xi}$ is the image of the class of a cocycle $\xi = (\xi_{i_0 \ldots i_p})$ in $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$.

Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf defined by the rule $$ \mathcal{I}'(V) = \{s \in \mathcal{I}|_Z(V) \mid \exists (U, t),~U \subset X\text{ open}, ~t \in \mathcal{I}(U),~V = Z \cap U,~s = t|_{Z \cap U} \} $$ Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$. Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$ such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is a section of $\mathcal{I}'$. Applying Topology, Lemma 5.13.5 we may after refining $\mathcal{V}$ assume that each $\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$.

Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$. Since $\mathcal{I}$ is flasque (Lemma 20.13.2) and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$ for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$ which restricts to $\xi_{i_0 \ldots i_p}$ on $V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$. (This appeal to injectives being flasque can be avoided by an additional application of Topology, Lemma 5.13.7.) Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain for the open covering $U = \bigcup U_i$. Since $\text{d}(\xi) = 0$ we see that the sections $\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial remarks of the proof, there exists open subsets $W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$ with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$ such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$. By Topology, Lemma 5.13.7 we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$ and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$. Then $s' = (s'_{i_0 \ldots i_p})$ with $s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$ is a cocycle for $\mathcal{I}$ for the open covering $U' = \bigcup U'_i$ of an open neighbourhood of $Z$. Since $\mathcal{I}$ has trivial higher Čech cohomology groups (Lemma 20.12.1) we conclude that $s'$ is a coboundary. It follows that the image of $\xi$ in the Čech complex for the open covering $Z = \bigcup Z \cap U'_i$ is a coboundary and we are done. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 2790–2806 (see updates for more information).

    \begin{lemma}
    \label{lemma-cohomology-of-closed}
    \begin{reference}
    \cite[Expose V bis, 4.1.3]{SGA4}
    \end{reference}
    Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset
    such that any two points of $Z$ have disjoint open neighbourhoods in $X$.
    For every abelian sheaf $\mathcal{F}$ on $X$ the canonical
    map
    $$
    \colim H^p(U, \mathcal{F})
    \longrightarrow
    H^p(Z, \mathcal{F}|_Z)
    $$
    where the colimit is over open neighbourhoods $U$ of $Z$ in $X$
    is an isomorphism.
    \end{lemma}
    
    \begin{proof}
    We first prove this for $p = 0$. Injectivity follows from
    the definition of $\mathcal{F}|_Z$ and holds in general
    (for any subset of any topological space $X$). Next, suppose that
    $s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$
    such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$
    comes from $s_i \in \mathcal{F}(U_i)$. It follows that
    there exist opens $W_{ij} \subset U_i \cap U_j$ with
    $W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that
    $s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying
    Topology, Lemma
    \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}
    we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and
    such that $V_i \cap V_j \subset W_{ij}$. Hence we see that
    $s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the
    open neighbourhood $\bigcup V_i$ of $Z$.
    
    \medskip\noindent
    To finish the proof, it suffices to show that if $\mathcal{I}$ is an
    injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$
    for $p > 0$. This follows using short exact sequences and dimension
    shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element
    of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$.
    By Lemma \ref{lemma-cech-Hausdorff-quasi-compact}
    the element $\overline{\xi}$ comes from
    $\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$
    for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$.
    Say $\overline{\xi}$ is the image of the class of a cocycle
    $\xi = (\xi_{i_0 \ldots i_p})$ in
    $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$.
    
    \medskip\noindent
    Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf
    defined by the rule
    $$
    \mathcal{I}'(V) =
    \{s \in \mathcal{I}|_Z(V) \mid
    \exists (U, t),\ U \subset X\text{ open},
    \ t \in \mathcal{I}(U),\ V = Z \cap U,\ s = t|_{Z \cap U} \}
    $$
    Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$.
    Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an
    open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$
    such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is
    a section of $\mathcal{I}'$. Applying
    Topology, Lemma \ref{topology-lemma-refine-covering}
    we may after refining $\mathcal{V}$ assume that each
    $\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$.
    
    \medskip\noindent
    Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$.
    Since $\mathcal{I}$ is flasque (Lemma \ref{lemma-injective-flasque})
    and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$
    for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose
    a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$
    which restricts to $\xi_{i_0 \ldots i_p}$ on
    $V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$.
    (This appeal to injectives being flasque can be avoided by an
    additional application of
    Topology, Lemma
    \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}.)
    Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain
    for the open covering $U = \bigcup U_i$.
    Since $\text{d}(\xi) = 0$ we see that the sections
    $\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero
    on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial
    remarks of the proof, there exists open subsets
    $W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$
    with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$
    such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$.
    By Topology, Lemma
    \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}
    we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$
    and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$.
    Then $s' = (s'_{i_0 \ldots i_p})$ with
    $s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$
    is a cocycle for $\mathcal{I}$ for the open covering
    $U' = \bigcup U'_i$ of an open neighbourhood of $Z$.
    Since $\mathcal{I}$ has trivial higher {\v C}ech cohomology groups
    (Lemma \ref{lemma-injective-trivial-cech})
    we conclude that $s'$ is a coboundary. It follows that the image of
    $\xi$ in the {\v C}ech complex for the open covering
    $Z = \bigcup Z \cap U'_i$ is a coboundary and we are done.
    \end{proof}

    References

    [SGA4, Expose V bis, 4.1.3]

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