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Tag 0A37

Chapter 20: Cohomology of Sheaves > Section 20.20: Cohomology and colimits

Lemma 20.20.2. In the situation discussed above. Let $i \in \mathop{\rm Ob}\nolimits(\mathcal{I})$ and let $U_i \subset X_i$ be quasi-compact open. Then $$ \mathop{\rm colim}\nolimits_{a : j \to i} H^p(f_a^{-1}(U_i), \mathcal{F}_j) = H^p(p_i^{-1}(U_i), \mathcal{F}) $$ for all $p \geq 0$. In particular we have $H^p(X, \mathcal{F}) = \mathop{\rm colim}\nolimits H^p(X_i, \mathcal{F}_i)$.

Proof. The case $p = 0$ is Sheaves, Lemma 6.29.4.

In this paragraph we show that we can find a map of systems $(\gamma_i) : (\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$ with $\mathcal{G}_i$ an injective abelian sheaf and $\gamma_i$ injective. For each $i$ we pick an injection $\mathcal{F}_i \to \mathcal{I}_i$ where $\mathcal{I}_i$ is an injective abelian sheaf on $X_i$. Then we can consider the family of maps $$ \gamma_i : \mathcal{F}_i \longrightarrow \prod\nolimits_{b : k \to i} f_{b, *}\mathcal{I}_k = \mathcal{G}_i $$ where the component maps are the maps adjoint to the maps $f_b^{-1}\mathcal{F}_i \to \mathcal{F}_k \to \mathcal{I}_k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map $$ \psi_a : f_a^{-1}\mathcal{G}_i \to \mathcal{G}_j $$ whose components are the canonical maps $f_b^{-1}f_{a \circ b, *}\mathcal{I}_k \to f_{b, *}\mathcal{I}_k$ for $b : k \to j$. Thus we find an injection $\{\gamma_i\} : \{\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$ of systems of abelian sheaves. Note that $\mathcal{G}_i$ is an injective sheaf of abelian groups on $\mathcal{C}_i$, see Lemma 20.12.11 and Homology, Lemma 12.24.3. This finishes the construction.

Arguing exactly as in the proof of Lemma 20.20.1 we see that it suffices to prove that $H^p(X, \mathop{\rm colim}\nolimits f_i^{-1}\mathcal{G}_i) = 0$ for $p > 0$.

Set $\mathcal{G} = \mathop{\rm colim}\nolimits f_i^{-1}\mathcal{G}_i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the Čech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma 20.12.9. Such a covering is the inverse by $p_i$ of such a covering $\mathcal{U}_i$ on the space $X_i$ for some $i$ by Topology, Lemma 5.24.6. We have $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) = \mathop{\rm colim}\nolimits_{a : j \to i} \check{\mathcal{C}}^\bullet(f_a^{-1}(\mathcal{U}_i), \mathcal{G}_j) $$ by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma 20.12.1. Thus we conclude by Algebra, Lemma 10.8.8. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 3269–3280 (see updates for more information).

    \begin{lemma}
    \label{lemma-colimit}
    In the situation discussed above.
    Let $i \in \Ob(\mathcal{I})$ and let $U_i \subset X_i$ be quasi-compact open.
    Then
    $$
    \colim_{a : j \to i} H^p(f_a^{-1}(U_i), \mathcal{F}_j) =
    H^p(p_i^{-1}(U_i), \mathcal{F})
    $$
    for all $p \geq 0$. In particular we have
    $H^p(X, \mathcal{F}) = \colim H^p(X_i, \mathcal{F}_i)$.
    \end{lemma}
    
    \begin{proof}
    The case $p = 0$ is Sheaves, Lemma \ref{sheaves-lemma-descend-opens}.
    
    \medskip\noindent
    In this paragraph we show that we can find a map of systems
    $(\gamma_i) : (\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$
    with $\mathcal{G}_i$ an injective abelian sheaf and $\gamma_i$ injective.
    For each $i$ we pick an injection $\mathcal{F}_i \to \mathcal{I}_i$
    where $\mathcal{I}_i$ is an injective abelian sheaf on $X_i$.
    Then we can consider the family of maps
    $$
    \gamma_i :
    \mathcal{F}_i
    \longrightarrow
    \prod\nolimits_{b : k \to i} f_{b, *}\mathcal{I}_k = \mathcal{G}_i
    $$
    where the component maps are the maps adjoint to the maps
    $f_b^{-1}\mathcal{F}_i \to \mathcal{F}_k \to \mathcal{I}_k$.
    For $a : j \to i$ in $\mathcal{I}$ there is a canonical map
    $$
    \psi_a : f_a^{-1}\mathcal{G}_i \to \mathcal{G}_j
    $$
    whose components are the canonical maps
    $f_b^{-1}f_{a \circ b, *}\mathcal{I}_k \to f_{b, *}\mathcal{I}_k$
    for $b : k \to j$. Thus we find an injection
    $\{\gamma_i\} : \{\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$
    of systems of abelian sheaves. Note that $\mathcal{G}_i$ is an injective
    sheaf of abelian groups on $\mathcal{C}_i$, see
    Lemma \ref{lemma-pushforward-injective-flat} and
    Homology, Lemma \ref{homology-lemma-product-injectives}.
    This finishes the construction.
    
    \medskip\noindent
    Arguing exactly as in the proof of
    Lemma \ref{lemma-quasi-separated-cohomology-colimit}
    we see that it suffices to prove that
    $H^p(X, \colim f_i^{-1}\mathcal{G}_i) = 0$ for $p > 0$.
    
    \medskip\noindent
    Set $\mathcal{G} = \colim f_i^{-1}\mathcal{G}_i$.
    To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact
    open of $X$, it suffices to show that the {\v C}ech cohomology of
    $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of
    $X$ by finitely many quasi-compact opens is zero, see
    Lemma \ref{lemma-cech-vanish-basis}.
    Such a covering is the inverse by $p_i$ of such a covering $\mathcal{U}_i$
    on the space $X_i$ for some $i$ by
    Topology, Lemma \ref{topology-lemma-descend-opens}. We have
    $$
    \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) =
    \colim_{a : j \to i}
    \check{\mathcal{C}}^\bullet(f_a^{-1}(\mathcal{U}_i), \mathcal{G}_j)
    $$
    by the case $p = 0$. The right hand side is a filtered colimit of
    complexes each of which is acyclic in positive degrees by
    Lemma \ref{lemma-injective-trivial-cech}. Thus we conclude by
    Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}.
    \end{proof}

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