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Tag 03FD

Chapter 21: Cohomology on Sites > Section 21.13: Cohomology of modules

Lemma 21.13.4. Let $\mathcal{C}$ be a site. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$. Let $\mathcal{F}$ be an $\mathcal{O}$-module, and denote $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups. Then we have $$ H^i(\mathcal{C}, \mathcal{F}_{ab}) = H^i(\mathcal{C}, \mathcal{F}) $$ and for any object $U$ of $\mathcal{C}$ we also have $$ H^i(U, \mathcal{F}_{ab}) = H^i(U, \mathcal{F}). $$ Here the left hand side is cohomology computed in $\textit{Ab}(\mathcal{C})$ and the right hand side is cohomology computed in $\textit{Mod}(\mathcal{O})$.

Proof. By Derived Categories, Lemma 13.20.4 the $\delta$-functor $(\mathcal{F} \mapsto H^p(U, \mathcal{F}))_{p \geq 0}$ is universal. The functor $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$, $\mathcal{F} \mapsto \mathcal{F}_{ab}$ is exact. Hence $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$ is a $\delta$-functor also. Suppose we show that $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$ is also universal. This will imply the second statement of the lemma by uniqueness of universal $\delta$-functors, see Homology, Lemma 12.11.5. Since $\textit{Mod}(\mathcal{O})$ has enough injectives, it suffices to show that $H^i(U, \mathcal{I}_{ab}) = 0$ for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O})$, see Homology, Lemma 12.11.4.

Let $\mathcal{I}$ be an injective object of $\textit{Mod}(\mathcal{O})$. Apply Lemma 21.11.9 with $\mathcal{F} = \mathcal{I}$, $\mathcal{B} = \mathcal{C}$ and $\text{Cov} = \text{Cov}_\mathcal{C}$. Assumption (3) of that lemma holds by Lemma 21.13.3. Hence we see that $H^i(U, \mathcal{I}_{ab}) = 0$ for every object $U$ of $\mathcal{C}$.

If $\mathcal{C}$ has a final object then this also implies the first equality. If not, then according to Sites, Lemma 7.28.5 we see that the ringed topos $(\mathop{\mathit{Sh}}\nolimits(\mathcal{C}), \mathcal{O})$ is equivalent to a ringed topos where the underlying site does have a final object. Hence the lemma follows. $\square$

    The code snippet corresponding to this tag is a part of the file sites-cohomology.tex and is located in lines 1882–1903 (see updates for more information).

    \begin{lemma}
    \label{lemma-cohomology-modules-abelian-agree}
    Let $\mathcal{C}$ be a site.
    Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}$.
    Let $\mathcal{F}$ be an $\mathcal{O}$-module, and denote
    $\mathcal{F}_{ab}$ the underlying sheaf of abelian groups.
    Then we have
    $$
    H^i(\mathcal{C}, \mathcal{F}_{ab})
    =
    H^i(\mathcal{C}, \mathcal{F})
    $$
    and for any object $U$ of $\mathcal{C}$ we also have
    $$
    H^i(U, \mathcal{F}_{ab})
    =
    H^i(U, \mathcal{F}).
    $$
    Here the left hand side is cohomology computed in
    $\textit{Ab}(\mathcal{C})$ and the right hand side
    is cohomology computed in $\textit{Mod}(\mathcal{O})$.
    \end{lemma}
    
    \begin{proof}
    By
    Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}
    the $\delta$-functor $(\mathcal{F} \mapsto H^p(U, \mathcal{F}))_{p \geq 0}$
    is universal. The functor
    $\textit{Mod}(\mathcal{O}) \to \textit{Ab}(\mathcal{C})$,
    $\mathcal{F} \mapsto \mathcal{F}_{ab}$ is exact. Hence
    $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$
    is a $\delta$-functor also. Suppose we show that
    $(\mathcal{F} \mapsto H^p(U, \mathcal{F}_{ab}))_{p \geq 0}$
    is also universal. This will imply the second statement of the lemma
    by uniqueness of universal $\delta$-functors, see
    Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor}.
    Since $\textit{Mod}(\mathcal{O})$ has enough injectives,
    it suffices to show that $H^i(U, \mathcal{I}_{ab}) = 0$
    for any injective object $\mathcal{I}$ in $\textit{Mod}(\mathcal{O})$, see
    Homology, Lemma \ref{homology-lemma-efface-implies-universal}.
    
    \medskip\noindent
    Let $\mathcal{I}$ be an injective object of $\textit{Mod}(\mathcal{O})$.
    Apply Lemma \ref{lemma-cech-vanish-collection}
    with $\mathcal{F} = \mathcal{I}$, $\mathcal{B} = \mathcal{C}$
    and $\text{Cov} = \text{Cov}_\mathcal{C}$.
    Assumption (3) of that lemma holds by
    Lemma \ref{lemma-injective-module-trivial-cech}.
    Hence we see that $H^i(U, \mathcal{I}_{ab}) = 0$
    for every object $U$ of $\mathcal{C}$.
    
    \medskip\noindent
    If $\mathcal{C}$ has a final
    object then this also implies the first equality. If not, then
    according to Sites, Lemma \ref{sites-lemma-topos-good-site} we see that
    the ringed topos $(\Sh(\mathcal{C}), \mathcal{O})$ is equivalent to a
    ringed topos where the underlying site does have a final object.
    Hence the lemma follows.
    \end{proof}

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