The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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 7.29.5 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. $\square$


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