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*}

21.3 Cohomology of sheaves

Let $\mathcal{C}$ be a site, see Sites, Definition 7.6.2. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. We know that the category of abelian sheaves on $\mathcal{C}$ has enough injectives, see Injectives, Theorem 19.7.4. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet $. For any object $U$ of the site $\mathcal{C}$ we define

21.3.0.1
\begin{equation} \label{sites-cohomology-equation-cohomology-object-site} H^ i(U, \mathcal{F}) = H^ i(\Gamma (U, \mathcal{I}^\bullet )) \end{equation}

to be the $i$th cohomology group of the abelian sheaf $\mathcal{F}$ over the object $U$. In other words, these are the right derived functors of the functor $\mathcal{F} \mapsto \mathcal{F}(U)$. The family of functors $H^ i(U, -)$ forms a universal $\delta $-functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$.

It sometimes happens that the site $\mathcal{C}$ does not have a final object. In this case we define the global sections of a presheaf of sets $\mathcal{F}$ over $\mathcal{C}$ to be the set

21.3.0.2
\begin{equation} \label{sites-cohomology-equation-global-sections} \Gamma (\mathcal{C}, \mathcal{F}) = \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(e, \mathcal{F}) \end{equation}

where $e$ is a final object in the category of presheaves on $\mathcal{C}$. In this case, given an abelian sheaf $\mathcal{F}$ on $\mathcal{C}$, we define the $i$th cohomology group of $\mathcal{F}$ on $\mathcal{C}$ as follows

21.3.0.3
\begin{equation} \label{sites-cohomology-equation-cohomology} H^ i(\mathcal{C}, \mathcal{F}) = H^ i(\Gamma (\mathcal{C}, \mathcal{I}^\bullet )) \end{equation}

in other words, it is the $i$th right derived functor of the global sections functor. The family of functors $H^ i(\mathcal{C}, -)$ forms a universal $\delta $-functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$.

Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi, see Sites, Definition 7.15.1. With $\mathcal{F}[0] \to \mathcal{I}^\bullet $ as above we define

21.3.0.4
\begin{equation} \label{sites-cohomology-equation-higher-direct-image} R^ if_*\mathcal{F} = H^ i(f_*\mathcal{I}^\bullet ) \end{equation}

to be the $i$th higher direct image of $\mathcal{F}$. These are the right derived functors of $f_*$. The family of functors $R^ if_*$ forms a universal $\delta $-functor from $\textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site, see Modules on Sites, Definition 18.6.1. Let $\mathcal{F}$ be an $\mathcal{O}$-module. We know that the category of $\mathcal{O}$-modules has enough injectives, see Injectives, Theorem 19.8.4. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet $. For any object $U$ of the site $\mathcal{C}$ we define

21.3.0.5
\begin{equation} \label{sites-cohomology-equation-cohomology-object-site-modules} H^ i(U, \mathcal{F}) = H^ i(\Gamma (U, \mathcal{I}^\bullet )) \end{equation}

to be the the $i$th cohomology group of $\mathcal{F}$ over $U$. The family of functors $H^ i(U, -)$ forms a universal $\delta $-functor $\textit{Mod}(\mathcal{O}) \to \text{Mod}_{\mathcal{O}(U)}$. Similarly

21.3.0.6
\begin{equation} \label{sites-cohomology-equation-cohomology-modules} H^ i(\mathcal{C}, \mathcal{F}) = H^ i(\Gamma (\mathcal{C}, \mathcal{I}^\bullet )) \end{equation}

it the $i$th cohomology group of $\mathcal{F}$ on $\mathcal{C}$. The family of functors $H^ i(\mathcal{C}, -)$ forms a universal $\delta $-functor $\textit{Mod}(\mathcal{C}) \to \text{Mod}_{\Gamma (\mathcal{C}, \mathcal{O})}$.

Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi, see Modules on Sites, Definition 18.7.1. With $\mathcal{F}[0] \to \mathcal{I}^\bullet $ as above we define

21.3.0.7
\begin{equation} \label{sites-cohomology-equation-higher-direct-image-modules} R^ if_*\mathcal{F} = H^ i(f_*\mathcal{I}^\bullet ) \end{equation}

to be the $i$th higher direct image of $\mathcal{F}$. These are the right derived functors of $f_*$. The family of functors $R^ if_*$ forms a universal $\delta $-functor from $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')$.


Comments (2)

Comment #3238 by Dario WeiƟmann on

In the second sentence fo the introduction: a abelian


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