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*}

20.4 Derived functors

We briefly explain an approach to right derived functors using resolution functors. Let $(X, \mathcal{O}_ X)$ be a ringed space. The category $\textit{Mod}(\mathcal{O}_ X)$ is abelian, see Modules, Lemma 17.3.1. In this chapter we will write

\[ K(X) = K(\mathcal{O}_ X) = K(\textit{Mod}(\mathcal{O}_ X)) \quad \text{and} \quad D(X) = D(\mathcal{O}_ X) = D(\textit{Mod}(\mathcal{O}_ X)). \]

and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition 13.8.1 and Definition 13.11.3. By Derived Categories, Remark 13.24.3 there exists a resolution functor

\[ j = j_ X : K^{+}(\textit{Mod}(\mathcal{O}_ X)) \longrightarrow K^{+}(\mathcal{I}) \]

where $\mathcal{I}$ is the strictly full additive subcategory of $\textit{Mod}(\mathcal{O}_ X)$ consisting of injective sheaves. For any left exact functor $F : \textit{Mod}(\mathcal{O}_ X) \to \mathcal{B}$ into any abelian category $\mathcal{B}$ we will denote $RF$ the right derived functor described in Derived Categories, Section 13.20 and constructed using the resolution functor $j_ X$ just described:

20.4.0.1
\begin{equation} \label{cohomology-equation-RF} RF = F \circ j_ X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) \end{equation}

see Derived Categories, Lemma 13.25.1 for notation. Note that we may think of $RF$ as defined on $\textit{Mod}(\mathcal{O}_ X)$, $\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_ X))$, $K^{+}(X)$, or $D^{+}(X)$ depending on the situation. According to Derived Categories, Definition 13.17.2 we obtain the $i$th right derived functor

20.4.0.2
\begin{equation} \label{cohomology-equation-RFi} R^ iF = H^ i \circ RF : \textit{Mod}(\mathcal{O}_ X) \longrightarrow \mathcal{B} \end{equation}

so that $R^0F = F$ and $\{ R^ iF, \delta \} _{i \geq 0}$ is universal $\delta $-functor, see Derived Categories, Lemma 13.20.4.

Here are two special cases of this construction. Given a ring $R$ we write $K(R) = K(\text{Mod}_ R)$ and $D(R) = D(\text{Mod}_ R)$ and similarly for bounded versions. For any open $U \subset X$ we have a left exact functor $ \Gamma (U, -) : \textit{Mod}(\mathcal{O}_ X) \longrightarrow \text{Mod}_{\mathcal{O}_ X(U)} $ which gives rise to

20.4.0.3
\begin{equation} \label{cohomology-equation-total-derived-cohomology} R\Gamma (U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_ X(U)) \end{equation}

by the discussion above. We set $H^ i(U, -) = R^ i\Gamma (U, -)$. If $U = X$ we recover (20.3.0.3). If $f : X \to Y$ is a morphism of ringed spaces, then we have the left exact functor $ f_* : \textit{Mod}(\mathcal{O}_ X) \longrightarrow \textit{Mod}(\mathcal{O}_ Y) $ which gives rise to the derived pushforward

20.4.0.4
\begin{equation} \label{cohomology-equation-total-derived-direct-image} Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) \end{equation}

The $i$th cohomology sheaf of $Rf_*\mathcal{F}^\bullet $ is denoted $R^ if_*\mathcal{F}^\bullet $ and called the $i$th higher direct image in accordance with (20.3.0.4). The two displayed functors above are exact functors of derived categories.

Abuse of notation: When the functor $Rf_*$, or any other derived functor, is applied to a sheaf $\mathcal{F}$ on $X$ or a complex of sheaves it is understood that $\mathcal{F}$ has been replaced by a suitable resolution of $\mathcal{F}$. To facilitate this kind of operation we will say, given an object $\mathcal{F}^\bullet \in D(X)$, that a bounded below complex $\mathcal{I}^\bullet $ of injectives of $\textit{Mod}(\mathcal{O}_ X)$ represents $\mathcal{F}^\bullet $ in the derived category if there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet $. In the same vein the phrase “let $\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet $ be a morphism of $D(X)$” does not mean that $\alpha $ is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.


Comments (1)

Comment #1810 by Keenan Kidwell on

In the second sentence of the text block following the displayed equation , "...are exact functor..." should be "...are exact functors..."


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