20.3 Derived functors
We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section 20.28.
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
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
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:
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 13.16.2 we obtain the $i$th right derived functor
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
by the discussion above. We set $H^ i(U, -) = R^ i\Gamma (U, -)$. If $U = X$ we recover (20.2.0.3). 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
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 (20.2.0.4). 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functor, is applied to a sheaf $\mathcal{F}$ on $X$ or a complex of sheaves it is understood that $\mathcal{F}$ has been replaced by a suitable resolution of $\mathcal{F}$.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 $\mathcal{F}^\bullet \in D(\mathcal{O}_ X)$, 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 $D(\mathcal{O}_ X)$” does not mean that $\alpha $ is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.
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Comment #1810 by Keenan Kidwell on