The Stacks project

21.3 Derived functors

We briefly explain an approach to right derived functors using resolution functors. Namely, suppose that $(\mathcal{C}, \mathcal{O})$ is a ringed site. 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 $\textit{Mod}(\mathcal{O})$ which consists of injective $\mathcal{O}$-modules. For any left exact functor $F : \textit{Mod}(\mathcal{O}) \to \mathcal{B}$ into any abelian category $\mathcal{B}$ we will denote $RF$ the right derived functor of Derived Categories, Section 13.20 constructed using the resolution functor $j$ just described:

see Derived Categories, Lemma 13.25.1 for notation. Note that we may think of $RF$ as defined on $\textit{Mod}(\mathcal{O})$, $\text{Comp}^{+}(\textit{Mod}(\mathcal{O}))$, or $K^{+}(\mathcal{O})$ depending on the situation. According to Derived Categories, Definition 13.16.2 we obtain the $i$the 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 the bounded versions. For any object $U$ of $\mathcal{C}$ have a left exact functor $ \Gamma (U, -) : \textit{Mod}(\mathcal{O}) \longrightarrow \text{Mod}_{\mathcal{O}(U)} $ which gives rise to

by the discussion above. Note that $H^ i(U, -) = R^ i\Gamma (U, -)$ is compatible with (21.2.0.5) above. We similarly have

compatible with (21.2.0.6). If $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is a morphism of ringed topoi then we get a left exact functor $f_* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')$ which gives rise to 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 (21.2.0.7). The displayed functors above are exact functor of derived categories.