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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

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

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

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

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:
\begin{equation} \label{sites-cohomology-equation-RF} RF = F \circ j' : D^{+}(\mathcal{O}) \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 $\textit{Mod}(\mathcal{O})$, $\text{Comp}^{+}(\textit{Mod}(\mathcal{O}))$, or $K^{+}(\mathcal{O})$ depending on the situation. According to Derived Categories, Definition 13.16.2 we obtain the $i$the right derived functor
\begin{equation} \label{sites-cohomology-equation-RFi} R^ iF = H^ i \circ RF : \textit{Mod}(\mathcal{O}) \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 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

\[ R\Gamma (U, -) : D^{+}(\mathcal{O}) \longrightarrow D^{+}(\mathcal{O}(U)) \]

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

\[ R\Gamma (\mathcal{C}, -) : D^{+}(\mathcal{O}) \longrightarrow D^{+}(\Gamma (\mathcal{C}, \mathcal{O})) \]

compatible with ( 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

\[ Rf_* : D^{+}(\mathcal{O}) \to D^+(\mathcal{O}') \]

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 ( The displayed functors above are exact functor of derived categories.

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