Section 20.3 (0716): Derived functors

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

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

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:

20.3.0.1

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

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 13.16.2 we obtain the $i$ th right derived functor

20.3.0.2

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

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

20.3.0.3

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

by the discussion above. We set $H^ i(U, -) = R^ i\Gamma (U, -)$ . If $U = X$ we recover (20.2.0.3). 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

20.3.0.4

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

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 (20.2.0.4). 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.

Comments (1)

Comment #1810 by Keenan Kidwell on January 30, 2016 at 17:53

In the second sentence of the text block following the displayed equation $Rf_*:D^+(X)\to D^+(Y)$ , "...are exact functor..." should be "...are exact functors..."

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20.3 Derived functors

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