13.20 Right derived functors and injective resolutions
At this point we can use the material above to define the right derived functors of an additive functor between an abelian category having enough injectives and a general abelian category.
Lemma 13.20.1. Let $\mathcal{A}$ be an abelian category. Let $I \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be an injective object. Let $I^\bullet $ be a bounded below complex of injectives in $\mathcal{A}$.
$I^\bullet $ computes $RF$ relative to $\text{Qis}^{+}(\mathcal{A})$ for any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into any triangulated category $\mathcal{D}$.
$I$ is right acyclic for any additive functor $F : \mathcal{A} \to \mathcal{B}$ into any abelian category $\mathcal{B}$.
Proof.
Part (2) is a direct consequences of part (1) and Definition 13.15.3. To prove (1) let $\alpha : I^\bullet \to K^\bullet $ be a quasi-isomorphism into a complex. By Lemma 13.18.6 we see that $\alpha $ has a left inverse. Hence the category $I^\bullet /\text{Qis}^{+}(\mathcal{A})$ is essentially constant with value $\text{id} : I^\bullet \to I^\bullet $. Thus also the ind-object
\[ I^\bullet /\text{Qis}^{+}(\mathcal{A}) \longrightarrow \mathcal{D}, \quad (I^\bullet \to K^\bullet ) \longmapsto F(K^\bullet ) \]
is essentially constant with value $F(I^\bullet )$. This proves (1), see Definitions 13.14.2 and 13.14.10.
$\square$
Lemma 13.20.2. Let $\mathcal{A}$ be an abelian category with enough injectives.
For any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into a triangulated category $\mathcal{D}$ the right derived functor
\[ RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D} \]
is everywhere defined.
For any additive functor $F : \mathcal{A} \to \mathcal{B}$ into an abelian category $\mathcal{B}$ the right derived functor
\[ RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B}) \]
is everywhere defined.
Proof.
Combine Lemma 13.20.1 and Proposition 13.16.8 for the second assertion. To see the first assertion combine Lemma 13.18.3, Lemma 13.20.1, Lemma 13.14.14, and Equation (13.14.9.1).
$\square$
Lemma 13.20.3. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor.
The functor $RF$ is an exact functor $D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.
The functor $RF$ induces an exact functor $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.
The functor $RF$ induces a $\delta $-functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.
The functor $RF$ induces a $\delta $-functor $\mathcal{A} \to D^{+}(\mathcal{B})$.
Proof.
This lemma simply reviews some of the results obtained so far. Note that by Lemma 13.20.2 $RF$ is everywhere defined. Here are some references:
The derived functor is exact: This boils down to Lemma 13.14.6.
This is true because $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is exact and compositions of exact functors are exact.
This is true because $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is a $\delta $-functor, see Lemma 13.12.1.
This is true because $\mathcal{A} \to \text{Comp}^{+}(\mathcal{A})$ is exact and precomposing a $\delta $-functor by an exact functor gives a $\delta $-functor.
$\square$
Lemma 13.20.4. Let $\mathcal{A}$ be an abelian category with enough injectives. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor.
For any short exact sequence $0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$ of complexes in $\text{Comp}^{+}(\mathcal{A})$ there is an associated long exact sequence
\[ \ldots \to H^ i(RF(A^\bullet )) \to H^ i(RF(B^\bullet )) \to H^ i(RF(C^\bullet )) \to H^{i + 1}(RF(A^\bullet )) \to \ldots \]
The functors $R^ iF : \mathcal{A} \to \mathcal{B}$ are zero for $i < 0$. Also $R^0F = F : \mathcal{A} \to \mathcal{B}$.
We have $R^ iF(I) = 0$ for $i > 0$ and $I$ injective.
The sequence $(R^ iF, \delta )$ forms a universal $\delta $-functor (see Homology, Definition 12.12.3) from $\mathcal{A}$ to $\mathcal{B}$.
Proof.
This lemma simply reviews some of the results obtained so far. Note that by Lemma 13.20.2 $RF$ is everywhere defined. Here are some references:
This follows from Lemma 13.20.3 part (3) combined with the long exact cohomology sequence (13.11.1.1) for $D^{+}(\mathcal{B})$.
This is Lemma 13.16.3.
This is the fact that injective objects are acyclic.
This is Lemma 13.16.6.
$\square$
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