The Stacks project

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.

  1. The functor $RF$ is an exact functor $D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

  2. The functor $RF$ induces an exact functor $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

  3. The functor $RF$ induces a $\delta $-functor $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$.

  4. 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:

  1. The derived functor is exact: This boils down to Lemma 13.14.6.

  2. This is true because $K^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is exact and compositions of exact functors are exact.

  3. This is true because $\text{Comp}^{+}(\mathcal{A}) \to D^{+}(\mathcal{A})$ is a $\delta $-functor, see Lemma 13.12.1.

  4. 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.


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