The Stacks project

Lemma 13.16.9. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor of abelian categories. Then

1. every object of $\mathcal{A}$ is right acyclic for $F$,

2. $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined,

3. $RF : D(\mathcal{A}) \to D(\mathcal{B})$ is everywhere defined,

4. every complex computes $RF$, in other words, the canonical map $F(K^\bullet ) \to RF(K^\bullet )$ is an isomorphism for all complexes, and

5. $R^ iF = 0$ for $i \not= 0$.