Lemma 13.16.5. Let $F : \mathcal{A} \to \mathcal{B}$ be a left exact functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $0 \to A \to B \to C \to 0$ be a short exact sequence of $\mathcal{A}$.

If $A$ and $C$ are right acyclic for $F$ then so is $B$.

If $A$ and $B$ are right acyclic for $F$ then so is $C$.

If $B$ and $C$ are right acyclic for $F$ and $F(B) \to F(C)$ is surjective then $A$ is right acyclic for $F$.

In each of the three cases

is a short exact sequence of $\mathcal{B}$.

## Comments (2)

Comment #8401 by ElĂas Guisado on

Comment #9013 by Stacks project on