Lemma 13.16.4. Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories and assume $RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B})$ is everywhere defined. Let $A$ be an object of $\mathcal{A}$.

$A$ is right acyclic for $F$ if and only if $F(A) \to R^0F(A)$ is an isomorphism and $R^ iF(A) = 0$ for all $i > 0$,

if $F$ is left exact, then $A$ is right acyclic for $F$ if and only if $R^ iF(A) = 0$ for all $i > 0$.

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