Definition:Acyclic Object

Definition

Let $\mathbf A$ be an abelian category with enough injectives.

Let $\mathbf B$ be an abelian category.

Let $F : \mathbf A \to \mathbf B$ be a left exact functor.

Let $X$ be an object of $\mathbf A$.

Then $X$ is $F$-acyclic if and only if $\mathrm R^i \map F X = 0$ for all positive integers $i \in \Z_{i \mathop \ge 1}$.

In the above $\mathrm R^i F$ denotes the $i$-th right derived functor of $F$.