nLab acyclic object

Definition

In homological algebra:

For $F : \mathcal{A} \to \mathcal{B}$ a left exact additive functor between abelian categories, an object $A \in \mathcal{A}$ is $F$-acyclic if the right derived functor of $F$ has no cohomology on $A$ in positive degree

$$(p \gt 0) \Rightarrow R^p F A = 0 \,.$$

Properties

A resolution by $F$-acyclic objects serves to compute the derived functor of $F$. See at derived functor in homological algebra – Via acyclic resolutions