nLab
acyclic object

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

$$(p>0)\Rightarrow R^{p}FA=0.$$(p \gt 0) \Rightarrow R^p F A = 0 \,.