Acyclic object

Summary

For a left exact functor $F$, an object $X$ is $F$-acyclic if all the higher right derived functors $R^n F(X)$ with $n \ge 1$ vanish. Injective objects are acyclic for every such $F$, but many more objects are acyclic for a given $F$. Derived functors can be computed with resolutions by $F$-acyclic objects, which are often much easier to find than injective resolutions.