Acyclic object

In mathematics, in the field of homological algebra, given an abelian category $$\mathcal{C}$$ having enough injectives and an additive (covariant) functor


 * $$F :\mathcal{C}\to\mathcal{D}$$,

an acyclic object with respect to $$F$$, or simply an $$F$$-acyclic object, is an object $$A$$ in $$\mathcal{C}$$ such that


 * $$ {\rm R}^i F (A) = 0 \,\!$$ for all $$ i>0 \,\!$$,

where $${\rm R}^i F$$ are the right derived functors of $$F$$.