Acyclic object

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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 $R i F$ are the right derived functors of $F$ .

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