acyclic object

In mathematics, in the field of homological algebra, given an abelian category

$\backslash mathcal\{C\}$ having enough injectives and an additive (covariant) functor

:$F\; :\backslash mathcal\{C\}\backslash to\backslash mathcal\{D\}$,

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

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

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

$F$.{{cite book | last=Caenepeel | first=Stefaan | title=Brauer groups, Hopf algebras and Galois theory | zbl=0898.16001 | series=Monographs in Mathematics | volume=4 | location=Dordrecht | publisher=Kluwer Academic Publishers | year=1998 | isbn=1-4020-0346-3 | page=454 }}