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 $F$.

References

• Caenepeel, Stefaan (1998). Brauer groups, Hopf algebras and Galois theory. Monographs in Mathematics 4. Dordrecht: Kluwer Academic Publishers. p. 454. ISBN 1-4020-0346-3. Zbl 0898.16001.