Acyclic object

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

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

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

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

where ${\displaystyle {\rm {R}}^{i}F}${\displaystyle {\rm {R}}^{i}F} are the right derived functors of ${\displaystyle F}${\displaystyle F}.^[1]

• Homological algebra