Acyclic object

Remarks

Example: every injective object $I$ is $F$-acyclic, because $0 \to I \to I \to 0$ is an injective resolution, and applying $F$ gives a complex with no cohomology in degrees $n \ge 1$. Non-example: for $F = \Hom_\Z(\Z/2\Z, -)$ on abelian groups, the group $\Z$ is not $F$-acyclic, since $R^1 F(\Z) = \operatorname{Ext}^1_\Z(\Z/2\Z, \Z) \cong \Z/2\Z \ne 0$. For $F = - \otimes_R N$, every flat module is $F$-acyclic, since $\operatorname{Tor}_n^R(M, N) = 0$ for flat $M$ and $n \ge 1$. If $X \to J^\bullet$ is a resolution by $F$-acyclic objects, then $R^n F(X) \cong H^n(F(J^\bullet))$ for every $n$.