nLab acyclic object

Definition

In homological algebra:

For $F : \mathcal{A} \to \mathcal{B}$ a left exact additive functor between abelian categories, an object $A \in \mathcal{A}$ is $F$-acyclic if the right derived functor of $F$ has no cohomology on $A$ in positive degree

$$(p \gt 0) \Rightarrow R^p F A = 0 \,.$$

Properties

A resolution by $F$-acyclic objects serves to compute the derived functor of $F$. See at derived functor in homological algebra – Via acyclic resolutions

References

See most references on homological algebra.

Also:

• Michael Barr, Acyclic models, Canadian J. Math. 48 2 (1996) 258–273 [doi:10.4153/CJM-1996-013-x, pdf, pdf]

• Michael Barr, Acyclic models, CRM Monograph Series 17 (2002) [ams:crmm-17, pdf, pdf]