DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-04

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An acyclic object for a left exact functor

Definition

Let $I$ be a supplied injective resolution datum on a class $D$ , and let $F : A \to B$ be an additive left exact functor between abelian categories.

An object $A \in D$ is $F$ -acyclic if $R_{I}^{n} F (A) = 0 for every n > 0.$

If $J$ is another supplied injective resolution datum on the same domain and one assumes the Axiom of Dependent Choice, then Two supplied injective resolution data define naturally isomorphic right derived functors gives natural isomorphisms $R_{I}^{n} F (A) ≅ R_{J}^{n} F (A)$ for every $n$ . Under that additional hypothesis, this vanishing condition is independent of the chosen supplied injective datum.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors`