DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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An acyclic object for a right exact functor

Definition

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

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

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

Depends on

Dependency tree · 2 levels

Sources

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