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

Used by

An F-acyclic resolution Definition
The acyclic-resolution theorem for left derived functors Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)