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

How statement and proof provenance work

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Left derived objects relative to supplied projective resolution data

Definition

Let $P$ be a supplied projective resolution datum on a class $D$ of objects in an abelian category $A$ , and let $F : A \to B$ be an additive functor to an abelian category $B$ .

For $A \in D$ and $n \in Z$ , the $n$ th left derived object of $F$ relative to $P$ at $A$ is $L_{n}^{P} F (A) : = H_{n} ⁣ (F (P (A)_{d e l})),$ where $P (A)_{d e l}$ is the deleted resolution from Deleted resolutions.

No exactness hypothesis on $F$ is needed for this definition. The datum $P$ supplies the chosen resolution whose image under $F$ is being measured by homology.

Depends on

Used by

L₀ of a non-right-exact functor need not recover the functor Counterexample
An acyclic object for a right exact functor Definition
The left derived map relative to supplied resolution data Definition
FALSE: every additive functor has L₀ naturally isomorphic to itself False statement
Objectwise comparison of two projective resolution data induces an isomorphism on derived objects Lemma
The induced homology map is independent of the chosen comparison lift Lemma
A bifunctor can be derived in either variable when the relevant resolution data are supplied Proposition
An exact functor has vanishing positive derived functors Proposition
Negative derived degrees vanish for one-sided resolutions Proposition
Positive left derived functors vanish on projective objects Proposition

Dependency tree · two levels

16 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)