TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Two supplied projective resolution data define naturally isomorphic left derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $P$ and $Q$ be supplied projective resolution data on the same domain, and let $F : A \to B$ be an additive functor. For every $n \in Z$ , the additive functors $L_{n}^{P} F$ and $L_{n}^{Q} F$ are naturally isomorphic.

Facts & Assumptions

Given: An integer $n$ .

[L1]

Both constructions define additive functors (Left derived functors relative to supplied data are additive functors).

[L2]

For each object, objectwise comparison of the two chosen resolutions induces an isomorphism on derived objects (Objectwise comparison of two projective resolution data induces an isomorphism on derived objects).

[L3]

Those isomorphisms are natural in the object (The change-of-projective-resolution isomorphisms are natural).

Proof

technique · direct

1.1

By [L2], each object $A$ carries an isomorphism $θ_{P, Q} (A) : L_{n}^{P} F (A) \tilde{\to} L_{n}^{Q} F (A)$ .

L2given

2.1

By [L3], the family from step 1.1 is natural. Together with [L1], this is exactly a natural isomorphism of additive functors $L_{n}^{P} F \Rightarrow L_{n}^{Q} F$ .

L1L3step 1.1∎

Depends on

Used by

An acyclic object for a right exact functor Definition
Two resolution data and their change isomorphism Example
Change-of-projective-resolution isomorphisms satisfy identity and cocycle laws Proposition
Positive left derived functors vanish on projective objects Proposition
Derived functors are well defined relative to supplied resolution data Remark
The acyclic-resolution theorem for left derived functors Theorem

Dependency tree · two levels

17 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)