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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→B be an additive functor. For every n∈Z, the additive functors LnPF and LnQF 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
1.1L2given

By [L2], each object A carries an isomorphism θP,Q(A):LnPF(A)→∼LnQF(A).

2.1L1L3step 1.1∎

By [L3], the family from step 1.1 is natural. Together with [L1], this is exactly a natural isomorphism of additive functors LnPF⇒LnQF.

Depends on

Dependency tree · 2 levels

Sources