LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The Ext balance isomorphism is independent of resolution comparison data

Statement

Assume the Axiom of Dependent Choice. Under the enough-projectives and enough-injectives hypotheses used for the two supplied derived constructions, the balance isomorphism ${Ext}_{P}^{n} (M, N) ≅ {Ext}_{I}^{n} (M, N)$ is independent of the comparison lifts used after changing either supplied resolution.

Facts & Assumptions

Given: The supplied projective and injective resolution constructions of $Ext$ .

Proof

technique · direct

1.1

For fixed resolutions, the two edge maps defining the balance zigzag are induced by the augmentations $P_{∙} \to M$ and $N \to I^{∙}$ , so they involve no comparison lift. After replacing a projective or injective resolution, choose a comparison map over or under the resolved object. These maps give a morphism between the two Hom double complexes and commute with both edge augmentations.

givenconstruct

2.1

Any two projective comparison maps are chain-homotopic, and any two injective comparison maps are cochain-homotopic, by the two comparison uniqueness theorems. Applying $Hom$ turns either homotopy into a homotopy of the corresponding total-complex maps. Hence the induced maps on all three cohomologies in the edge-to-total zigzag are independent of the chosen lifts, and the balance isomorphism is independent of those choices.

step 1.1algebra∎

Depends on

Used by

The balanced Ext bifunctor Definition
Ext balance isomorphisms satisfy change-of-resolution cocycle laws Proposition
The Ext balance isomorphism is natural in both variables 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 (standard reference, not scraped)