PropositionStatement: 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 natural in both variables

Statement

Assume the Axiom of Dependent Choice and the supplied resolution hypotheses of the balance theorem. For every $n \geq 0$ , the balance maps $β_{M, N}^{n} : {Ext}_{P}^{n} (M, N) \to {Ext}_{I}^{n} (M, N)$ are natural in both $M$ and $N$ .

Facts & Assumptions

Given: Morphisms $u : M^{'} \to M$ and $v : N \to N^{'}$ in the resolved class.

Proof

technique · direct

1.1

Choose the projective and injective comparison maps for $u$ and $v$ . Projective-resolution Ext has the stated bifunctor variance and Injective-resolution Ext has the stated bifunctor variance give the two routes around the naturality square.

givenconstruct

2.1

The chosen comparison maps induce a morphism of Hom double complexes. Because its squares with the projective augmentation $P_{∙} \to M$ and the injective coaugmentation $N \to I^{∙}$ commute, both edge-to-total quasi-isomorphisms commute with it. Taking cohomology makes the balance square commute. The independence lemma removes the chosen comparison maps, proving naturality in both variables.

step 1.1algebra∎

Depends on

Used by

The balanced Ext bifunctor Definition
Naturality of the balance isomorphism Example
Ext balance isomorphisms satisfy change-of-resolution cocycle laws Proposition
The two Ext long exact sequences agree under balance 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)