PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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

1.1givenconstruct

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.

2.1step 1.1algebra∎

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.

Depends on

Dependency tree · 2 levels

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2