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

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)