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

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Yoneda Ext one is naturally isomorphic to derived Ext one

Statement

Assume the Axiom of Dependent Choice, the balanced-Ext hypotheses of The balanced Ext bifunctor, and that the extension classes in question form a set. Sending an extension of $M$ by $N$ to the connecting image of $1_{M}$ gives a natural isomorphism of abelian groups ${YExt}^{1} (M, N) ≅ {Ext}^{1} (M, N) .$

Facts & Assumptions

Given: Objects $M, N$ under the stated resolution hypotheses.

Proof

technique · direct

1.1givenconstruct

The assignment is well-defined by Equivalent extensions have the same Ext class, surjective by Every Ext-one class is represented by an extension, and injective by Two extensions with the same Ext class are equivalent.

2.1step 1.1algebra∎

For $a : M^{'} \to M$ , the pullback extension and the original extension form a morphism of short exact sequences; naturality of the connecting morphism sends $1_{M}$ to the pullback of its class along $a$ . For $b : N \to N^{'}$ , the analogous pushout square sends the class forward along $b$ . These are exactly the two maps in Extension classes are contravariant in the quotient and covariant in the subobject, so the bijection is natural in both variables. The diagonal-pullback and codiagonal-pushout defining Baer sum correspond, by the same two naturality squares, to addition in Ext; hence the bijection is additive.

Depends on

Used by

Dependency tree · two levels

24 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, Chapters 3–4 (standard reference, not scraped)