TheoremStatement: 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.

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

1.1

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.

givenconstruct

2.1

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.

step 1.1algebra∎

Depends on

Used by

Ext one of Z modulo n by an abelian group as extension classes Example
FALSE: an extension of M by N represents Ext one of N by M False statement

Dependency tree · two levels

20 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)