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

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.

Every Ext-one class is represented by an extension

Statement

Assume the Axiom of Dependent Choice and the balanced-Ext hypotheses of The balanced Ext bifunctor. Given a projective presentation $0 \to Ω M \to P \to M \to 0$ , every element of ${Ext}^{1} (M, N)$ is the class of an extension of $M$ by $N$ .

Facts & Assumptions

Given: An object $N$ , an object $M$ with the stated projective presentation, and $ξ \in {Ext}^{1} (M, N)$ .

Proof

technique · direct

1.1givenconstruct

The low-degree dimension-shifting sequence from Ext dimension shifting in the first variable represents $ξ$ by a morphism $f : Ω M \to N$ , modulo morphisms extending across $P$ . Push out $0 \to Ω M \to P \to M \to 0$ along $f$ , as defined in Pullback and pushout of an extension.

2.1step 1.1algebra∎

The resulting bottom row is an extension of $M$ by $N$ . Naturality of its connecting homomorphism sends $1_{M}$ to the class represented by $f$ , which is $ξ$ by the definition in The derived Ext-one class of an extension.

Depends on

Used by

Dependency tree · two levels

19 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)