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.

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Equivalent extensions have the same Ext class

Statement

Equivalent extensions of $M$ by $N$ determine the same class in ${Ext}^{1} (M, N)$ .

Facts & Assumptions

Given: An endpoint-identity morphism between two short exact sequences $0 \to N \to E \to M \to 0$ .

Proof

technique · direct

1.1givenconstruct

Naturality of the long exact Ext sequence gives a commutative square between the two connecting maps $Hom (M, M) \to {Ext}^{1} (M, N)$ .

2.1step 1.1algebra∎

The endpoint maps are identities, so the square sends $1_{M}$ to the same element on both sides. These two images are the two extension classes.

Depends on

Used by

Dependency tree · two levels

6 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)