TheoremStatement: 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.

Ext dimension shifting in the second variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, and fix supplied projective and injective resolution data on all its objects. If $0 \to N \to I \to Σ N \to 0$ is an injective copresentation, then for $q \geq 1$ there are natural isomorphisms ${Ext}^{q + 1} (M, N) ≅ {Ext}^{q} (M, Σ N)$ ; its low-degree part is $0 \to Hom (M, N) \to Hom (M, I) \to Hom (M, Σ N) \to {Ext}^{1} (M, N) \to 0$ .

Facts & Assumptions

Given: The displayed short exact sequence with $I$ injective.

Proof

technique · direct

1.1given

The long exact sequence in the second variable contains ${Ext}^{q} (M, I) \to {Ext}^{q} (M, Σ N) \to {Ext}^{q + 1} (M, N) \to {Ext}^{q + 1} (M, I)$ .

2.1step 1.1algebra∎

The outer groups vanish for $q \geq 1$ because $I$ is injective, so the middle arrow is an isomorphism. At $q = 0$ the same sequence gives exactly the printed low-degree segment.

Depends on

Used by

Dependency tree · two levels

11 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)