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.

Ext dimension shifting in the first variable

Statement

Assume the Axiom of Dependent Choice. Let $A$ be abelian with enough projectives and enough injectives, fix supplied projective and injective resolution data on all its objects, and let $0 ⟶ Ω M \overset{i}{\to} P_{0} \overset{ε}{\to} M ⟶ 0$ be the first stage of a projective resolution. For every object $N$ there is an exact sequence $0 \to Hom (M, N) \to Hom (P_{0}, N) \to Hom (Ω M, N) \to {Ext}^{1} (M, N) \to 0,$ and for every $q \geq 1$ there is a natural isomorphism ${Ext}^{q} (Ω M, N) ≅ {Ext}^{q + 1} (M, N) .$

Facts & Assumptions

Given: The displayed first stage and the remaining projective resolution $\dots \to P_{2} \to P_{1} \to P_{0} \to M \to 0$ .

[L1]

A short exact sequence in the first variable gives the long exact Ext sequence (The long exact Ext sequence in the first variable).

Proof

technique · direct

1.1

Apply [L1] to $0 \to Ω M \to P_{0} \to M \to 0$ . Its relevant terms are ${Ext}^{q} (P_{0}, N) \to {Ext}^{q} (Ω M, N) \to {Ext}^{q + 1} (M, N) \to {Ext}^{q + 1} (P_{0}, N)$ .

L1givenconstruct

2.1

Since $P_{0}$ is projective, the outer groups vanish for $q \geq 1$ by Positive projective-resolution Ext vanishes on a projective first variable, giving the displayed natural isomorphism. The degree-zero end of the same long exact sequence is exactly the displayed five-term sequence.

step 1.1algebra∎

Depends on

Used by

An Ext dimension shift Example
Every Ext-one class is represented by an extension Lemma
Projective dimension at most n iff higher Ext vanishes Theorem

Dependency tree · two levels

17 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)