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.

The long exact Ext sequence 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. For $0 \to N^{'} \to N \to N^{''} \to 0$ and every $M$ , there is a natural exact sequence $0 \to Hom (M, N^{'}) \to Hom (M, N) \to Hom (M, N^{''}) \overset{δ^{0}}{\to} {Ext}^{1} (M, N^{'}) \to {Ext}^{1} (M, N) \to \dots,$ where $δ^{q} : {Ext}^{q} (M, N^{''}) \to {Ext}^{q + 1} (M, N^{'})$ ; it is natural in the short exact sequence and contravariantly natural in $M$ .

Facts & Assumptions

Given: A short exact sequence $0 \to N^{'} \to N \to N^{''} \to 0$ and an object $M$ .

Proof

technique · direct

1.1

Apply Right derived functors form a cohomological delta functor to the left exact functor $Hom (M, -)$ ; it supplies the displayed long exact sequence for the injective construction.

givenconstruct

2.1

Replace its terms by balanced Ext using The balanced Ext bifunctor. Delta-functor naturality gives naturality in the short exact sequence, while precomposition in $M$ gives the stated contravariant naturality.

step 1.1algebra∎

Depends on

Used by

The derived Ext-one class of an extension Definition
Equivalent extensions have the same Ext class Lemma
The two Ext long exact sequences agree under balance Proposition
Ext dimension shifting in the second variable Theorem

Dependency tree · two levels

15 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)