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 Tor sequence in the left-module variable

Statement

Assume the Axiom of Dependent Choice. For $0 \to M^{'} \to M \to M^{''} \to 0$ of left $R$ -modules and a right module $N$ , there is the natural long exact sequence $\dots \to {Tor}_{i} (N, M^{'}) \to {Tor}_{i} (N, M) \to {Tor}_{i} (N, M^{''}) \to {Tor}_{i - 1} (N, M^{'}) \to \dots$ .

Proof

Given: the stated short exact sequence, a right module $N$ , and supplied projective resolutions of its three left modules.

1.1

By The horseshoe lemma for projective resolutions, choose a projective horseshoe resolution of the middle module that fits with resolutions of the outer modules into a degreewise split short exact sequence. Tensoring it with $N$ preserves the degreewise splittings, hence gives a short exact sequence of chain complexes.

given

2.1

The homology long-exact-sequence construction supplies the displayed connecting maps and exactness.

step 1.1algebra

3.1

Change-of-resolution coherence identifies the three homology families with the fixed balanced Tor functor and makes the sequence natural. Its degree-zero tail is $N \otimes_{R} M^{'} \to N \otimes_{R} M \to N \otimes_{R} M^{''} \to 0$ , so no unclaimed left exactness is introduced.

step 2.1algebra∎

Depends on

Used by

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 (standard reference, not scraped)