CounterexampleConstruction: Literature-sourcedVerification: 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.

A noncommutative handedness error in Tor

Statement refuted

Let $R = M_{2} (k)$ and let $V = k^{2}$ be supplied only as the usual left column $R$ -module. Two copies of the supplied left module $_{R} V$ do not by themselves provide an expression $_{R} V \otimes_{R}_{R} V$ or ${Tor}_{i}^{R} (_{R} V,_{R} V)$ .

Counterexample

Given: the left matrix action of $R$ on column vectors, with no right $R$ -action included in the data.

1.1

A balanced tensor relation needs a right action on the first factor: $(x r) \otimes y = x \otimes (r y)$ .

given

2.1

The notation $_{R} V$ specifies only $r \cdot v$ for the first copy; it supplies no value for $v \cdot r$ . One could ask for additional right-module or bimodule data, but it is not part of the two given left modules.

step 1.1algebra

3.1

Therefore the proposed tensor and Tor expressions have missing type data; this is a concrete handedness counterexample.

step 2.1algebra∎

Depends on

Tor does not take two left modules over an arbitrary ring without extra bimodule structure

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)