CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

An invariant line need not have an invariant complement over a Laurent ring

Statement refuted

Every invariant line in a finite free module over the Laurent ring $Λ_{1} = Z [t^{\pm 1}]$ admits an invariant complement.

Facts & Assumptions

Given: the ring $Λ_{1} = Z [t^{\pm 1}]$ ; an integer $n \geq 2$ ; the free module $W = Λ_{1}^{n}$ with standard basis $e_{1}, \dots, e_{n}$ and the unreduced Burau action $ρ_{n}^{m a t}$ of $B_{n}$ ; the column vector $v = (1, \dots, 1)^{T}$ and the row vector $σ = (1, t, \dots, t^{n - 1})$ .

[A1]

$B_{i} v = v$ for every $i$ , hence $ρ_{n}^{m a t} (β) v = v$ for every $β \in B_{n}$ ; and a row vector $τ$ satisfies $τ B_{i} = τ$ for every $i$ if and only if $τ = c σ$ for some $c \in Λ_{1}$ (The invariant vector and the invariant covectors of the unreduced Burau, clauses (a) and (b); the action is defined on generator matrices by The unreduced Burau matrices and extended by The unreduced Burau matrices satisfy the Artin relations).

[A2]

For $n \geq 2$ the element $1 + t + \dots + t^{n - 1}$ is not a unit of $Λ_{1}$ (Units, powers and the domain property of the Laurent polynomial ring, clause (d)); in particular it is not a unit of the form $\pm t^{m}$ .

Counterexample

Given: the same data as above.

Proof technique: direct.

1.1A1

The invariant line. By [A1], $ρ_{n}^{m a t} (β) v = v$ for every $β \in B_{n}$ ; hence $Λ_{1} v = {a v : a \in Λ_{1}}$ is a $B_{n}$ -invariant line in the finite free module $W$ .

1.2A1algebra

No invariant complement. Suppose, for contradiction, that $C \subseteq W$ is a $Λ_{1}$ -submodule with $W = Λ_{1} v \oplus C$ and $ρ_{n}^{m a t} (β) (C) \subseteq C$ for every $β \in B_{n}$ . Let $π : W \to Λ_{1} v$ be the projection along $C$ , so $π$ is $Λ_{1}$ -linear, $π (v) = v$ , and $π$ is $B_{n}$ -equivariant: writing $x = a v + c$ with $a \in Λ_{1}$ , $c \in C$ , invariance of $C$ and $v$ give $ρ_{n}^{m a t} (β) x = a v + ρ_{n}^{m a t} (β) c$ with $ρ_{n}^{m a t} (β) c \in C$ , so $π (ρ_{n}^{m a t} (β) x) = a v = π (x)$ . Writing $π (x) = λ (x) v$ defines a $Λ_{1}$ -linear functional $λ : W \to Λ_{1}$ with $λ (v) = 1$ and $λ (ρ_{n}^{m a t} (β) x) = λ (x)$ for all $β \in B_{n}$ and $x \in W$ .

2.1A1A2step 1.2∎

The contradiction. Since $λ (ρ_{n}^{m a t} (β) x) = λ (x)$ for every $β$ , evaluating on $β = σ_{i}$ gives $λ B_{i} = λ$ for every $i$ ; by the classification in [A1] there is $c \in Λ_{1}$ with $λ = c σ$ as row vectors. Then $1 = λ (v) = c σ (v) = c (1 + t + \dots + t^{n - 1})$ , so $1 + t + \dots + t^{n - 1}$ has the multiplicative inverse $c$ in $Λ_{1}$ . This contradicts [A2] for $n \geq 2$ . Hence the invariant line $Λ_{1} v$ has no $B_{n}$ -invariant complement, and the refuted statement fails already for $n = 2$ . This is the integral obstruction behind the caveat of The reduced and unreduced Burau representations have the same kernel that its splitting $K^{n} = \ker σ \oplus K v$ is only a field statement. AC is inherited from the cited same-kernel proposition; the module and matrix computations are choice free.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

30 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

Joan S. Birman and Tara E. Brendle, Braids: A Survey (background on Burau matrices, the cyclic cover and absolute homology) (standard reference, not scraped)
Vasudha Bharathram, Joan S. Birman and Tara E. Brendle, The Burau representation is faithful for n = 4, arXiv:2607.05283v1 (6 July 2026), Introduction and section 2 (printed pp. 1-5) (standard reference, not scraped)