LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pending

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.

The Euler-twisted Hall product is associative and unital

Statement

The operation of The Euler-twisted Hall algebra is a well-defined associative $Q (ν)$ -bilinear product with unit $[0]$ .

Facts & Assumptions

Given: A finite-field hereditary Hall category and the twisted product $[M] ⋆ [N] = ν^{⟨ M, N ⟩} \sum_{L} g_{M, N}^{L} [L]$ .

[F1]

The untwisted Hall product is associative with unit $[0]$ (The untwisted Hall product is associative and unital).

[F2]

The Euler form is bilinear on $K_{0}$ (The hereditary Euler form is additive on the Grothendieck group).

[F3]

The twisted basis products have finite support (The Euler-twisted Hall product is well-defined).

Proof

technique · direct

1.1givenF2algebra

Fix basis objects $M, N, P$ . Every middle object $L$ in $[M] ⋆ [N]$ has class $[L] = [M] + [N]$ , and every middle object $U$ in $[N] ⋆ [P]$ has class $[U] = [N] + [P]$ , by short exact sequence additivity in $K_{0}$ . Thus the scalar exponent in $([M] ⋆ [N]) ⋆ [P]$ is $⟨ M, N ⟩ + ⟨ M + N, P ⟩$ , while in $[M] ⋆ ([N] ⋆ [P])$ it is $⟨ N, P ⟩ + ⟨ M, N + P ⟩$ .

1.2F1F2algebra

The zero object has class $0$ and $⟨ 0, α ⟩ = ⟨ α, 0 ⟩ = 0$ . Its twisted products therefore have the same coefficients as the untwisted products in [F1], so $[0]$ is a two-sided unit.

2.1F1F2step 1.1algebra

Bilinearity [F2] makes those exponents equal. After factoring out their common scalar, the remaining coefficient sums are the corresponding untwisted Hall coefficients, which agree by [F1]. Therefore the two parenthesizations agree on basis triples.

3.1F3step 2.1step 1.2algebra∎

Finite support [F3] and bilinearity extend the basis calculation to all vectors in the direct-sum Hall space. Step 1.2 supplies the unit.

Depends on

Used by

Dependency tree · two levels

11 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

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2 (standard reference, not scraped)
Matthew B. Young, Degenerate versions of Green’s theorem for Hall modules, arXiv:1810.08238v2 (standard reference, not scraped)