TheoremStatement: 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.

Green’s coproduct is an algebra map into the twisted tensor completion

Statement

For a finite-field hereditary Hall category, products of Green images converge coefficientwise and $Δ^{'} ([M] ⋆ [N]) = Δ^{'} ([M]) \cdot Δ^{'} ([N])$ in the completion of The Green Hall completion and coproduct, where the target multiplication is the stated Euler-twisted tensor multiplication. The coproduct is coassociative and counital by The Green Hall coproduct is coassociative. It is not asserted that the whole completion is an algebra.

Facts & Assumptions

Given: A finite-field hereditary Hall category and basis objects $M, N$ .

[F1]

The basis formulas for $Δ^{'}$ , the twisted Hall product and twisted tensor multiplication are those of The Green Hall completion and coproduct.

[F2]

Products $Δ^{'} ([M]) \cdot Δ^{'} ([N])$ have finite coefficient sums (Products of Green coproducts converge coefficientwise).

[F3]

The cross–square identity holds for the four external objects (The Green cross–square counting identity).

[F4]

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

[F5]

Endpoint-labelled exact-sequence counts and Hall numbers satisfy $P_{A, B}^{R} = a_{A} a_{B} g_{A, B}^{R}$ (Exact-sequence counts equal endpoint automorphism counts times subobject counts).

Proof

technique · direct coefficient comparison

1.1F1given

Fix output objects $K, L$ . From [F1], the coefficient of $[K] \otimes [L]$ in $Δ^{'} ([M] ⋆ [N])$ is $\frac{ν^{⟨ M, N ⟩ + ⟨ K, L ⟩}}{a_{M} a_{N}} \sum_{[J]} \frac{P_{M, N}^{J} P_{K, L}^{J}}{a_{J}} .$ The sum is finite because only middle terms of extensions of $M$ by $N$ occur.

1.2F1F2F5algebra

In $Δ^{'} ([M]) \cdot Δ^{'} ([N])$ , denote the decompositions of $M$ by $(K_{2}, L_{2})$ and those of $N$ by $(K_{1}, L_{1})$ . The Hall products in the two tensor factors contribute $P_{K_{2}, K_{1}}^{K}$ and $P_{L_{2}, L_{1}}^{L}$ . Consequently the coefficient of $[K] \otimes [L]$ is $\frac{1}{a_{M} a_{N}} \sum_{K_{1}, K_{2}, L_{1}, L_{2}} ν^{E} \frac{P_{K_{2}, L_{2}}^{M} P_{K_{1}, L_{1}}^{N} P_{K_{2}, K_{1}}^{K} P_{L_{2}, L_{1}}^{L}}{a_{K_{1}} a_{K_{2}} a_{L_{1}} a_{L_{2}}},$ where $E = ⟨ K_{2}, L_{2} ⟩ + ⟨ K_{1}, L_{1} ⟩ + (L_{2}, K_{1}) + ⟨ K_{2}, K_{1} ⟩ + ⟨ L_{2}, L_{1} ⟩ .$ The summation is finite by [F2]. Converting each Hall coefficient to the endpoint-labelled sequence normalization uses [F5].

2.1F4step 1.2algebra

Each nonzero summand in step 1.2 has $[M] = [K_{2}] + [L_{2}]$ , $[N] = [K_{1}] + [L_{1}]$ , $[K] = [K_{2}] + [K_{1}]$ and $[L] = [L_{2}] + [L_{1}]$ . Expanding the two desired Euler exponents by [F4] gives $⟨ M, N ⟩ + ⟨ K, L ⟩ = E + 2 ⟨ K_{2}, L_{1} ⟩ .$ All terms in the expansion occur once in $E$ except the two displayed copies of $⟨ K_{2}, L_{1} ⟩$ .

3.1F3F4step 1.1step 1.2step 2.1algebra

By [F3], the weighted square count in step 1.2 with factor $∣ {YExt}^{1} (K_{2}, L_{1}) ∣ / ∣ Hom (K_{2}, L_{1}) ∣$ equals the cross count in step 1.1. By [F4] and finite-dimensionality, $\frac{∣ {YExt}^{1} (K_{2}, L_{1}) ∣}{∣ Hom (K_{2}, L_{1}) ∣} = q^{- ⟨ K_{2}, L_{1} ⟩} = ν^{- 2 ⟨ K_{2}, L_{1} ⟩} .$ Thus the exponent $E$ in step 1.2 contributes exactly $ν^{⟨ M, N ⟩ + ⟨ K, L ⟩}$ times the required Ext/Hom ratio. Substitution of the identity [F3] makes the coefficients in steps 1.1 and 1.2 equal.

4.1F2step 3.1algebra∎

Equality holds at every output basis pair, proving multiplicativity on basis elements. Bilinearity and finite support of Hall products extend it to all finite Hall vectors. Coefficientwise convergence is [F2], and coassociativity and the counit are supplied by the flag proof in The Green Hall coproduct is coassociative.

Depends on

Used by

Dependency tree · two levels

19 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)
Claus Michael Ringel, Hall algebras and quantum groups, Inventiones Mathematicae 101 (1990), 583–591 (standard reference, not scraped)