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The Euler-twisted Hall product is well-defined

Statement

For a finite-field hereditary Hall category, the basis formula in The Euler-twisted Hall algebra has finite support and defines a $Q (ν)$ -bilinear operation on the direct-sum Hall space.

Facts & Assumptions

Given: The category, $ν = \sqrt{q}$ , and the Euler-twisted basis formula.

[F1]

The untwisted coefficient $g_{M, N}^{L}$ is finite and has finite support in $[L]$ for fixed $M, N$ (Untwisted Hall structure constants and products are finite).

[F2]

The twist scalar $ν^{χ_{0} (M, N)}$ is a well-defined element of $Q (ν)$ because $χ_{0} (M, N)$ is an integer (The Euler-twisted Hall algebra).

Proof

1.1F1F2given

For fixed $M, N$ , [F1] gives a finite set of isomorphism classes $[L]$ with nonzero Hall coefficient. Multiplication by the single scalar in [F2] does not enlarge that support, so the displayed basis product is a finite linear combination in the direct sum.

2.1step 1.1algebra∎

Every Hall vector has finite support. Bilinearity therefore expands any product of two vectors into a finite sum of the finite basis products in step 1.1, producing an element of the same direct sum.

Depends on

Used by

The Green Hall completion and coproduct Definition
The Euler-twisted Hall product is associative and unital Lemma

Cited to discharge well-definedness by The Euler-twisted Hall algebra.

Dependency tree · two levels

6 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)