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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.
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Green coproduct coefficients are finite

Statement

For fixed objects $R, M, N$ in a finite-field hereditary Hall category, the endpoint-labelled count $P_{M, N}^{R}$ and hence the coefficient of $[M] \otimes [N]$ in The Green Hall completion and coproduct are finite. This assertion does not claim finite support in $[M], [N]$ for a fixed $R$ .

Facts & Assumptions

Given: Fixed objects $R, M, N$ in the category.

[F1]

Hom sets are finite in the stated category (Finite-field hereditary Hall categories).

[F2]

The coefficient is the scalar $ν^{χ_{0} (M, N)} P_{M, N}^{R} / a_{R}$ (The Green Hall completion and coproduct).

Proof

technique · direct

1.1F1given

An endpoint-labelled sequence $0 \to N \overset{i}{\to} R \overset{p}{\to} M \to 0$ is a pair $(i, p)$ in $Hom (N, R) \times Hom (R, M)$ satisfying exactness. This product is finite by [F1], so its exact-sequence subset and the count $P_{M, N}^{R}$ are finite.

2.1F2step 1.1algebra∎

The automorphism number $a_{R}$ is a positive finite integer, and $ν^{χ_{0} (M, N)}$ is a defined scalar in $Q (ν)$ . Thus the coefficient in [F2] is well-defined. Since the argument fixed $M, N$ , it proves no finite-support claim as those objects vary.

Depends on

Used by

Cited to discharge well-definedness by The Green Hall completion and coproduct.

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)