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Motivic stack Hall and untwisted finitary Hall have different carriers
Facts & Assumptions
Given: The finitary and motivic products with their stated coefficient carriers and quotient-first sequence convention.
On its declared domain of a small finitary abelian category and a commutative unital coefficient ring, the finitary product has basis symbols indexed by object isomorphism classes and its coefficient counts subobjects in (The untwisted Hall algebra of a finitary abelian category). This remark contrasts the defining carriers; it does not assert that every finite-quiver representation category meets that supplier's finitary hypotheses.
The motivic product uses relative stack classes and pullback/pushforward along the exact-sequence stack (The motivic Hall algebra of a finite quiver).
Proof
By [F1], the finitary carrier is a direct sum of copies of the coefficient ring indexed by object isomorphism classes, and each product coefficient is a natural-number subobject count. By [F2], the motivic carrier consists of relative classes of stacks mapping to representation stacks, and its product pulls those families back to a stack of exact sequences before pushing to the middle-object stack. These are different defining carriers and coefficient operations, even though both use the same quotient-first short-exact-sequence order.
The definitions provide no map from those natural-number counts to the relative stack classes. Any comparison therefore needs a specified realization and normalization; neither associativity proof alone supplies such a comparison.
Remarks
The actual current PA-19 Definition was inspected as the upstream interface; it remains a draft carrier in its owning pair until ordinary readiness and review. This item uses only its stated coefficient carrier and product formula.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
12 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, §§1.1–1.3 (standard reference, not scraped)
- Tom Bridgeland, An Introduction to Motivic Hall Algebras, arXiv:1002.4372v1, §§4.1–4.2 (standard reference, not scraped)