Multiplicity in triangulated categories

TL;DR

This paper develops a theory of multiplicity in triangulated categories with a central ring action, generalizing classical invariants and applying to cohomology vanishing and algebraic structures.

Contribution

It introduces a new multiplicity invariant in R-linear triangulated categories, extending classical invariants like Hochster's theta and Buchweitz's Herbrand difference.

Findings

Invariant determined by leading coefficients of Hilbert polynomials.

Applications to vanishing of cohomology.

Relevance to modules over local complete intersections and group algebras.

Abstract

We lay out the theory of a multiplicity in the setting of a triangulated category having a central ring action from a graded-commutative ring $R$, in other words, an $R$-linear triangulated category. The invariant we consider is modelled on those for graded modules over a commutative graded ring. We show that this invariant is determined by the leading coefficients of the Hilbert polynomials expressing the lengths of certain Hom sets. Our theory is a natural analogue of Hochster's theta invariant for homology and Buchweitz's Herbrand difference for cohomology. Moreover, we give applications to vanishing of cohomology and modules over local complete intersection rings, group algebras of a finite group, and certain finite dimensional algebras.