Multiplicity in triangulated categories

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.