Conditions for vanishing of the s-multiplicity e^s(X,Y)

Determine conditions on a pair of objects X and Y in an R-linear triangulated category T with a central action from a graded-commutative Noetherian ring R under which the s-multiplicity e^s(X,Y), defined via alternating sums of lengths of the graded R^0-modules Hom_T(X,Σ^nY) and their difference operators of index d, vanishes.

Background

The paper introduces a multiplicity e^s(X,Y) for pairs of objects X and Y in an R-linear triangulated category T, where R is a graded-commutative Noetherian ring acting centrally. This invariant is modeled on Hilbert polynomial growth of Hom_T(X,Σ^nY), and is a triangulated-category analogue of Hochster’s theta invariant and Buchweitz’s Herbrand difference.

They prove structural properties of e^s(X,Y), including computation from leading coefficients of Hilbert polynomials, sign behavior under shifts, additivity on distinguished triangles, and a Koszul–Euler characteristic interpretation. Motivated by a vanishing theorem that derives strong cohomology vanishing consequences from e^s(X,Y)=0, the authors explicitly pose the open question of characterizing when e^s(X,Y) itself vanishes.