Dual $F$-signature of special Cohen-Macaulay modules over cyclic quotient surface singularities

Abstract

The notion of $F$-signature was defined by Huneke and Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules by Sannai and is called dual $F$-signature. In this paper, we determine the dual $F$-signature of a certain class of Cohen-Macaulay modules (so-called ``special") over cyclic quotient surface singularities. Also, we compare the dual $F$-signature of a special Cohen-Macaulay module with that of its Auslander-Reiten translation. This gives a new characterization of the Gorensteinness.