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

Abstract

The notion of F -signature was defined by C. Huneke and G. Leuschke and this numerical invariant characterizes some singularities. This notion is extended to finitely generated modules by A. 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 Gorensteiness.

The theory of F-rational signature

“…In [San15] Sannai computed dual Fsignature of the Veronese subrings of k[x, y] (Example 3.2). More generally, the results of Nakajima in [Nak18] can be used for computations in cyclic quotients of k[x, y]. In [Has17] Hashimoto studied the dual F-signature of invariant subrings and was able to characterize vanishing of s dual (ω R ) representation-theoretically even in the non Cohen-Macaulay case.…”

“…However, outside of a small number of examples (cf. [Nak18,Has17]), it has remained open whether the limit defining the dual F-signature exists. Not only is this problematic when attempting to compute or estimate s dual (R), it is also at the heart of the difficulty in attempting to show that the dual F-signature defines a lower semicontinuous function on Spec(R).…”