F-signature of graded Gorenstein rings

Abstract

a b s t r a c tFor a commutative ring R, the F -signature was defined by Huneke and Leuschke [Math. Ann. 324 (2) (2002) [391][392][393][394][395][396][397][398][399][400][401][402][403][404]. It is an invariant that measures the order of the rank of the free direct summand of R (e) . Here, R (e) is R itself, regarded as an R-module through e-times Frobenius action F e .In this paper, we show a connection of the F -signature of a graded ring with other invariants. More precisely, for a graded F -finite Gorenstein ring R of dimension d, we give an inequality among the F -signature s(R), a-invariant a(R) and Poincaré polynomial P(R, t).Moreover, we show that R (e) has only one free direct summand for any e, if and only if R is F -pure and a(R) = 0. This gives a characterization of such rings.