F-signature of graded Gorenstein rings

AbstractFor a commutative ring R, the F-signature was defined by Huneke and Leuschke [Math. Ann. 324 (2) (2002) 391–404]. 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 Fe.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). s(R)≤(−a(R))d2d−1d!limt→1(1−t)dP(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