FF-signature of graded Gorenstein rings

For a commutative ring RR, the FF-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)R(e). Here, R(e)R(e) is RR itself, regarded as an RR-module through ee-times Frobenius action FeFe.In this paper, we show a connection of the FF-signature of a graded ring with other invariants. More precisely, for a graded FF-finite Gorenstein ring RR of dimension dd, we give an inequality among the FF-signature s(R)s(R), aa-invariant a(R)a(R) and Poincaré polynomial P(R,t)P(R,t). s(R)≤(−a(R))d2d−1d!limt→1(1−t)dP(R,t).Moreover, we show that R(e)R(e) has only one free direct summand for any ee, if and only if RR is FF-pure and a(R)=0a(R)=0. This gives a characterization of such rings.