F-signature of graded Gorenstein rings

For 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 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 Poincare polynomial P(R, t). S(R) <= (-a(R))(d)/2(d-1)d! t ->(lim)(1-t)(d)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. (C) 2011 Published by Elsevier B.V.