Tight Hilbert Polynomial and F-rational local rings

Abstract

Let (R, m) be a Noetherian local ring of prime characteristic p and Q be an m-primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function. We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalises a result of Goto and Nakamura.We show that if dim R = 2, the Hochster-Huneke graph of R is connected and this lower bound is achieved then R is F-rational. Craig Huneke asked if the F -rationality of unmixed local rings may be characterized by the vanishing of e * 1 (Q). We construct examples to show that without additional conditions, this is not possible. Let R be an excellent, reduced, equidimensional Noetherian local ring and Q be generated by parameter test elements. We find formulas for e * 1 (Q), e * 2 (Q), . . . , e * d (Q) in terms of Hilbert coefficients of Q, lengths of local cohomology modules of R, and the length of the tight closure of the zero submodule of H d m (R). Using these we prove: R is F-rational ⇐⇒ e * 1 (Q) = e 1 (Q) ⇐⇒ depth R ≥ 2 and e * 1 (Q) = 0.