Tight Hilbert polynomial and F-rational local rings

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 H^*_Q(n)=ℓ (R/(Q^n)^*) and the coefficient e_1^*(Q) of the tight Hilbert polynomial P^*_Q(n)=∑ _i=0^d(-1)^ie_i^*(Q)( [ n+d-1-i; d-i ]) . We obtain a lower bound for the tight Hilbert function of Q for equidimensional excellent local rings that generalizes a result of Goto and Nakamura. We show that if 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) depthR≥ 2 and e_1^*(Q)=0.