Tight Hilbert polynomial and F-rational local rings

Abstract

Let $$(R,\mathfrak {m})$$ be a Noetherian local ring of prime characteristic p and Q be an $$\mathfrak {m}$$ -primary parameter ideal. We give criteria for F-rationality of R using the tight Hilbert function $$H^*_Q(n)=\ell (R/(Q^n)^*)$$ and the coefficient $$e_1^*(Q)$$ of the tight Hilbert polynomial $$P^*_Q(n)=\sum _{i=0}^d(-1)^ie_i^*(Q)\left( {\begin{array}{c}n+d-1-i\\ d-i\end{array}}\right) .$$ 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 $$\dim R=2 $$ , 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 $$e_1^*(Q), e_2^*(Q), \ldots , e_d^*(Q)$$ 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 $$H^d_{\mathfrak {m}}(R).$$ Using these, we prove: R is F-rational $$\iff e_1^*(Q)=e_1(Q) \iff {\text {depth}}R\ge 2$$ and $$e_1^*(Q)=0.$$