Tight Hilbert Polynomial and F-rational local rings
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 $P^*_Q(n)=\sum_{i=0}^d(-1)^ie_i^*(Q)\binom{n+d-1-i}{d-i}.$ 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 $e_1^*(Q), e_2^*(Q), \ldots, e_d^*(Q)$ 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 $\Leftrightarrow e_1^*(Q)=e_1(Q) \Leftrightarrow$ depth $R\geq 2$ and $e_1^*(Q)=0.$