Tight Hilbert Polynomials

In this talk, we will explore the Hilbert function and the Hilbert polynomial of filtrations of ideals arising from the tight closure of ideals. We compute the tight Hilbert polynomial in some diagonal hypersurface rings. In most cases the associated graded ring with respect to tight closure filtration turns out to be Cohen-Macaulay. This helps us find the tight Hilbert polynomial in these diagonal hypersurfaces. Let (R, m) be a Noetherian local ring of prime characteristic p, and Q be an m-primary parameter ideal. We provide 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)^i e_i^*(Q). 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 iff e_1^*(Q) = e_1(Q) iff depth R >= 2 and e_1^*(Q) = 0.

Let I be an ideal generated by a system of parameters in an excellent Cohen-Macaulay local domain then the associated graded ring G^*(I) of the filtration { (I^n)^* } is Cohen-Macaulay.