Huneke’s conjecture on vanishing first tight Hilbert coefficient implying F-rationality

Prove that if R is an excellent local ring of characteristic p > 0, is reduced and satisfies Serre’s condition (S2), and the first tight Hilbert coefficient e∗1(Q) equals zero for some (and hence for all) parameter ideal Q ⊆ R, then R is F-rational.

Background

The paper studies tight Hilbert coefficients in local rings of prime characteristic, focusing on Buchsbaum rings. In this context, the tight Hilbert function associated to powers of parameter ideals has coefficients e∗0(Q),..., e∗d(Q), with e∗1(Q) being the first tight Hilbert coefficient.

Prior work established non-negativity of e∗1(Q) under mild assumptions, and characterized F-rationality via the vanishing of e∗1(Q) in the Cohen–Macaulay case. Huneke’s conjecture proposes that vanishing of e∗1(Q) (for some parameter ideal) implies F-rationality more generally under (S2) and excellence. This paper proves the conjecture for reduced Buchsbaum rings, leaving the full generality of the conjecture open.