One-dimensional Gorenstein local rings with decreasing Hilbert function

Abstract

In this paper we solve a problem posed by M.E. Rossi: Is the Hilbert function of a Gorenstein local ring of dimension one not decreasing? More precisely, for any integer h>1, hâ\u88\u8914+22k,35+46k|kâ\u88\u88N, we construct infinitely many one-dimensional Gorenstein local rings, included integral domains, reduced and non-reduced rings, whose Hilbert function decreases at level h; moreover, we prove that there are no bounds to the decrease of the Hilbert function. The key tools are numerical semigroup theory, especially some necessary conditions to obtain decreasing Hilbert functions found by the first and the third author, and a construction developed by V. Barucci, M. D'Anna and the second author, that gives a family of quotients of the Rees algebra. Many examples are included

Complete Intersection Monomial Curves and the Cohen—Macaulayness of Their Tangent Cones

“…Here a 1 = 4, a 2 = 3, a 3 = 3, a 4 = 5, u 1 = 9, u 2 = 3, u 3 = 2 and u 4 = 7. Consider the vector v 1 = (9,12,15,9). For all w ≥ 0 the ideal I(n + wv 1 ) is a complete intersection on Theorem 2.8.…”

Minimal free resolutions of the tangent cones for Gorenstein monomial curves

“…Rossi [10] asks whether the Hilbert function of a Gorenstein local ring of dimension one is nondecreasing. Recently, it has been shown that there are many families of monomial curves giving negative answer to this problem [9]…”

“…Rossi [10] asks whether the Hilbert function of a Gorenstein local ring of dimension one is nondecreasing. Recently, it has been shown that there are many families of monomial curves giving negative answer to this problem [9]. However, it is still open for Gorenstein local rings associated to monomial curves in affine d−space for 3 < d < 10 and our main aim is to understand the Hilbert function when d = 4 .…”

On pseudo symmetric monomial curves

“…[11,12,18,30] and references therein. On the other hand, counterexamples were given only in affine 10-space by Herzog and Waldi in [22] and in affine 12-space by Eakin and Sathaye in [13], and most recently, Oneto et al [27,28] announced some methods for producing Gorenstein monomial curves whose tangent cones have decreasing Hilbert functions. However, the problem is still open for monomial curves in n-space, where 3 < n < 10.…”