On semigroup rings with decreasing Hilbert function

Abstract

Given a one-dimensional semigroup ring R=k[[S]], in this article we study the behaviour of the Hilbert function HR. By means of the notion of support of the elements in S, for some classes of semigroup rings we give conditions on the generators of S in order to have decreasing HR. When the embedding dimension v and the multiplicity e verify v+3â\u89¤eâ\u89¤v+4, the decrease of HRgives an explicit description of the Apéry set of S. In particular for e=v+3, we prove that HRis non-decreasing if eâ\u89¤12 and we classify the semigroups with e=13 and HRdecreasing. Finally we deduce that HRis non-decreasing for every Gorenstein semigroup ring with eâ\u89¤v+4. This fact is not true in general: through numerical duplication and some of the above results another recent paper shows the existence of infinitely many one-dimensional Gorenstein rings with decreasing Hilbert function

One-dimensional Gorenstein local rings with decreasing Hilbert function

“…It follows that ord(α m ) > 1 and so α m = 2n 2 . On the other hand by [21,Proposition 4.3.1] we have 3n 2 − e ∈ Ap, that is a contradiction because 3n 2 − e > 2n 2 .…”

“…Further we recall that, by [21,Corollary 4.11], in a symmetric semigroup with decreasing Hilbert function the difference between the multiplicity and the embedding dimension has to be greater or equal to 5: in the following example is 6. Example 3.7.…”

“…• In [21] the first and the third author, for numerical semigroup rings such that ν ≥ e − 4, where ν and e denote respectively the embedding dimension and the multiplicity of R.…”

“…If e and ν are the multiplicity and embedding dimension of S (or equivalently of k[[S]]) we first show that we need e − v ≥ 4. By using a result of [21] and a theorem of H. Nari [20] we give an explicit construction of a family of almost symmetric semigroups with the required properties. We also show other examples with the above properties.…”

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.…”