On the perodicity of the first Betti number of the semigroup rings under translations

TL;DR

This paper proves that for certain families of monomial curves, the Betti numbers of their semigroup rings become periodic with a specific period, confirming a conjecture for large parameters.

Contribution

It establishes the periodicity of Betti numbers for specific classes of monomial curves, confirming the Herzog-Srinivasan conjecture in these cases.

Findings

Betti numbers are periodic with period a+b+c for large j.

Complete intersection property occurs if and only if (a+b+c)|j.

Periodic behavior aligns with the conjectured period.

Abstract

Let k be a field of characteristic zero. Given an ordered 3-tuple of positive integers a=(a,b,c) and for j in N, a family of sequences a_j = (j,a+j,a+b+j, a+b+c+j), we consider the collection of monomial curves in A^{4} associated with a_j. The Betti numbers of the Semigroup rings collection associated with a_j are conjectured to be eventually periodic with period a+b+c by Herzog and Srinivasan. Let p in N, in this paper, we prove that for a = (p(b+c),b, c) or a = (a, b, p(a+b)) in the collection of defining ideals associated with a_j, for large j the ideals are complete intersections if and only if (a + b + c)|j. Moreover, the complete intersections are periodic with the conjectured period.