Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences

Abstract

Let $\mm=(m_0,...,m_n)$ be an arithmetic sequence, i.e., a sequence of integers $m_0<...<m_n$ with no common factor that minimally generate the numerical semigroup ${\sum }_{i=0}^n{m}_i\mathbb{N}$ and such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in \{1,...,n-1\}$. The homogeneous coordinate ring $Γ_\mm$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},...,X_n=t^{m_n}$ is a graded $R$-module where $R$ is the polynomial ring $k[X_0,...,X_n]$ with the grading obtained by setting $°X_i:=m_i$. In this paper, we construct an explicit minimal graded free resolution for $Γ_\mm$ and show that its Betti numbers depend only on the value of $m_0$ modulo $n$. As a consequence, we prove a conjecture of Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under translation for the monomial curves defined by an arithmetic sequence.

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