Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences

By Philippe Gimenez, I. Sengupta, H. Srinivasan. 2011

57 citations recorded by the source corpus.

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 $\sum_{i=0}^{n}m_i\N$ and such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in\{1,...,n-1\}$. The homogeneous coordinate ring $\Gamma_\mm$ 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 $\deg{X_i}:=m_i$. In this paper, we construct an explicit minimal graded free resolution for $\Gamma_\mm$ 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/379c257832215db7216a014116210028e8403cca

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Citations in the corpus

On the defining equations of the tangent cone of a numerical semigroup ring (2013, methodology): “We recall here the formula given in [8] for the Betti numbers of K[H ].”
The trace of the canonical module (2016, result): “We also characterize when such semigroups are almost symmetric, taking into account that the symmetric case was known from work of Gimenez, Sengupta and Srinivasan in [13].”