Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences
Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences
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由算术序列定义的单项曲线的最小分级自由分辨率
DOI：
10.1016/j.jalgebra.2013.04.026
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
H. Srinivasan
中科院分区：
文献类型：
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作者：
P. Gimenez;I. Sengupta;H. Srinivasan
设m=（m0，…，mn）是等差数列，即一个整数序列m0<⋯< mn0，没有能最小地生成数值半群∑i=0nmiN的公因子，且对于所有i∈{1，…，n−1},mi−mi−1=mi+1−mi。由[公式：见文]参数定义的仿射单项式曲线的齐次坐标环Γmof是一个分级的R模，其中R为多项式环k[X0，…，Xn]，通过设置degXi:=mi得到分级。本文构造了Γmand的显式最小梯度自由分辨率，证明了它的Betti数仅依赖于模n的值，从而证明了Herzog和Srinivasan关于等差数列定义的单项式曲线在平移下半群环Betti数的最终周期性的一个猜想。
Let m=(m0,…,mn) be an arithmetic sequence, i.e., a sequence of integers m0<⋯<mnwith no common factor that minimally generate the numerical semigroup ∑i=0nmiN and such that mi−mi−1=mi+1−mifor all i∈{1,…,n−1}. The homogeneous coordinate ring Γmof the affine monomial curve parametrically defined by [Formula: see text] is a graded R-module where R is the polynomial ring k[X0,…,Xn] with the grading obtained by setting degXi:=mi. In this paper, we construct an explicit minimal graded free resolution for Γmand show that its Betti numbers depend only on the value of m0modulo 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.