Minimal graded free resolutions for monomial curves defined by arithmetic sequences

JOURNAL OF ALGEBRA

卷:388

Article

Gimenez, Philippe^1,2  Sengupta, Indranath³  Srinivasan, Hema⁴ 

[1] Univ Valladolid, Fac Sci, Dept Algebra Geometry & Topol, E-47011 Valladolid, Spain

[2] Univ Valladolid, IMUVA, E-47011 Valladolid, Spain

[3] Jadavpur Univ, Dept Math, Kolkata 700032, WB, India

[4] Univ Missouri, Dept Math, Columbia, MO 65211 USA

关键词: Monomial curves;    Arithmetic sequences;    Determinantal ideals;    Betti numbers;    Minimal free resolution;   

DOI  :  10.1016/j.jalgebra.2013.04.026

来源: Elsevier

【 摘 要 】

Let m = (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 Sigma(n)(i=0) m(i)N and such that m(i) - m(i-1) = m(i+1) - m(i) for all i is an element of {1, ..., n - 1}. The homogeneous coordinate ring Gamma(m) of the affine monomial curve parametrically defined by X-0 = t(m0), ..., X-n = t(mn) 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(m) 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. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free