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

2013/03/05 by Adriano Marzullo, Marzullo, Adriano
Chemistry · Mathematics · #Commutative Algebra and Its Applications #Graph theory and applications #Synthesis of Indole Derivatives #math.AC

paper · pdf · doi:10.48550/arxiv.1303.1116

arxiv created 2013/03/05 · arxiv updated 2013/03/06

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 A⁴ 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.

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