UPPER BOUND FOR THE FIRST EULER-POINCARÉ CHARACTERISTICS OF KOZUL COMPLEXES RELATIVE TO PARAMETER IDEALS

Abstract

Let  be a Noetherian local ring and be a finitely generated -module of dimension  Let  be a parameter ideal of . Denote by  the  homology module of the Koszul  generated by the system  Set    and call it the first Euler-Poincaré characteristic of  relative to In this paper, we study the upper bound for the first Euler-Poincaré characteristics of  relative to the parameter ideal  By using the unmixed degree, the multiplicity, and the superficial elements we give the upper bound for  given by where are the unmixed degree, the multiplicity of   with respect to  respectively.