ON THE PARTIAL EULER–POINCARÉ CHARACTERISTICS OF KOSZUL COMPLEXES OF IDEALIZATION

Abstract

Let $(R,\mathfrak{𝔪})$ be a Noetherian local ring and $M$ be a finitely generated $R$-module. In this paper, we give precise formulas computing all the partial Euler–Poincaré characteristics of Koszul complexes of the idealization $R⋉M$ with respect to an almost p-standard system of parameters (a strict subclass of d-sequences) in terms of multiplicities of certain subquotients of $R$ and $M$. As an application, we clarify the invariants $p_k(R⋉M)$ of the idealization $R⋉M$ that were introduced by N. T. Cuong and Khoi (1996).