Fractional ideals and integration with respect to the generalised Euler characteristic

TL;DR

This paper explores fractional ideals in one-dimensional Cohen-Macaulay local rings, introduces motivic Poincaré series related to ideal filtrations, and analyzes their functional equations, advancing understanding of algebraic and motivic properties.

Contribution

It introduces new motivic Poincaré series for ideal filtrations associated with fractional ideals and studies their functional equations, linking algebraic and motivic aspects.

Findings

Introduction of new motivic Poincaré series

Functional equations for these series derived

Enhanced understanding of fractional ideals in Cohen-Macaulay rings

Abstract

Let $b$ be a fractional ideal of a one-dimensional Cohen-Macaulay local ring $O$ containing a perfect field $k$. This paper is devoted to the study some $O$-modules associated with $b$. In addition, different motivic Poincar\'e series are introduced by considering ideal filtrations associated with $b$; the corresponding functional equations of these Poincar\'e series are also described.