Ideal related K-theory with coefficients

TL;DR

This paper introduces a new invariant called ideal related K-theory with coefficients, designed to serve as a substitute for total K-theory when a distinguished ideal exists, and explores its relationships with existing K-groups.

Contribution

It defines the ideal related K-theory with coefficients and constructs diagrams linking it to ordinary K-groups, facilitating computations and theoretical developments.

Findings

New invariant for K-theory with a distinguished ideal

Diagrams relating new groups to existing K-groups

Foundation for a universal multi-coefficient theorem

Abstract

In this paper, we define an invariant, which we believe should be the substitute for total K-theory in the case when there is one distinguished ideal. Moreover, some diagrams relating the new groups to the ordinary K-groups with coefficients are constructed. These diagrams will in most cases help to determine the new groups, and will in a companion paper be used to prove a universal multi-coefficient theorem for the one distinguished ideal case for a large class of algebras.