Higher algebraic $K$-groups and $\mathcal D$-split sequences

TL;DR

This paper develops formulas for calculating higher algebraic K-groups of various rings using D-split sequences and derived equivalences, extending previous results to more general classes of rings without homological restrictions.

Contribution

It introduces a novel approach using D-split sequences and derived equivalences to compute K-groups for a broad class of rings, including matrix subrings and tiled orders.

Findings

Formulas for higher algebraic K-groups of certain rings.

Extension of existing K-theory results to more general rings.

No homological restrictions required on rings or ideals.

Abstract

In this paper, we use $\mathcal D$-split sequences and derived equivalences to provide formulas for calculation of higher algebraic $K$-groups (or mod-$p$ $K$-groups) of certain matrix subrings which cover tiled orders, rings related to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic $K$-theory literature to a more general context.