Recollements of derived categories II: Algebraic K-theory

TL;DR

This paper establishes conditions for the additivity of higher algebraic K-groups in derived categories, providing new long exact sequences and applications to noncommutative localizations, free products, and specific group K-theories.

Contribution

It introduces sufficient conditions for K-group additivity in recollements and constructs long Mayer-Vietoris sequences, extending algebraic K-theory computations to new contexts.

Findings

Derived categories satisfy K-theory additivity under certain conditions.

Constructed long Mayer-Vietoris sequences for algebraic K-groups.

Described K-theory of free products and infinite dihedral groups.

Abstract

For a recollement of derived module categories of rings, we provide sufficient conditions to guarantee the additivity formula of higher algebraic K-groups of the rings involved, and establish a long Mayer-Vietoris exact sequence of higher algebraic K-groups for homological exact contexts introduced in the first paper of this series. Our results are then applied to recollements induced from homological ring epimorphisms and noncommutative localizations. Consequently, we get an infinitely long Mayer-Vietoris exact sequence of K-theory for Milnor squares, re-obtain a result of Karoubi on localizations and a result on generalized free products pioneered by Waldhausen and developed by Neeman and Ranicki. In particular, we describe algebraic $K$-groups of the free product of two groups over a regular coherent ring as the ones of the noncommutative tensor product of an exact context. This yields a new description of algebraic $K$-theory of infinite dihedral group.