Recollements of derived categories II: Algebraic K-theory

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...

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Bibliographic Details
Published inarXiv.org
Main Authors Chen, Hongxing, Changchang Xi
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 20.05.2014
Subjects
Algebra
Group theory
Homology
Rings (mathematics)
Tensors
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Summary: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.
ISSN:2331-8422