Recollements of derived categories II: Algebraic K-theory

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.

38 pages

References in corpus (4)

• Algebraic K-theory of endomorphism rings
• Ringel modules and homological subcategories
• Higher algebraic $K$-groups and $\mathcal{D}$-split sequences
• Stratifications of derived categories from tilting modules over tame hereditary algebras

Cited by in corpus (9)

• Ladders and simplicity of derived module categories
• Jordan-Hölder theorems for derived categories of derived discrete algebras
• Recollements of derived categories III: finitistic dimensions
• Reducing homological conjectures by n-recollements
• Recollements and stratifying ideals
• Ladders of compactly generated triangulated categories and preprojective algebras
• Ladders of recollements, categories of monomorphisms and singularity categories
• A remark on the Farrell-Jones conjecture
• Recollements and homological dimensions