\(K\)-theory and patching for categories of complexes
type: Item
instance of: scholarly article
title: \(K\)-theory and patching for categories of complexes
zbMATH Open document ID: 0747.18012
DOI: 10.1215/S0012-7094-91-06214-9
author: Steven E. Landsburg
published in: Duke Mathematical Journal
publication date: 1992-06-25T00:00:00Z
review text: A well-known theorem of Milnor's tells us that the category of finitely- generated projective \(R\)-modules, where \(R\) is the pullback of a square of surjective homomorphisms of commutative noetherian rings \[ \begin{matrix} R & \longrightarrow & R_ 1 \\ \downarrow && \downarrow \\ R_ 2 & \longrightarrow & R_{12} \end{matrix} \] is the fiber product of the corresponding categories over \(R_ 1\) and \(R_ 2\), and that there is an associated Mayer-Vietoris exact sequence involving the \(K_ 1\) and \(K_ 0\)'s of these categories. These theorems extend to various categories of modules of finite projective dimension only under very restrictive conditions. In the paper, the author gives an analogous description of certain categories of bounded complexes of projective modules over the rings in question, and constructs a \(K_ 1\)-\(K_ 0\) Mayer-Vietoris sequence.
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13D15
zbMATH DE Number: 10354
zbMATH Keywords: categories of bounded complexes of projective modules
zbMATH Keywords: \(K_ 1\)-\(K_ 0\) Mayer-Vietoris sequence
reviewed by: Michael R. Stein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 10354