\(K\)-theory and patching for categories of complexes
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.
categories of bounded complexes of projective modules
\(K_ 1\)-\(K_ 0\) Mayer-Vietoris sequence
scientific article; zbMATH DE number 10354
scientific article