The \(K\)-theory of vector bundles with endomorphisms over a scheme (Q1922888)

By using the powerful localization theorem of \textit{R. W. Thomason} and \textit{T. Trobaugh} [Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)], the author of the present paper studies the \(K\)-theory of the category of vector bundles with endomorphisms over a scheme. The results generalize those of \textit{D. R. Grayson} [J. Algebra 48, 439-446 (1977; Zbl 0413.18010)]. An example is given to show that the Mayer-Vietoris sequence does not hold for the \(K\)-theory of vector bundles with endomorphisms, which indicates that the \(K\)-theory of vector bundles with endomorphisms is not determined by local data.