Group rings over hereditary orders (Q1820845)

Let R be a complete discrete valuation ring with quotient field K, let \(\Lambda\) be an R-order in a semisimple K-algebra A and let G be a finite group. For any ring \(\Gamma\), let \(G_ 0(\Gamma)\) be the Grothendieck group of finitely generated \(\Gamma\)-modules and let \(SG_ 0(\Lambda)\) be the kernel of the natural surjection \(G_ 0(\Lambda)\to G_ 0(A)\), given by [M]\(\to [K\otimes_{R}M]\). Owing to \textit{A. Kuku} [J. Algebra 91, 18-31 (1984; Zbl 0548.16009)], if D is a noncommutative K-division algebra and \(\Delta\) its unique maximal order, then one can choose G so that \(SG_ 0(\Delta G)\neq 0\). The aim of this paper is to exhibit G explicitly in the case where the residue class field of R is finite. It turns out that G can be taken as a cyclic group naturally connected with the arithmetic of D. The paper is very well written and contains a number of results pertaining to group rings which are of independent interest.