A note on Grothendieck groups of group rings and orders
Let R be a Dedekind domain with field of fractions K, and let \(\Lambda\) be an R-order in the separable K-algebra A. Furthermore, let \(_{\Lambda}{\mathfrak M}^ t\) denote the category of all finitely generated \(\Lambda\)-modules that are R-torsion, and let \(\alpha_{\Lambda}: K_ 0(_{\Lambda}{\mathfrak M}^ t)\to G_ 0(\Lambda)\) denote the homomorphism arising from the natural inclusion of \(_{\Lambda}{\mathfrak M}^ t\) into the category of all finitely generated \(\Lambda\)-modules. The main result of this work is the following: Let \(\Gamma\) be an R-order in A containing \(\Lambda\). Then the commutative square \(\alpha_{\Lambda}\circ Res^{\Gamma}_{\Lambda}=Res^{\Gamma}_{\Lambda}\circ \alpha_{\Gamma}: K_ 0(_{\Gamma}{\mathfrak M}^ t)\to G_ 0(\Lambda)\) is a pushout diagram. As an application, the author offers a proof of \textit{R. G. Swan}'s theorem \(G_ 0(RG)\cong G_ 0(KG)\) for R semilocal of characteristic 0 and G a finite group [Topology 2, 85-110 (1963; Zbl 0119.029)].
finitely generated \(\Lambda \) -modules
scientific article; zbMATH DE number 4200408
scientific article