The Grothendieck group of a classical order of finite lattice type
Let R be a complete d.v.r with valuation ideal p and quotient field K, and \(\Lambda\) a representation-finite R-order in a separable K-algebra A. For a maximal order \(\Gamma\supset \Lambda\), define the relative decomposition map d': \(G_ 0(\Gamma /p\Gamma)\to G_ 0(\Lambda /p\Lambda)\) by \(d'[V_{\nu}]:=\sum^{e}_{i=1}v_{i\nu}[S_ i]\), where \(V_{\nu}\), \(S_ i\) are the simple modules over the R/p-algebras \(\Gamma\) /p\(\Gamma\) and \(\Lambda\) /p\(\Lambda\), respectively, and \(v_{i\nu}\) denotes the multiplicity of \(S_ i\) in the \(\Lambda\)-module \(V_{\nu}\). It is shown that \(G_ 0(\Lambda)\cong {\mathbb{Z}}^{(\sigma)}\oplus co\ker d'\). A proof is given by means of the Igusa-Todorov algorithm in its integral form which has been established by the author [J. Lond. Math. Soc., II. Ser. 32, 75-87 (1985; Zbl 0577.16019)]. Recently, he affirmed that his formula for \(G_ 0(\Lambda)\) carries over to representation-infinite orders. Some applications to special R-orders are given in the paper.
scientific article; zbMATH DE number 3991713