Calculation of the group \(K_1\) of a generalized matrix ring
The main purpose of the paper under review is to calculate the \(K_1\)-group of a \(2{\times}2\)-generalized matrix ring. More precisely those are rings of \(2{\times}2\)-matrices with diagonal elements in rings \(R\) and \(S\) and the off diagonal elements in \(M\) and \(N\) respectively, where \(R\) and \(S\) are rings, \(M\) an \(R\)-\(S\)-bimodule, \(N\) and \(S\)-\(R\)-bimodule equipped with bimodule homomorphisms \[ {\phi}: M{\otimes}_SN \to R, \quad {\psi}: N{\otimes}_RM \to S. \] The product im \(K\) is defined using multiplication of matrices and the homomorphisms \(\phi\) and \(\psi\). Let \(I = \text{im}({\phi})\) and \(J = \text{Im}({\psi})\) be the corresponding ideals, called the trace ideals. The author uses the description of \(K_1(K)\) as the abelian group generated by pairs \((P, {\alpha})\) where \(P\) is a finitely generated projective \(K\)-module that satisfies certain relations. The idea for the calculation of \(K_1(K)\) is that the category of those modules can be described explicitly in some cases. In general, such a module can be described as a pairs \((X, Y)\) with \(X\) and \(R\)-module and \(Y\) and \(S\)-module that have extra structure. The first result of the paper is that, if \(I \subset J(R)\) and \(R\) is complete in the \(I\)-adic topology and \(J \subset J(S)\) and \(S\) is complete in the \(J\)-adic topology then the categories of finitely generated \((R{\times}S)\)-modules is equivalent to the category of finitely generated projective \(K\)-modules. As a corollary, he derives that \(K_1(K) \cong K_1(R){\oplus}K_1(S)\). For the second result, the author defines projections maps from \(K_1(K)\) to \(K_1(R)\) and \(K_1(S)\). Let \({\pi}: K_1(K) \to K_1(R) \oplus K_1(S)\) be the sum of the maps. The second result in the paper, describes the quotient \(K_1(K)/\text{ker}({\pi})\). Using this description, the author gives conditions under which \(K_1(K) \cong K_1(R){\oplus}K_1(S){\oplus}K_1(\text{ker}({\pi}))\). One such condition is when the structure map \(\phi\) is a monomorphism and \(I\) is nilpotent. In the case when \(R = S\) then \(K\) is determined by a central element \(s\in R\). In this case \(K_1(K_s)\) is isomorphic to \(K_1(R)\) when \(s\) is invertible and to two copies to \(K_1(R)\) when \(s\) is nilpotent.
scientific article; zbMATH DE number 6381099