Modules with chain conditions for finite matrix subgroups
The author presents new ways of describing modules with maximum resp. minimum condition for finite matrix subgroups and shows that these chain conditions can be characterized by two well-known natural transformations. The first one is \(\mu\colon\text{Hom}_S(P,M)\otimes_RA\to\text{Hom}_S(P,M\otimes_R A)\) given by \(\mu(\varphi\otimes a)(x)=\varphi(x)\otimes a\) for \(\varphi\in\text{Hom}_S(P,M)\), \(a\in A\) and \(x\in P\). Here \(_SM_R\) is a bimodule and \(_SP\) and \(_RA\) are modules. Dual results are proven by studying the homomorphism \(\nu\colon c\otimes_R\text{Hom}_S(M,U)\to\text{Hom}_S(\text{Hom}_R(C,M),U)\) with \(\nu(c\otimes\varphi)(g)=\varphi(g(c))\) for \(c\in C\), \(\varphi\in\text{Hom}_S(M,U)\), and \(g\in\text{Hom}_R(C,M)\), where \(_SM_R\) is a bimodule and \(C_R\) and \(_SU\) are modules.
scientific article; zbMATH DE number 1013965
scientific article