Modules graded by \(G\)-sets: Duality and finiteness conditions
Let \(G\) be a group, \(A\) a finite \(G\)-set, \(R=\bigoplus_{g\in G}R_g\) a \(G\)-graded ring, and denote by \((G,A,R)\text{-gr}\) the category of \(A\)-graded left \(R\)-modules. If \(M\) is an object of this category and \(N\) is an \(R\)-submodule of \(M\), then let \((N)^g\) denote the smallest \(A\)-graded submodule of \(M\) containing \(N\). The paper deals with the following question: which finiteness conditions on \(N\) (that is, ascending and descending chain conditions, existence of Krull dimension) are inherited by \((N)^g\)? In case \(A=G\), such an investigation uses a well-known duality theorem of M. Cohen and S. Montgomery. The authors show that in general such a technique does not provide much, so they prove, by using different methods, the above properties on \(N\) are inherited by \((N)^g\).
scientific article; zbMATH DE number 1088619
scientific article