On commutative invariants for modules over crossed products of minimax nilpotent linear groups

Abstract

Let $N$ be a minimax nilpotent torsion-free normal subgroup of a soluble group $G$ of finite rank, $R$ be a finitely generated commutative domain and $R\ast N$ be a crossed product of $R$ and $N$. In the paper we construct a correspondence between an $R\ast N$-module $W$ and a finite set $M$ of equivalent classes of prime ideals minimal over $An{n}_{kA}(W/WI)$, where $kA$ is a group algebra of an abelian minimax group $A$ and $I$ is an appropriative $G$-invariant ideal of $RG$. It is shown that if $Wg\cong W$ for all $g\in g$ then the action of the group $G$ by conjugations on $N$ can be extended to an action of the group $G$ on the set $M$. The results allow us to apply methods of commutative algebra to the study of $W$.

14 pages