Structure theorems over polynomial rings (Q854107)

Given a polynomial ring \(R\) over a field \(k\) and a finite group \(G\), let \(S\) be a finitely generated \(RG\)-module. A structure theorem for \(S\) is a set, indexed by subsets \(I\) of \(\{1,\dots,n\}\), of finitely generated \(kG\)-submodules \(X_I\) such that \(S\) is the direct sum of tensor products \(k[d_i:\,i\in I]\otimes_k X_I\). The author shows the equivalence of the existence of a structure theory for \(S\) and various other conditions, the most transparent being that only finitely many isomorphism classes of indecomposable \(kG\)-modules occur as summands of \(S\).