A characterization of linearly reductive groups by their invariants (Q1972293)

In this clearly written paper, the author proves that if \(G\) is a reductive algebraic group over a field \(K\) and the invariant ring \(K[V]^G\) is Cohen-Macaulay for every \(G\)-module \(V\), then \(G\) is linearly reductive. A consequence is that if \(G\) is linearly algebraic, then \(G\) is linearly reductive if and only if for every affine \(G\)--scheme \(X\), the invariant ring \(K[X]^G\) is finitely generated and for every \(G\)--module \(V\), the invariant ring \(K[V]^G\) is Cohen-Macaulay.