15.44 (2002)

a) Let $G$ be a reductive group over an algebraically closed field $K$ of arbitrary characteristic. Let $X$ be an affine $G$-variety such that, for a fixed Borel subgroup $B \leqslant G$, the coordinate algebra $K[X]$ as a $G$-module is the union of an ascending chain of submodules each of whose factors is an induced module $\text{Ind}_B^G V$ of some one-dimensional $B$-module $V$. (See S. Donkin, Rational representations of algebraic groups. Tensor products and filtration (Lect. Notes Math., 1140), Springer, Berlin, 1985). Suppose in addition that $K[X]$ is a Cohen–Macaulay ring, that is, a free module over the subalgebra generated by any homogeneous system of parameters. Is then the ring of invariants $K[X]^G$ Cohen–Macaulay?

b) Is the ring of invariants $K[M(n)^m]^{GL(n)}$ Cohen–Macaulay in all characteristics? (Here $M(n)^m$ is the direct sum of $m$ copies of the space of $n \times n$ matrices.)