Let G be a reductive algebraic group over an algebraically closed field k of characteristic zero, and let W be a finite-dimensional representation of G. Let U be another finite-dimensional G-representation. A natural generalization of the Hochster-Roberts theorem would be that \((U\otimes_ kk[W])^ G\) is a Cohen-Macaulay \(k[W]^ G\)-module. Examples show that this cannot be true in general; however, a conjecture due to R. P. Stanley [Proc. Symp. Pure Math. 34, 345-355 (1979; Zbl 0411.22006)] gives at least some cases under which the above statement is true. The author replaces this conjecture with a slightly weaker one and proves this new conjecture for certain pairs (G,W) (namely for those where the locus of G-unstable points in W//G is constructible, i.e. it can be build up from smaller manageable parts in some sense, explained explicitly in the paper). It is proved that the conjecture holds if \(G=SL(2)\).
If \(G=SL(V)\), \(W=End(V)^{m*}\), \(U=End(V)\), then \(T_{m,n}=(U\otimes_ kk[W])^ G\) is the noncommutative trace ring of m generic (n\(\times n)\)- matrices. The conjecture is satisfied in this case, which makes it possible for the author to prove that \(T_{m,n}\) is Cohen-Macaulay.