Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups

Abstract:

Theorem. Let R be a commutative Noetherian ring with identity. Let M = $M = ({c_{ij}})$ be an s by s symmetric matrix with entries in R. Let I the be ideal of $t + 1$ by $t + 1$ minors of M. Suppose that the grade of I is as large as possible, namely, gr $I = g = s(s + 1)/2 - st + t(t - 1)/2$. Then I is a perfect ideal, so that $R/I$ is Cohen Macaulay if R is. Let G be a linear algebraic group acting rationally on $R = K[{x_1}, \ldots ,{x_n}]$. Hochster has conjectured that if G is reductive, then ${R^G}$ is Cohen-Macaulay, where ${R^G}$ denotes the ring of invariants of the action of G. The above theorem provides a special case of this conjecture. For $G = O(t,K)$, the orthogonal group, and K a field of characteristic zero, the above yields: Corollary. For R and G as above, ${R^G}$ is Cohen-Macaulay for an appropriate action of G. In order to obtain these results it was necessary to prove a more general form of the theorem stated above, which in turn yields a more general form of the corollary.