ISSN 1088-6850(online) ISSN 0002-9947(print)

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

Author: Ronald E. Kutz
Journal: Trans. Amer. Math. Soc. 194 (1974), 115-129
MSC: Primary 13C15; Secondary 13H10, 14M15, 15A72, 20G15
DOI: https://doi.org/10.1090/S0002-9947-1974-0352082-2
MathSciNet review: 0352082
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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.

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