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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0352082-2
Keywords: Cohen-Macaulay rings, ideals generated by minors of matrices, perfectideals, rings of invariants, orthogonal group
Article copyright: © Copyright 1974 American Mathematical Society

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