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* = *be an s by s symmetric matrix with entries in R. Let I the be ideal of* *by* *minors of M. Suppose that the grade of I is as large as possible, namely*, gr . *Then I is a perfect ideal, so that* *is Cohen Macaulay if R is*.

Let *G* be a linear algebraic group acting rationally on . Hochster has conjectured that if *G* is reductive, then is Cohen-Macaulay, where denotes the ring of invariants of the action of *G*. The above theorem provides a special case of this conjecture. For , the orthogonal group, and *K* a field of characteristic zero, the above yields:

**Corollary.** *For R and G as above*, *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