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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Maurice Auslander and David A. Buchsbaum,*Codimension and multiplicity*, Ann. of Math. (2)**68**(1958), 625–657. MR**0099978**, https://doi.org/10.2307/1970159**[2]**Armand Borel,*Linear algebraic groups*, Notes taken by Hyman Bass, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0251042****[3]**David A. Buchsbaum and Dock S. Rim,*A generalized Koszul complex. II. Depth and multiplicity*, Trans. Amer. Math. Soc.**111**(1964), 197–224. MR**0159860**, https://doi.org/10.1090/S0002-9947-1964-0159860-7**[4]**Wei Liang Chow,*On unmixedness theorem*, Amer. J. Math.**86**(1964), 799–822. MR**0171804**, https://doi.org/10.2307/2373158**[5]**J. A. Eagon,*Ideals generated by the subdeterminants of a matrix*, Thesis, University of Chicago, Chicago, Ill., 1961.**[6]**J. A. Eagon and D. G. Northcott,*Ideals defined by matrices and a certain complex associated with them.*, Proc. Roy. Soc. Ser. A**269**(1962), 188–204. MR**0142592****[7]**J. A. Eagon and D. G. Northcott,*Generically acyclic complexes and generically perfect ideals*, Proc. Roy. Soc. Ser. A**299**(1967), 147–172. MR**0214586****[8]**John Fogarty,*Invariant theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0240104****[9]**M. Hochster,*Generically perfect modules are strongly generically perfect*, Proc. London Math. Soc. (3)**23**(1971), 477–488. MR**0301002****[10]**M. Hochster,*Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes*, Ann. of Math. (2)**96**(1972), 318–337. MR**0304376**, https://doi.org/10.2307/1970791**[11]**M. Hochster and John A. Eagon,*Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci*, Amer. J. Math.**93**(1971), 1020–1058. MR**0302643**, https://doi.org/10.2307/2373744**[12]**Irving Kaplansky,*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021****[13]**F. S. Macaulay,*The algebraic theory of modular systems*, Cambridge Tracts**19**(1916).**[14]**Masayoshi Nagata,*Local rings*, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR**0155856****[15]**D. W. Sharpe,*On certain polynomial ideals defined by matrices*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 155–175. MR**0163927**, https://doi.org/10.1093/qmath/15.1.155**[16]**Hermann Weyl,*The classical groups*, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR**1488158****[17]**O. Zariski and P. Samuel,*Commutative algebra*. Vols. I, II, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1958, 1960. MR**19**, 833;**22**#11006.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
13C15,
13H10,
14M15,
15A72,
20G15

Retrieve articles in all journals with MSC: 13C15, 13H10, 14M15, 15A72, 20G15

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