CohenMacaulay rings and ideal theory in rings of invariants of algebraic groups
Author:
Ronald E. Kutz
Journal:
Trans. Amer. Math. Soc. 194 (1974), 115129
MSC:
Primary 13C15; Secondary 13H10, 14M15, 15A72, 20G15
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 CohenMacaulay, 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 CohenMacaulay 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
(20 #6414)
 [2]
Armand
Borel, Linear algebraic groups, Notes taken by Hyman Bass, W.
A. Benjamin, Inc., New YorkAmsterdam, 1969. MR 0251042
(40 #4273)
 [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
(28 #3076), http://dx.doi.org/10.1090/S00029947196401598607
 [4]
Wei
Liang Chow, On unmixedness theorem, Amer. J. Math.
86 (1964), 799–822. MR 0171804
(30 #2031)
 [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
(26 #161)
 [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
(35 #5435)
 [8]
John
Fogarty, Invariant theory, W. A. Benjamin, Inc., New
YorkAmsterdam, 1969. MR 0240104
(39 #1458)
 [9]
M.
Hochster, Generically perfect modules are strongly generically
perfect, Proc. London Math. Soc. (3) 23 (1971),
477–488. MR 0301002
(46 #162)
 [10]
M.
Hochster, Rings of invariants of tori, CohenMacaulay rings
generated by monomials, and polytopes, Ann. of Math. (2)
96 (1972), 318–337. MR 0304376
(46 #3511)
 [11]
M.
Hochster and John
A. Eagon, CohenMacaulay rings, invariant theory, and the generic
perfection of determinantal loci, Amer. J. Math. 93
(1971), 1020–1058. MR 0302643
(46 #1787)
 [12]
Irving
Kaplansky, Commutative rings, Allyn and Bacon Inc., Boston,
Mass., 1970. MR
0254021 (40 #7234)
 [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 YorkLondon, 1962. MR 0155856
(27 #5790)
 [15]
D.
W. Sharpe, On certain polynomial ideals defined by matrices,
Quart. J. Math. Oxford Ser. (2) 15 (1964), 155–175.
MR
0163927 (29 #1226)
 [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 (98k:01049)
 [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.
 [1]
 M. Auslander and D. A. Buchsbaum, Codimension and multiplicity, Ann. of Math. 68 (1958), 625657. MR 20 #6414. MR 0099978 (20:6414)
 [2]
 A. Borel, Linear algebraic groups, Benjamin, New York, 1969. MR 40 #4273. MR 0251042 (40:4273)
 [3]
 D. A. Buchsbaum and D. S. Rim, A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197224. MR 28 #3076. MR 0159860 (28:3076)
 [4]
 W. L. Chow, On unmixedness theorem, Amer. J. Math. 86 (1964), 799822. MR 30 #2031. MR 0171804 (30:2031)
 [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), 188204. MR 26 #161. MR 0142592 (26:161)
 [7]
 J. A. Eagon and D. G. Northcott, Generically acyclic complexes and genetically perfect ideals, Proc. Roy. Soc. Ser. A 299 (1967), 147172. MR 35 #5435 MR 0214586 (35:5435)
 [8]
 J. Fogarty, Invariant theory, Benjamin, New York, 1969. MR 39 # 1458. MR 0240104 (39:1458)
 [9]
 M. Hochster, Generically perfect modules are strongly generically perfect, Proc. London Math. Soc. (3) 23 (1971), 477488. MR 0301002 (46:162)
 [10]
 , Rings of invariants of tori, CohenMacaulay rings generated by monomials, and polytopes 96 (1972), 318337. MR 0304376 (46:3511)
 [11]
 M. Hochster and J. A. Eagon, CohenMacaulay rings, invariant theory, and the generic perfection of determinantal loci, Amer. J. Math. 93 (1971), 10201058. MR 0302643 (46:1787)
 [12]
 I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
 [13]
 F. S. Macaulay, The algebraic theory of modular systems, Cambridge Tracts 19 (1916).
 [14]
 M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
 [15]
 D. W. Sharpe, On certain polynomial ideals, defined by matrices. Quart. J. Math. Oxford Ser. (2) 15 (1964), 155175. MR 29 #1226. MR 0163927 (29:1226)
 [16]
 H. Weyl, The classical groups. Their invariants and representations, 2nd ed., Princeton Univ. Press, Princeton, N. J., 1946. MR 1488158 (98k:01049)
 [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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197403520822
PII:
S 00029947(1974)03520822
Keywords:
CohenMacaulay rings,
ideals generated by minors of matrices,
perfectideals,
rings of invariants,
orthogonal group
Article copyright:
© Copyright 1974 American Mathematical Society
