A remark on the monomial conjecture and Cohen-Macaulay canonical modules

Nhan, Le Thanh

doi:10.1090/S0002-9939-06-08573-X

A remark on the monomial conjecture and Cohen-Macaulay canonical modules
HTML articles powered by AMS MathViewer

by Le Thanh Nhan

Proc. Amer. Math. Soc. 134 (2006), 2785-2794

DOI: https://doi.org/10.1090/S0002-9939-06-08573-X

Published electronically: May 4, 2006

PDF | Request permission

Abstract:

In this paper, some sufficient conditions for rings and modules to satisfy the monomial conjecture are given. A characterization of Cohen-Macaulay canonical modules is presented.

References

Nguyễn Tụ’ Cu’ò’ng, On the dimension of the non-Cohen-Macaulay locus of local rings admitting dualizing complexes, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 479–488. MR 1094747, DOI 10.1017/S0305004100069929
Nguyen Tu Cuong and Nguyen Duc Minh, Lengths of generalized fractions of modules having small polynomial type, Math. Proc. Cambridge Philos. Soc. 128 (2000), no. 2, 269–282. MR 1735305, DOI 10.1017/S0305004199003941
Nguyen Tu Cuong, Nguyen Thai Hoa, and Nguyen Thi Hong Loan, On certain length functions associated to a system of parameters in local rings, Vietnam J. Math. 27 (1999), no. 3, 259–272. MR 1811339
Nguyen Tu Cuong and Le Thanh Nhan, Pseudo Cohen-Macaulay and pseudo generalized Cohen-Macaulay modules, J. Algebra 267 (2003), no. 1, 156–177. MR 1993472, DOI 10.1016/S0021-8693(03)00225-4
Nguyen Tu Cuong, Marcel Morales, and Le Thanh Nhan, The finiteness of certain sets of attached prime ideals and the length of generalized fractions, J. Pure Appl. Algebra 189 (2004), no. 1-3, 109–121. MR 2038567, DOI 10.1016/j.jpaa.2003.11.006
S. P. Dutta, A note on the monomial conjecture, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2871–2878. MR 1466948, DOI 10.1090/S0002-9947-98-02158-8
Shiro Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra 85 (1983), no. 2, 490–534. MR 725097, DOI 10.1016/0021-8693(83)90109-6
M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25–43. MR 349656, DOI 10.1017/S0027763000015701
Melvin Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503–553. MR 723406, DOI 10.1016/0021-8693(83)90092-3
Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
I. G. Macdonald, Secondary representation of modules over a commutative ring, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972) Academic Press, London, 1973, pp. 23–43. MR 0342506
Peter Schenzel, On birational Macaulayfications and Cohen-Macaulay canonical modules, J. Algebra 275 (2004), no. 2, 751–770. MR 2052635, DOI 10.1016/j.jalgebra.2003.12.016
R. Y. Sharp and M. A. Hamieh, Lengths of certain generalized fractions, J. Pure Appl. Algebra 38 (1985), no. 2-3, 323–336. MR 814188, DOI 10.1016/0022-4049(85)90020-9
R. Y. Sharp and H. Zakeri, Modules of generalized fractions, Mathematika 29 (1982), no. 1, 32–41. MR 673503, DOI 10.1112/S0025579300012134
R. Y. Sharp and H. Zakeri, Generalized fractions and the monomial conjecture, J. Algebra 92 (1985), no. 2, 380–388. MR 778456, DOI 10.1016/0021-8693(85)90128-0
Jan R. Strooker, Homological questions in local algebra, London Mathematical Society Lecture Note Series, vol. 145, Cambridge University Press, Cambridge, 1990. MR 1074178, DOI 10.1017/CBO9780511629242
Jan R. Strooker and Jürgen Stückrad, Monomial conjecture and complete intersections, Manuscripta Math. 79 (1993), no. 2, 153–159. MR 1216771, DOI 10.1007/BF02568334