Affine semigroups and Cohen-Macaulay rings generated by monomials
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- by Ngô Viêt Trung and Lê Tuân Hoa PDF
- Trans. Amer. Math. Soc. 298 (1986), 145-167 Request permission
Corrigendum: Trans. Amer. Math. Soc. 305 (1988), 857.
Abstract:
We give a criterion for an arbitrary ring generated by monomials to be Cohen-Macaulay in terms of certain numerical and topological properties of the additive semigroup generated by the exponents of the monomials. As a consequence, the Cohen-Macaulayness of such a ring is dependent upon the characteristic of the ground field.References
- George E. Cooke and Ross L. Finney, Homology of cell complexes, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1967. Based on lectures by Norman E. Steenrod. MR 0219059
- Peter Schenzel, Ngô Viêt Trung, and Nguyễn Tụ’ Cu’ò’ng, Verallgemeinerte Cohen-Macaulay-Moduln, Math. Nachr. 85 (1978), 57–73 (German). MR 517641, DOI 10.1002/mana.19780850106
- Shiro Goto, On the Cohen-Macaulayfication of certain Buchsbaum rings, Nagoya Math. J. 80 (1980), 107–116. MR 596526
- Shiro Goto and Yasuhiro Shimoda, On the Rees algebras of Cohen-Macaulay local rings, Commutative algebra (Fairfax, Va., 1979) Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 201–231. MR 655805
- Shiro Goto, Naoyoshi Suzuki, and Keiichi Watanabe, On affine semigroup rings, Japan. J. Math. (N.S.) 2 (1976), no. 1, 1–12. MR 450257, DOI 10.4099/math1924.2.1
- Shiro Goto and Keiichi Watanabe, On graded rings. II. ($\textbf {Z}^{n}$-graded rings), Tokyo J. Math. 1 (1978), no. 2, 237–261. MR 519194, DOI 10.3836/tjm/1270216496
- W. Gröbner, Über Veronesesche Varietäten und deren Projektionen, Arch. Math. (Basel) 16 (1965), 257–264 (German). MR 183712, DOI 10.1007/BF01220031
- Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620
- Branko Grünbaum, Convex polytopes, Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. MR 0226496 T. Hibi, Affine semigroup rings and Hodge algebras. Some recent developments in the theory of commutative rings, Sûrikaisekikenkyûsho Kôkyûroku 484 (1983), 42-51.
- M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, DOI 10.2307/1970791
- Melvin Hochster and Joel L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Advances in Math. 13 (1974), 115–175. MR 347810, DOI 10.1016/0001-8708(74)90067-X
- Masa-Nori Ishida, Torus embeddings and dualizing complexes, Tohoku Math. J. (2) 32 (1980), no. 1, 111–146. MR 567836, DOI 10.2748/tmj/1178229687
- Shin Ikeda, The Cohen-Macaulayness of the Rees algebras of local rings, Nagoya Math. J. 89 (1983), 47–63. MR 692342
- G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
- Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879 F. S. Macaulay, Algebraic theory of modular systems, Cambridge Tracts in Math., vol. 19, 1916.
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911 T. Oda, Torus embeddings and applications, Lecture Notes of Tata Institute, Springer, 1982.
- Gerald Allen Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49. MR 407036, DOI 10.1016/0001-8708(76)90114-6
- Peter Schenzel, On Veronesean embeddings and projections of Veronesean varieties, Arch. Math. (Basel) 30 (1978), no. 4, 391–397. MR 485849, DOI 10.1007/BF01226072
- Rodney Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356. MR 263800, DOI 10.1007/BF01110229
- Rodney Y. Sharp, Local cohomology and the Cousin complex for a commutative Noetherian ring, Math. Z. 153 (1977), no. 1, 19–22. MR 442062, DOI 10.1007/BF01214729
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2
- Richard P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982), no. 2, 175–193. MR 666158, DOI 10.1007/BF01394054
- Jürgen Stückrad and Wolfgang Vogel, Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie, J. Math. Kyoto Univ. 13 (1973), 513–528 (German). MR 335504, DOI 10.1215/kjm/1250523322
- Jürgen Stückrad and Wolfgang Vogel, Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), no. 4, 727–746. MR 509072, DOI 10.2307/2373908
- Ngô Viêt Trung, Classification of the double projections of Veronese varieties, J. Math. Kyoto Univ. 22 (1982/83), no. 4, 567–581. MR 685519
- Ngô Việt Trung, Projections of one-dimensional Veronese varieties, Math. Nachr. 118 (1984), 47–67. MR 773610, DOI 10.1002/mana.19841180104
- Lê Tuấn Hoa, Classification of the triple projections of Veronese varieties, Math. Nachr. 128 (1986), 185–197. MR 855954, DOI 10.1002/mana.19861280116
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 298 (1986), 145-167
- MSC: Primary 13H10; Secondary 14M05
- DOI: https://doi.org/10.1090/S0002-9947-1986-0857437-3
- MathSciNet review: 857437