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Canonical ideals of Cohen-Macaulay partially ordered sets

Published online by Cambridge University Press:  22 January 2016

Takayuki Hibi
Takayuki Hibi*
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University, Chikusa-ku, Nagoya 464, Japan
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Our dream is to revive the ideal theory in partially ordered sets from a viewpoint of commutative algebra.

Historically, the concept of ideals in commutative algebra was first studied by Dedekind, who considered the ring of algebraic integers in an algebraic number field.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 112 , December 1988 , pp. 1 - 24
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[Bac] Baclawski, K., Cohen-Macaulay ordered sets, J. Algebra, 63 (1980), 226–258.CrossRefGoogle Scholar
[Bir] Birkhoff, G., “Lattice Theory”, third ed., Amer. Math. Soc. Colloq. Publ., No. 25, Amer. Math. Soc., Providence, R.I., 1967.Google Scholar
[Bjö] Björner, A., Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., 260 (1980), 159–183.CrossRefGoogle Scholar
[D-E-P] De Concini, C., Eisenbud, D. and Procesi, C., Hodge Algebras, Astérisque, 91 (1982).Google Scholar
[Eis] Eisenbud, D., Introduction to algebras with straightening laws, in “Ring Theory and Algebra III”. (McDonald, B. R., ed.), Lect. Notes in Pure and Appl. Math., No. 55, Dekker, New York, 1980, pp. 243–268.Google Scholar
[Fri] Frink, O., Ideals in partially ordered sets, Amer. Math. Monthly, 61 (1954), 223–234.CrossRefGoogle Scholar
[Gar] Garsia, A., Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math., 38 (1980), 229–266.CrossRefGoogle Scholar
[G-W] Goto, S. and Watanabe, K.-i., On graded rings, I, J. Math. Soc. Japan, 30 (1978), 179–213.CrossRefGoogle Scholar
[H-K] Herzog, J. and Kunz, E., Der Kanonische Modul eines Cohen-Macaulay-Rings, Lect. Notes in Math., No. 238, Springer-Verlag, Berlin/Heidelberg/New York, 1971.Google Scholar
[H1] Hibi, T., For which finite groups G is the lattice L(G) of subgroups Gorenstein?, Nagoya Math. J., 105 (1987), 147–151.CrossRefGoogle Scholar
[H2] Hibi, T., Distributive lattices, affine semigroup rings and algebras with straightening laws, in “Commutative Algebra and Combinatorics” (Nagata, M. and Matsumura, H., eds.), Advanced Studies in Pure Math., Vol. 11, North-Holland, Amsterdam, 1987, pp. 93–109.Google Scholar
[H3] Hibi, T., Union and glueing of a family of Cohen-Macaulay partially ordered sets, Nagoya Math. J., 107 (1987), 91–119.CrossRefGoogle Scholar
[H4] Hibi, T., Level rings and algebras with straightening laws, J. Algebra, 117 (1988), 343–362.CrossRefGoogle Scholar
[H5] Hibi, T., Classification of integral trees, Order, 3 (1987), 383–389.Google Scholar
[H6] Hibi, T., Plane graphs and Cohen-Macaulay posets, European J. Combin., to appear.Google Scholar
[H7] Hibi, T., A numerical characterization of Gorenstein complexes, submitted.Google Scholar
[H8] Hibi, T., Toroidal posets, preprint.Google Scholar
[H9] Hibi, T., Linear diophantine equations and Stanley’s P-partitions, in preparation.Google Scholar
[H10] Hibi, T., The Ehrhart polynomial of a convex polytope, in preparation.Google Scholar
[H-W] Hibi, T. and Watanabe, K.-i., Study of three-dimensional algebras with straightening laws which are Gorenstein domains I, Hiroshima Math. J., 15 (1985), 27–54.Google Scholar
[Hoc1] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math., 96 (1972), 318–337.CrossRefGoogle Scholar
[Hoc2] Hochster, M., Cohen-Macaulay rings, combinatorics, and simplicial complexes, in “Ring Theory II” (McDonald, B. R. and Morris, R., eds.), Lect. Notes in Pure and Appl. Math., No. 26, Dekker, New York, 1977, pp. 171–223.Google Scholar
[Rei] Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. in Math., 21 (1976), 30–49.CrossRefGoogle Scholar
[Sta1] Stanley, R., Ordered structures and partitions, Mem. Amer. Math. Soc., 119 (1972).Google Scholar
[Sta2] Stanley, R., Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J., 40 (1973), 607–632.CrossRefGoogle Scholar
[Sta3] Stanley, R., Finite lattices and Jordan-Holder sets, Alg. Univ., 4 (1974), 361–371.CrossRefGoogle Scholar
[Sta4] Stanley, R., Combinatorial reciprocity theorems, Adv. in Math., 14 (1974), 194–253.CrossRefGoogle Scholar
[Sta5] Stanley, R., The upper bound conjecture and Cohen-Macaulay rings, Stud. Appl. Math., 54 (1975), 135–142.CrossRefGoogle Scholar
[Sta6] Stanley, R., Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings, Duke Math. J., 43 (1976), 511–531.CrossRefGoogle Scholar
[Sta7] Stanley, R., Hilbert functions of graded algebras, Adv. in Math., 28 (1978), 57–83.CrossRefGoogle Scholar
[Sta8] Stanley, R., Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc., 249 (1979), 139–157.CrossRefGoogle Scholar
[Sta9] Stanley, R., The number of faces of a simplicial convex polytope, Adv. in Math., 35 (1980), 236–238.CrossRefGoogle Scholar
[Sta10] Stanley, R., Linear diophantine equations and local cohomology, Invent. Math., 68 (1982), 175–193.CrossRefGoogle Scholar
[Sta11] Stanley, R., “Combinatorics and Commutative Algebra”, Progress in Math., Vol. 41, Birkhäuser, Boston/Basel/Stuttgart, 1983.Google Scholar
[Sta12] Stanley, R., “Enumerative Combinatorics, Volume I”, Wadsworth, Monterey, Calif., 1986.Google Scholar
[Sta13] Stanley, R., On the number of faces of centrally-symmetric simplicial polytopes, Graphs Combin., 3 (1987), 55–66.Google Scholar
[Sto] Stone, M. H., The theory of representations for Boolean algebras, Trans. Amer. Math. Soc, 40 (1936), 37–111.Google Scholar
[Wat] Watanabe, K.-i., Study of algebras with straightening laws of dimension 2, in “Algebraic and Topological Theories—to the memory of Dr. Takehiko Miyata” (Nagata, M. et al., eds.), Kinokuniya, Tokyo, 1985, pp. 622–639.Google Scholar