Application of the generalized Weierstrass preparation theorem to the study of homogeneous ideals
HTML articles powered by AMS MathViewer
- by Mutsumi Amasaki PDF
- Trans. Amer. Math. Soc. 317 (1990), 1-43 Request permission
Abstract:
The system of Weierstrass polynomials, defined originally for ideals in convergent power series rings, together with its sequence of degrees allows us to analyze a homogeneous ideal directly. Making use of it, we study local cohomology modules, syzygies, and then graded Buchsbaum rings. Our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of Weierstrass polynomials, a rough classification of graded Buchsbaum rings in the general case and a complete classification of graded Buchsbaum integral domains of codimension two.References
- Mutsumi Amasaki, Preparatory structure theorem for ideals defining space curves, Publ. Res. Inst. Math. Sci. 19 (1983), no. 2, 493–518. MR 716966, DOI 10.2977/prims/1195182442
- Mutsumi Amasaki, On the structure of arithmetically Buchsbaum curves in $\textbf {P}^3_k$, Publ. Res. Inst. Math. Sci. 20 (1984), no. 4, 793–837. MR 762953, DOI 10.2977/prims/1195181111
- Mutsumi Amasaki, Examples of nonsingular irreducible curves which give reducible singular points of $\textrm {red}(H_{d,g})$, Publ. Res. Inst. Math. Sci. 21 (1985), no. 4, 761–786. MR 817163, DOI 10.2977/prims/1195178928
- Mutsumi Amasaki, Curves in $\textbf {P}^3$ whose ideals are simple in a certain numerical sense, Publ. Res. Inst. Math. Sci. 23 (1987), no. 6, 1017–1052. MR 935714, DOI 10.2977/prims/1195175871
- Mutsumi Amasaki, Integral arithmetically Buchsbaum curves in $\textbf {P}^3$, J. Math. Soc. Japan 41 (1989), no. 1, 1–8. MR 972161, DOI 10.2969/jmsj/04110001
- Maurice Auslander and David A. Buchsbaum, Codimension and multiplicity, Ann. of Math. (2) 68 (1958), 625–657. MR 99978, DOI 10.2307/1970159
- Edoardo Ballico, Giorgio Bolondi, and Rosa M. Miró-Roig, Numerical invariants of rank-$2$ arithmetically Buchsbaum sheaves, J. Pure Appl. Algebra 58 (1989), no. 2, 107–125. MR 1001470, DOI 10.1016/0022-4049(89)90153-9
- Giorgio Bolondi and Juan Migliore, Classification of maximal rank curves in the liaison class $L_n$, Math. Ann. 277 (1987), no. 4, 585–603. MR 901706, DOI 10.1007/BF01457859 —, Buchsbaum liaison classes, preprint, (1987). —, The Lazarsfeld-Rao and Zeuthen problems for Buchsbaum curves, preprint, U.T.M. 228 (1987). —, The structure of an even liaison class, preprint, U.T.M. 239 (1988).
- David Bayer and Michael Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710, DOI 10.1007/BF01389151 M. Chang, Characterization of arithmetically Buchsbaum subschemes of codimension $2$ in ${{\mathbf {P}}^n}$, preprint, (1988). P. Ellia et M. Fiorentini, Quelques remarques sur les courbes arithmetiquement Buchsbaum de l’espace projectif, preprint 94, Università di Ferrara.
- David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
- André Galligo, À propos du théorème de-préparation de Weierstrass, Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau), Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974, pp. 543–579 (French). Thèse de 3ème cycle soutenue le 16 mai 1973 à l’Institut de Mathématique et Sciences Physiques de l’Université de Nice. MR 0402102
- A. V. Geramita and J. C. Migliore, On the ideal of an arithmetically Buchsbaum curve, J. Pure Appl. Algebra 54 (1988), no. 2-3, 215–247. MR 963546, DOI 10.1016/0022-4049(88)90032-1 —, Generators for the ideal of an arithmetically Buchsbaum curve, preprint, Queen’s University 17 (1987). S. Goto, Maximal Buchsbaum modules over regular local rings, Proc. 7th Sympos. Commutative Algebra, Kyoto 1985, pp. 82-89.
- Shiro Goto, Maximal Buchsbaum modules over regular local rings and a structure theorem for generalized Cohen-Macaulay modules, Commutative algebra and combinatorics (Kyoto, 1985) Adv. Stud. Pure Math., vol. 11, North-Holland, Amsterdam, 1987, pp. 39–64. MR 951196, DOI 10.2969/aspm/01110039
- Laurent Gruson and Christian Peskine, Genre des courbes de l’espace projectif, Algebraic geometry (Proc. Sympos., Univ. Tromsø, Tromsø, 1977) Lecture Notes in Math., vol. 687, Springer, Berlin, 1978, pp. 31–59 (French). MR 527229
- Hans Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math. 15 (1972), 171–198 (German). MR 293127, DOI 10.1007/BF01404124 H. Hironaka, Bimeromorphic smoothing of complex analytic spaces, Notes, Warwick University, 1971.
- Heisuke Hironaka and Tosuke Urabe, Kaiseki kukan nyumon, 2nd ed., Suri Kagaku Raiburari [Mathematical Science Library], vol. 1, Asakura Publishing Co. Ltd., Tokyo, 1983 (Japanese). MR 995846
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- H. Michael Möller and Ferdinando Mora, New constructive methods in classical ideal theory, J. Algebra 100 (1986), no. 1, 138–178. MR 839576, DOI 10.1016/0021-8693(86)90071-2
- Jürgen Stückrad and Wolfgang Vogel, Buchsbaum rings and applications, Springer-Verlag, Berlin, 1986. An interaction between algebra, geometry and topology. MR 881220, DOI 10.1007/978-3-662-02500-0
- Tohsuke Urabe, On Hironaka’s monoideal, Publ. Res. Inst. Math. Sci. 15 (1979), no. 1, 279–287. MR 533349, DOI 10.2977/prims/1195188430
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 1-43
- MSC: Primary 13A15; Secondary 13C05, 13D25, 13H10, 14M05
- DOI: https://doi.org/10.1090/S0002-9947-1990-0992603-9
- MathSciNet review: 992603