Dehomogenization of gradings to Zariskian filtrations and applications to invertible ideals
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- by Hui Shi Li and Freddy Van Oystaeyen PDF
- Proc. Amer. Math. Soc. 115 (1992), 1-11 Request permission
Abstract:
The method of dehomogenizing graded rings has been used successfully in algebraic geometry, e.g., a determinental ring is a dehomogenization of a Schubert cycle. We extend this method to noncommutative graded rings, dehomogenizing suitably graded rings to Zariski filtered rings and deriving, in a very elementary way, homological properties related to Auslander regularity and the Gorenstein property for noncommutative rings. As an application we study the lifting of such properties from a quotient modulo an invertible ideal.References
- Maria Jesus Asensio, Michel Van den Bergh, and Freddy Van Oystaeyen, A new algebraic approach to microlocalization of filtered rings, Trans. Amer. Math. Soc. 316 (1989), no. 2, 537–553. MR 958890, DOI 10.1090/S0002-9947-1989-0958890-X A Borel, Algebraic $D$-modules, Academic Press, London and New York, 1987.
- Winfried Bruns and Udo Vetter, Determinantal rings, Lecture Notes in Mathematics, vol. 1327, Springer-Verlag, Berlin, 1988. MR 953963, DOI 10.1007/BFb0080378
- L. Le Bruyn, M. Van den Bergh, and F. Van Oystaeyen, Graded orders, Birkhäuser Boston, Inc., Boston, MA, 1988. MR 1003605, DOI 10.1007/978-1-4612-3944-4 L. Le Bruyn and F. Van Oystaeyen, Generalized Rees rings satisfying polynomial identities, J. Algebra 83 (1983), 404-436. Hui-Shi Li, Non-commutative Zariskian rings, thesis, Antwerp, 1990.
- Hui Shi Li, M. Van den Bergh, and F. Van Oystaeyen, Global dimension and regularity of Rees rings for non-Zariskian filtrations, Comm. Algebra 18 (1990), no. 10, 3195–3208. MR 1063972, DOI 10.1080/00927879008824069
- Hui Shi Li and F. Van Oystaeyen, Filtrations of simple Artinian rings, J. Algebra 132 (1990), no. 2, 361–376. MR 1061485, DOI 10.1016/0021-8693(90)90135-B —, Global dimension and Auslander regularity of graded rings and Rees rings, Bull. Soc. Math. Belg. XLIII, 1991, pp. 59-87.
- Hui Shi Li and F. Van Oystaeyen, Zariskian filtrations, Comm. Algebra 17 (1989), no. 12, 2945–2970. MR 1030604, DOI 10.1080/00927878908823888 —, Zariskian Filtrations, Monograph, (to appear).
- Hui Shi Li, F. Van Oystaeyen, and E. Wexler-Kreindler, Zariski rings and flatness of completion, J. Algebra 138 (1991), no. 2, 327–339. MR 1102814, DOI 10.1016/0021-8693(91)90178-B C. Nǎstǎsescu and F. Van Oystaeyen, The dimensions of ring theory, Reidel, Dordrecht, Holland, 1987. —, Graded rings theory, Math. Library, vol. 28, North Holland, Amsterdam, 1982.
- D. G. Northcott, An introduction to homological algebra, Cambridge University Press, New York, 1960. MR 0118752
- Joseph J. Rotman, An introduction to homological algebra, Pure and Applied Mathematics, vol. 85, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 538169
- M. Van den Bergh and F. Van Oystaeyen, Lifting maximal orders, Comm. Algebra 17 (1989), no. 2, 341–349. MR 978479, DOI 10.1080/00927878908823732
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 1-11
- MSC: Primary 16W50; Secondary 16W60
- DOI: https://doi.org/10.1090/S0002-9939-1992-1081698-X
- MathSciNet review: 1081698