Transactions of the American Mathematical Society

Author(s): Eero Hyry
Journal: Trans. Amer. Math. Soc. 351 (1999), 2213-2232.
MSC (1991): Primary 13A30; Secondary 14B15, 14M05
Posted: February 23, 1999
Retrieve article in: PDF
This article is available free of charge

Abstract: Let

be a multigraded ring defined over a local ring $(A,\mathfrak{m})$ . This paper deals with the question how the Cohen-Macaulay property of

is related to that of its diagonal subring $T^\Delta$ . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of

. If $I_1,\dotsc,I_r\subset A$ are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra $R_A(I_1,\dotsc,I_r)$ with the Cohen-Macaulay property of the usual Rees algebra $R_A(I_1\cdots I_r)$ . We also obtain a bound for the joint reduction numbers of two $\mathfrak{m}$ -primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

1.: W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge University Press, Cambridge, 1993. MR 95h:13020
2.: S. Goto and K. Watanabe, On graded rings I, J. Math. Soc. Japan 30 (1978), 179-213. MR 81m:13021
3.: S. Goto and K. Watanabe, On graded rings II, Tokyo J. Math. 1 (1978), 237-260. MR 81m:13022
4.: A. Grothendieck and J. Dieudonné, Eléments de Géométrie Algébrique II, Publ. Math. I.H.E.S. 8 (1961). MR 36:177b
5.: R. Hartshorne, Residues and Duality, Springer Lecture Notes 20, Springer-Verlag, Berlin, Heidelberg, New York, 1966. MR 36:5145
6.: M. Herrmann, E. Hyry, and J. Ribbe, On the Cohen-Macaulay and Gorenstein properties of multi-Rees algebras, Manuscripta Math. 79 (1993), 343-377. MR 94h:13003
7.: M. Herrmann, E. Hyry, and J. Ribbe, On multi-Rees algebras (with an appendix by N. V. Trung), Math. Ann. 301 (1995), 249-279. MR 96c:13005
8.: M. Herrmann, E. Hyry, J. Ribbe, and Z. Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 (1997), 311-341. MR 98k:13006
9.: M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up, Springer-Verlag, Berlin, Heidelberg, New York, 1988. MR 89g:13012
10.: I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, 1970. MR 40:7234
11.: D. Kirby and D. Rees, Multiplicities in graded rings I: The general theory, Contemp. Math. 159 (1994), 209-267. MR 95b:13002
12.: S. Kleiman and A. Thorup, A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra 167 (1994), 168-231. MR 96a:14007
13.: J. Lipman, Desingularization of two-dimensional schemes, Ann. of Math. 107 (1978), 151-207. MR 58:10924
14.: J. Lipman, Cohen-Macaulayness in graded algebras, Math. Res. Letters 1 (1994), 149-157. MR 95d:13006
15.: D. G. Northcott and D. Rees, Reduction of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), 145-158. MR 15:596a
16.: L. O'Carroll, On two theorems concerning reductions in local rings, J. Math. Kyoto Univ. 27 (1987), 61-67. MR 87m:13034
17.: D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. (2) 29 (1984), 397-414. MR 86e:13023
18.: P. Schenzel, N. V. Trung, and N. T. Cuong, Verallgemeinerte Cohen-Macaulay Moduln, Math. Nachr. 85 (1978), 57-73. MR 80i:13008
19.: A. Simis, The diagonal subalgebra of a multigraded $k$ -algebra, in Commutative Algebra: extended abstracts of an international conference, July 27-August 1, 1994, Vechta, Germany (ed. by Winfried Bruns), Vechtaer Universitätsschriften 13, Runge, Cloppenburg, 1994.
20.: N. V. Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229-237. MR 89i:13031
21.: N. V. Trung and S. Ikeda, When is the Rees algebra Cohen-Macaulay?, Comm. Algebra 17 (1989), 2893-2922. MR 91a:13009
22.: P. Valabrega and G. Valla, Form rings and regular sequences, Nagoya Math. J. 72 (1978), 93-101. MR 80d:14010
23.: W. V. Vasconcelos, Arithmetic of Blowup Algebras, LMS Lecture Note Ser. 195, Cambridge University Press, Cambridge, 1994. MR 95g:13005
24.: J. K. Verma, Joint reductions of complete ideals, Nagoya Math. J. 118 (1990), 155-163. MR 91c:13018
25.: J. K. Verma, Joint reductions and Rees algebras, Math. Proc. Cambridge Philos. Soc. 109 (1991), 335-343. MR 92h:13007
26.: J. K. Verma, Multigraded Rees algebras and mixed multiplicities, J. Pure Appl. Algebra 77 (1992), 219-228. MR 93e:13005

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A30, 14B15, 14M05

Eero Hyry
Affiliation: National Defence College, Santahamina, FIN-00860 Helsinki, Finland
Email: eero.hyry@helsinki.fi

DOI: 10.1090/S0002-9947-99-02143-1
PII: S 0002-9947(99)02143-1
Received by editor(s): June 1, 1996
Posted: February 23, 1999
Copyright of article: Copyright 1999, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS