Transactions of the American Mathematical Society

Author(s): Takesi Kawasaki
Journal: Trans. Amer. Math. Soc. 354 (2002), 123-149.
MSC (1991): Primary 13A30; Secondary 13D45, 13H10
Posted: June 6, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp's conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

1.: Ian M. Aberbach, Arithmetic Macaulayfications using ideals of dimension one, Illinois J. Math. 40 (1996), 518-526. MR 97h:13002
2.: Yoichi Aoyama and Shiro Goto, A brief summary of the elements of the theory of dualizing complexes and Sharp's conjecture, The Curve Seminar at Queen's, Vol. 4, Queen's Papers in Pure and Appl. Math., vol. 76, 1986. MR 88g:13013
3.: -, Some special cases of a conjecture of Sharp, J. Math. Kyoto Univ. 26 (1986), 613-634. MR 88h:13013
4.: -, A conjecture of Sharp--the case of local rings with $\dim \operatorname{non-CM}\leq 1$ or $\dim \leq 5$ , Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Kinokuniya, 1988, pp. 27-34. MR 90b:13018
5.: Jacob Barshay, Graded algebras of powers of ideals generated by -sequences, J. Algebra 25 (1973), 90-99. MR 48:11074
6.: Dave Bayer and Michael Stillman, Macaulay: A system for computation in algebraic geometry and commutative algebra, 1982-1994, Source and object code available for Unix and Macintosh computers. Contact the authors, or download from math.harvard.edu via anonymous ftp.
7.: Markus Brodmann, Local cohomology of certain Rees- and form-rings I, J. Algebra 81 (1983), 29-57. MR 85b:13030
8.: Nguyen Tu Cuong, P-standard systems of parameters and p-standard ideals in local rings, Acta Math. Vietnam. 20 (1995), 145-161. MR 96h:13064
9.: Shiro Goto, Blowing-up of Buchsbaum rings, Proceedings, Durham symposium on Commutative Algebra, London Math. Soc. Lect. Notes, vol. 72, Cambridge Univ. Press, 1982, pp. 140-162. MR 84h:13032
10.: -, On the associated graded rings of Buchsbaum rings, J. Algebra 85 (1983), 490-534. MR 85d:13032
11.: Shiro Goto and Yasuhiro Shimoda, On Rees algebras over Buchsbaum rings, J. Math. Kyoto Univ. 20 (1980), 691-708. MR 82c:13028
12.: Shiro Goto and Kikumichi Yamagishi, The theory of unconditioned strong d-sequences and modules of finite local cohomology, preprint.
13.: Alexander Grothendieck, Local cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, Berlin, Heiderberg, New-York, 1967. MR 37:219
14.: Robin Hartshorne, Residue and duality, Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin, Heidelberg, New York, 1966. MR 36:5145
15.: Manfred Herrmann, Eero Hyry, and Jürgen Ribbe, On the Cohen-Macaulay and Gorenstein properties of multigraded Rees algebra, Manuscripta Math. 79 (1993), 343-377. MR 94h:13003
16.: Eero Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), 2213-2232. MR 99i:13005
17.: Takesi Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc. 352 (2000), 2517-2552. MR 2000j:14077
18.: -, On arithmetic Macaulayfication of certain local rings, Comm. Algebra 26 (1998), 4385-4396. MR 99m:13007
19.: Kazuhiko Kurano, On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal, J. Algebra 190 (1997), 405-434, with an appendix by Yamagishi, Kikumich. MR 98f:13004
20.: Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Math., vol. 8, Cambridge University Press, 1986, First paperback edition, 1989. MR 90i:13001; MR 88h:13001
21.: Masayoshi Nagata, Local rings, Interscience Tracts, vol. 13, Interscience, New York-London-Sydney, 1962. MR 27:5790
22.: Tetsushi Ogoma, Existence of dualizing complexes, J. Math. Kyoto Univ. 24 (1984), 27-48. MR 85j:13028
23.: -, Associated primes of fibre product rings and a conjecture of Sharp in lower dimensional cases, Mem. Fac. Sci. Kochi Univ. (Math.) 6 (1985), 1-9. MR 86e:13009
24.: -, Fibre products of Noetherian rings and their applications, Math. Proc. Cambridge Philos. Soc. 97 (1985), 231-241. MR 86e:13008
25.: Idun Reiten, The converse to a theorem of Sharp in Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417-420. MR 45:5128
26.: Peter Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture Notes in Math., vol. 907, Springer, Berlin-Heidelberg-New York, 1982. MR 83i:13013
27.: -, Standard systems of parameters and their blowing-up rings, J. Reine Angew. Math. 344 (1983), 201-220. MR 84m:13025
28.: Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197-278.
29.: Rodney Y. Sharp, Acceptable rings and homomorphic images of Gorenstein rings, J. Algebra 44 (1977), 246-261. MR 56:348
30.: -, Necessary conditions for the existence of dualizing complexes in commutative algebra, Sém. Algèbre P. Dubreil 1977/78, Lecture Notes in Mathematics, vol. 740, Springer-Verlag, 1979, pp. 213-229. MR 81d:13013
31.: Yasuhiro Shimoda, A note on Rees algebras of two dimensional local domains, J. Math. Kyoto Univ. 19 (1979), 327-333. MR 80k:13011
32.: Giuseppe Valla, Certain graded algebras are always Cohen-Macaulay, J. Algebra 42 (1976), 537-548. MR 54:10240

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 13A30, 13D45, 13H10

Takesi Kawasaki
Affiliation: Department of Mathematics, Tokyo Metropolitan University, Hachioji-shi Minami-Ohsawa 1-1, Tokyo 192-0397, Japan
Email: kawasaki@comp.metro-u.ac.jp

DOI: 10.1090/S0002-9947-01-02817-3
PII: S 0002-9947(01)02817-3
Keywords: Arithmetic Macaulayfication, Cohen-Macaulay rings, dualizing complex, excellent rings, formal fibers, local cohomology, Macaulayfication, Rees algebra
Received by editor(s): February 15, 2000
Posted: June 6, 2001
Additional Notes: The author is partially supported by Grant-in-Aid for Co-Operative Research
Copyright of article: Copyright 2001, 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 2008, American Mathematical Society Privacy Statement		Search the AMS