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On arithmetic Macaulayfication of Noetherian rings


Author: Takesi Kawasaki
Journal: Trans. Amer. Math. Soc. 354 (2002), 123-149
MSC (1991): Primary 13A30; Secondary 13D45, 13H10
DOI: https://doi.org/10.1090/S0002-9947-01-02817-3
Published electronically: June 6, 2001
MathSciNet review: 1859029
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Abstract:

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.


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  • 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 $A$-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

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Additional Information

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: https://doi.org/10.1090/S0002-9947-01-02817-3
Keywords: Arithmetic Macaulayfication, Cohen-Macaulay rings, dualizing complex, excellent rings, formal fibers, local cohomology, Macaulayfication, Rees algebra
Received by editor(s): February 15, 2000
Published electronically: June 6, 2001
Additional Notes: The author is partially supported by Grant-in-Aid for Co-Operative Research
Article copyright: © Copyright 2001 American Mathematical Society

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