On arithmetic Macaulayfication of Noetherian rings

Kawasaki, Takesi

doi:10.1090/S0002-9947-01-02817-3

On arithmetic Macaulayfication of Noetherian rings
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by Takesi Kawasaki

Trans. Amer. Math. Soc. 354 (2002), 123-149

DOI: https://doi.org/10.1090/S0002-9947-01-02817-3

Published electronically: June 6, 2001

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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.

References

Ian M. Aberbach, Arithmetic Macaulayfications using ideals of dimension one, Illinois J. Math. 40 (1996), no. 3, 518–526. MR 1407634
Y. Aoyama and S. Goto, A brief summary of the elements of the theory of dualizing complexes and Sharp’s conjecture, The curves seminar at Queen’s, Vol. IV (Kingston, Ont., 1985–1986) Queen’s Papers in Pure and Appl. Math., vol. 76, Queen’s Univ., Kingston, ON, 1986, pp. Exp. No. A, 24. MR 883966
Y\B{o}ichi Aoyama and Shiro Goto, Some special cases of a conjecture of Sharp, J. Math. Kyoto Univ. 26 (1986), no. 4, 613–634. MR 864465, DOI 10.1215/kjm/1250520830
Y\B{o}ichi Aoyama and Shiro Goto, A conjecture of Sharp—the case of local rings with $\textrm {dim}\,\textrm {nonCM}\leq 1$ or $\textrm {dim}\leq 5$, Algebraic geometry and commutative algebra, Vol. I, Kinokuniya, Tokyo, 1988, pp. 27–34. MR 977750
Jacob Barshay, Graded algebras of powers of ideals generated by $A$-sequences, J. Algebra 25 (1973), 90–99. MR 332748, DOI 10.1016/0021-8693(73)90076-8
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.
Markus Brodmann, Local cohomology of certain Rees- and form-rings. I, J. Algebra 81 (1983), no. 1, 29–57. MR 696125, DOI 10.1016/0021-8693(83)90207-7
Nguyễn Tụ’ Cu’ò’ng, $p$-standard systems of parameters and $p$-standard ideals in local rings, Acta Math. Vietnam. 20 (1995), no. 1, 145–161. MR 1346354
Shiro Goto, Blowing-up of Buchsbaum rings, Commutative algebra: Durham 1981 (Durham, 1981) London Math. Soc. Lecture Note Ser., vol. 72, Cambridge Univ. Press, Cambridge-New York, 1982, pp. 140–162. MR 693633
Shiro Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra 85 (1983), no. 2, 490–534. MR 725097, DOI 10.1016/0021-8693(83)90109-6
Shiro Goto and Yasuhiro Shimoda, On Rees algebras over Buchsbaum rings, J. Math. Kyoto Univ. 20 (1980), no. 4, 691–708. MR 592354, DOI 10.1215/kjm/1250522166
Shiro Goto and Kikumichi Yamagishi, The theory of unconditioned strong d-sequences and modules of finite local cohomology, preprint.
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
Manfred Herrmann, Eero Hyry, and Jürgen Ribbe, On the Cohen-Macaulay and Gorenstein properties of multigraded Rees algebras, Manuscripta Math. 79 (1993), no. 3-4, 343–377. MR 1223028, DOI 10.1007/BF02568351
Eero Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2213–2232. MR 1467469, DOI 10.1090/S0002-9947-99-02143-1
Takesi Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2517–2552. MR 1707481, DOI 10.1090/S0002-9947-00-02603-9
Takesi Kawasaki, On arithmetic Macaulayfication of certain local rings, Comm. Algebra 26 (1998), no. 12, 4385–4396. MR 1661269, DOI 10.1080/00927879808826417
Kazuhiko Kurano, On Macaulayfication obtained by a blow-up whose center is an equi-multiple ideal, J. Algebra 190 (1997), no. 2, 405–434. With an appendix by Kikumichi Yamagishi. MR 1441955, DOI 10.1006/jabr.1996.6904
John W. Green, Harmonic functions in domains with multiple boundary points, Amer. J. Math. 61 (1939), 609–632. MR 90, DOI 10.2307/2371316
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Tetsushi Ogoma, Existence of dualizing complexes, J. Math. Kyoto Univ. 24 (1984), no. 1, 27–48. MR 737823, DOI 10.1215/kjm/1250521383
Tetsushi Ogoma, Associated primes of fibre product rings and a conjecture of Sharp in lower-dimensional cases, Mem. Fac. Sci. Kochi Univ. Ser. A Math. 6 (1985), 1–9. MR 772492
Tetsushi Ogoma, Fibre products of Noetherian rings and their applications, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 2, 231–241. MR 771818, DOI 10.1017/S0305004100062794
Idun Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. MR 296067, DOI 10.1090/S0002-9939-1972-0296067-7
Peter Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture Notes in Mathematics, vol. 907, Springer-Verlag, Berlin-New York, 1982 (German). With an English summary. MR 654151, DOI 10.1007/BFb0094123
Peter Schenzel, Standard systems of parameters and their blowing-up rings, J. Reine Angew. Math. 344 (1983), 201–220. MR 716256, DOI 10.1515/crll.1983.344.201
Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278.
Rodney Y. Sharp, Acceptable rings and homomorphic images of Gorenstein rings, J. Algebra 44 (1977), no. 1, 246–261. MR 441957, DOI 10.1016/0021-8693(77)90180-6
Rodney Y. Sharp, Necessary conditions for the existence of dualizing complexes in commutative algebra, Séminaire d’Algèbre Paul Dubreil 31ème année (Paris, 1977–1978) Lecture Notes in Math., vol. 740, Springer, Berlin, 1979, pp. 213–229. MR 563505
Yasuhiro Shimoda, A note on Rees algebras of two-dimensional local domains, J. Math. Kyoto Univ. 19 (1979), no. 2, 327–333. MR 545713, DOI 10.1215/kjm/1250522436
G. Valla, Certain graded algebras are always Cohen-Macaulay, J. Algebra 42 (1976), no. 2, 537–548. MR 422249, DOI 10.1016/0021-8693(76)90112-5