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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Linkage and sums of ideals


Author: Mark R. Johnson
Journal: Trans. Amer. Math. Soc. 350 (1998), 1913-1930
MSC (1991): Primary 13C40, 13C14
DOI: https://doi.org/10.1090/S0002-9947-98-01976-X
MathSciNet review: 1432202
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Abstract: It is shown (under mild conditions) that the sum of transversal ideals in a regular local ring cannot lie in the linkage class of a complete intersection. For a sum of geometrically linked Cohen-Macaulay ideals, we compute the depths of the conormal module and the first Koszul homology. As applications, we construct general examples of ideals which are strongly Cohen-Macaulay, strongly nonobstructed but not in the linkage class of a complete intersection, and Gorenstein ideals which are strongly nonobstructed but not syzygetic.


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  • 1. R. Apéry, Sur les courbes de première espèce de l'espace de trois dimensions, C. R. Acad. Sci. Paris 220 (1945), 271-272. MR 7:170c
  • 2. M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 632-647. MR 86i:16032
  • 3. R.-O. Buchweitz, Contributions à la thèorie des singularités, Thesis, l'Université de Paris, 1981.
  • 4. R.-O. Buchweitz and B. Ulrich, Homological properties which are invariant under linkage, preprint.
  • 5. F. Gaeta, Détermination de la chaine syzygétique des idéaux matriciels parfaits et son application à la postulation de leurs variétés algébriques associées, C. R. Acad. Sci. Paris 234 (1954), 1833-1835. MR 13:978a
  • 6. J. Herzog, Deformationen von Cohen-Macaulay Algebren, J. Reine Angew. Math. 318 (1980), 83-105. MR 81m:13012
  • 7. J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative Algebra, Proceedings, Trento, 1981, Lecture Notes in Pure and Appl. Math. 84, Marcel Dekker, 1983, pp. 79-169. MR 84a:13002
  • 8. C. Huneke, Linkage and Koszul homology of ideals, Amer. J. Math. 104 (1982), 1043-1062. MR 84f:13019
  • 9. C. Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), 739-763. MR 84m:13023
  • 10. C. Huneke and B. Ulrich, Divisor class groups and deformations, Amer. J. Math. 107 (1985), 1265-1303. MR 87f:13010
  • 11. C. Huneke and B. Ulrich, The structure of linkage, Ann. of Math. 126 (1987), 277-334. MR 88k:13020
  • 12. C. Huneke and B. Ulrich, Algebraic linkage, Duke Math. J. 56 (1988), 415-429. MR 89e:13023
  • 13. C. Huneke and B. Ulrich, Residual intersections, J. Reine Angew. Math. 390 (1988), 1-20. MR 89j:13024
  • 14. C. Huneke and B. Ulrich, Powers of licci ideals, in Commutative Algebra (Berkeley), Math. Sci. Res. Int. Publ. 15, Springer, 1989, pp. 339-346. MR 90i:13008
  • 15. C. Huneke and B. Ulrich, Local properties of licci ideals, Math. Z. 211 (1992), 129-154. MR 93j:13018
  • 16. M. Johnson and B. Ulrich, Artin-Nagata properties and Cohen-Macaulay associated graded rings, Compositio Math. 103 (1996), 7-29. MR 97f:13006
  • 17. A. Kustin and M. Miller, A general resolution for grade four Gorenstein ideals, Manuscripta Math. 35 (1981), 221-269. MR 83g:14026
  • 18. A. Kustin and M. Miller, Deformation and linkage of Gorenstein algebras, Trans. Amer. Math. Soc. 284 (1984), 501-533. MR 85k:13015
  • 19. S. Lichtenbaum, On the vanishing of Tor in regular local rings, Illinois J. Math. 10 (1966), 220-226. MR 32:5688
  • 20. E. Lopez, Licci Gorenstein ideals of deviation two, Ph.D. thesis, Michigan State University (1988).
  • 21. C. Peskine and L. Szpiro, Liaison des variétés algébriques, Invent. Math. 26 (1974), 271-302. MR 51:526
  • 22. A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), 203-224. MR 82i:13016
  • 23. A. Simis, W. V. Vasconcelos, and R. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), 389-416. MR 95e:13002
  • 24. H. Srinivasan, A grade five Gorenstein algebra with no minimal algebra resolutions, J. Algebra 179 (1996), 362-379. MR 96j:13012
  • 25. B. Ulrich, Theory and applications of universal linkage, in Commutative Algebra and Combinatorics, M. Nagata and H. Matsumura (eds.), Adv. Studies in Pure Math. 11, North-Holland, Amsterdam, 1987, pp. 285-301. MR 89b:13001
  • 26. B. Ulrich, Sums of linked ideals, Trans. Amer. Math. Soc. 318 (1990), 1-42. MR 90f:13012
  • 27. B. Ulrich, Artin-Nagata properties and reductions of ideals, Contemp. Math. 159 (1994), 373-400. MR 95a:13017
  • 28. B. Ulrich, Parafactoriality and small divisor class groups, in preparation.
  • 29. W. V. Vasconcelos, Koszul homology and the structure of low codimensional Cohen-Macaulay ideals, Trans. Amer. Math. Soc. 301 (1987), 591-613. MR 88i:13031
  • 30. W. V. Vasconcelos, Arithmetic of Blowup Algebras, London Math. Soc. Lecture Note Series, 1993. MR 95g:13005
  • 31. W. V. Vasconcelos and R. Villarreal, On Gorenstein ideals of codimension four, Proc. Amer. Math. Soc. 98 (1986), 205-210. MR 87k:13041
  • 32. R. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), 277-293. MR 91b:13031
  • 33. J. Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227-232. MR 47:8526

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Additional Information

Mark R. Johnson
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: mark@math.uark.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01976-X
Received by editor(s): June 10, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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