Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



Koszul homology and the structure of low codimension Cohen-Macaulay ideals

Author: Wolmer V. Vasconcelos
Journal: Trans. Amer. Math. Soc. 301 (1987), 591-613
MSC: Primary 13H10; Secondary 13C15
MathSciNet review: 882705
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The relationship between the properties of the Koszul homology modules of two ideals connected by linkage is studied. If the ideal $ I$ is either (i) a Cohen-Macaulay ideal of codimension 3, or (ii) a Gorenstein ideal of codimension 4, the one-dimensional Koszul module carries considerable information on the structural nature of the linkage class of $ I$ in case (i), or on the conormal module of $ I$ in case (ii). Emphasis is given to the verification of the properties by computation.

References [Enhancements On Off] (What's this?)

  • [A] R. Apery, Sur les courbes de première espèce de l'espace à trois dimensions, C. R. Acad. Sci. Paris 220 (1945), 271-272. MR 0013559 (7:170c)
  • [AB] M. Auslander and M. Bridger, Stable module theory, Mem. Amer. Math. Soc. No. 94, 1969. MR 0269685 (42:4580)
  • [AH] L. Avramov and J. Herzog, The Koszul algebra of a codimension 2 embedding, Math. Z. 175 (1980), 249-260. MR 602637 (82g:13011)
  • [BSV] P. Brumatti, A. Simis and W. V. Vasconcelos, Normal Rees algebras, J. Algebra (to appear). MR 921962 (89b:13011)
  • [Bu] B. Buchberger, An algorithmic method in polynomial ideal theory, Recent Trends in Mathematical Systems Theory (N. K. Bose, ed.), Reidel, Dordrecht, 1985.
  • [BE] D. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), 447-485. MR 0453723 (56:11983)
  • [BU] R.-O. Buchweitz and B. Ulrich, Homological properties which are invriant under linkage, preprint, 1983.
  • [Ga] F. Gaeta, Quelques progrès récents dans la classification des variétés algébriques d'un espace projective, Deuxième Colloque de Géométrie Algébrique, Thone, Liège, 1952, pp. 145-183. MR 0052828 (14:682b)
  • [Go] S. Goto, The divisor class group of a certain Krull domain, J. Math. Kyoto Univ. 17 (1977), 47-50. MR 0435063 (55:8025)
  • [HO] R. Hartshorne and A. Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415-437. MR 0347821 (50:322)
  • [He$ _{1}$] J. Herzog, Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul, Math. Z. 163 (1978), 149-162. MR 512469 (80a:13025)
  • [He$ _{2}$] -, Komplexe, Auflösungen und Dualität in der lokalen Algebra, Habilitationsschrift, Regensburg Universität, 1974.
  • [He$ _{3}$] -, Homological properties of the module of differentials, Atas VI Escola de Algebra, Soc. Brasileira de Matematica, 1981, pp. 33-64.
  • [HK] J. Herzog and M. Kühl, On the Betti numbers of finite pure and linear resolutions, Comm. Algebra 12 (1984), 1627-1646. MR 743307 (85e:13021)
  • [HM] J. Herzog and M. Miller, Gorenstein ideals of deviation two, Comm. Algebra 13 (1985), 1977-1990. MR 795487 (87b:13027)
  • [HSV$ _{1}$] J. Herzog, A. Simis and W. V. Vasconcelos, Koszul homology and blowing up rings, Proc. Commutative Algebra Trento Conf., Lecture Notes in Pure and Applied Math., vol. 84, Marcel Dekker, New York, 1983, pp. 79-169. MR 686942 (84k:13015)
  • [HSV$ _{2}$] -, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), 661-683. MR 737891 (86a:13015)
  • [HVV] J. Herzog, W. V. Vasconcelos and R. Villarreal, Ideals with sliding depth, Nagoya Math. J. 99 (1985), 159-172. MR 805087 (86k:13022)
  • [Hu$ _{1}$] C. Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), 1043-1062 MR 675309 (84f:13019)
  • [Hu$ _{2}$] -, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), 739-763. MR 694386 (84m:13023)
  • [Hu$ _{3}$] -, Determinantal ideals of linear type, Arch. Math. 47 (1986), 324-329. MR 866520 (87k:13037)
  • [K] E. Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111-115. MR 0330158 (48:8496)
  • [KMU] A. R. Kustin, M. Miller and B. Ulrich, Linkage theory for algebras with pure resolutions, J. Algebra 102 (1986), 199-228. MR 853240 (88a:13021)
  • [Le] K. Lebelt, Freie Auflösungen äußeser Potenzen, Manuscripta Math. 21 (1977), 341-355. MR 0450253 (56:8549)
  • [PS] C. Peskine and L. Szpiro, Liaison des variétés algébriques, Invent. Math. 26 (1974), 271-302. MR 0364271 (51:526)
  • [SV] A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), 203-224. MR 610474 (82i:13016)
  • [V] W. V. Vasconcelos, Ideals generated by $ R$-sequences, J. Algebra 6 (1967), 309-316. MR 0213345 (35:4209)
  • [VV] W. V. Vasconcelos and R. Villarreal, On Gorenstein ideals of codimension four, Proc. Amer. Math. Soc. 98 (1986), 205-210. MR 854019 (87k:13041)
  • [Vi] R. Villarreal, Koszul homology of Cohen-Macaulay ideals, Ph. D. Thesis, Rutgers University, 1986.
  • [Wa] J. Watanabe, A note on Gorenstein rings of embedding codimension 3, Nagoya Math. J. 50 (1973), 227-232. MR 0319985 (47:8526)
  • [We] J. Weyman, Resolutions of the exterior and symmetric powers of a module, J. Algebra 58 (1979), 333-34l. MR 540642 (80i:13005)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13H10, 13C15

Retrieve articles in all journals with MSC: 13H10, 13C15

Additional Information

Keywords: Linkage, Cohen-Macaulay ideal, Gorenstein ideal, Koszul homology
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society