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

Vasconcelos, Wolmer V.

doi:10.1090/S0002-9947-1987-0882705-X

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
DOI: https://doi.org/10.1090/S0002-9947-1987-0882705-X
MathSciNet review: 882705
Full-text PDF Free Access

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 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 in case (i), or on the conormal module of in case (ii). Emphasis is given to the verification of the properties by computation.

[A] Roger Apéry, Sur les courbes de première espèce de l’espace à trois dimensions, C. R. Acad. Sci. Paris 220 (1945), 271–272 (French). MR 13559
[AB] Maurice Auslander and Mark Bridger, Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969. MR 0269685
[AH] Lâcezar Avramov and Jürgen Herzog, The Koszul algebra of a codimension 2 embedding, Math. Z. 175 (1980), no. 3, 249–260. MR 602637, https://doi.org/10.1007/BF01163026
[BSV] P. Brumatti, A. Simis, and W. V. Vasconcelos, Normal Rees algebras, J. Algebra 112 (1988), no. 1, 26–48. MR 921962, https://doi.org/10.1016/0021-8693(88)90130-5
[Bu] B. Buchberger, An algorithmic method in polynomial ideal theory, Recent Trends in Mathematical Systems Theory (N. K. Bose, ed.), Reidel, Dordrecht, 1985.
[BE] David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, https://doi.org/10.2307/2373926
[BU] R.-O. Buchweitz and B. Ulrich, Homological properties which are invriant under linkage, preprint, 1983.
[Ga] Federico Gaeta, Quelques progrès récents dans la classification des variétés algébriques d’un espace projectif, Deuxième Colloque de Géométrie Algébrique, Liège, 1952, Georges Thone, Liège; Masson & Cie, Paris, 1952, pp. 145–183 (French). MR 0052828
[Go] Shiro Goto, The divisor class group of a certain Krull domain, J. Math. Kyoto Univ. 17 (1977), no. 1, 47–50. MR 435063, https://doi.org/10.1215/kjm/1250522811
[HO] Robin Hartshorne and Arthur Ogus, On the factoriality of local rings of small embedding codimension, Comm. Algebra 1 (1974), 415–437. MR 347821, https://doi.org/10.1080/00927877408548627
[He $_{1}$ ] Jürgen Herzog, Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul, Math. Z. 163 (1978), no. 2, 149–162 (German). MR 512469, https://doi.org/10.1007/BF01214062
[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), no. 13-14, 1627–1646. MR 743307, https://doi.org/10.1080/00927878408823070
[HM] Jürgen Herzog and Matthew Miller, Gorenstein ideals of deviation two, Comm. Algebra 13 (1985), no. 9, 1977–1990. MR 795487, https://doi.org/10.1080/00927878508823261
[HSV $_{1}$ ] J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
[HSV $_{2}$ ] J. Herzog, A. Simis, and W. V. Vasconcelos, On the arithmetic and homology of algebras of linear type, Trans. Amer. Math. Soc. 283 (1984), no. 2, 661–683. MR 737891, https://doi.org/10.1090/S0002-9947-1984-0737891-6
[HVV] J. Herzog, W. V. Vasconcelos, and R. Villarreal, Ideals with sliding depth, Nagoya Math. J. 99 (1985), 159–172. MR 805087, https://doi.org/10.1017/S0027763000021553
[Hu $_{1}$ ] Craig Huneke, Linkage and the Koszul homology of ideals, Amer. J. Math. 104 (1982), no. 5, 1043–1062. MR 675309, https://doi.org/10.2307/2374083
[Hu $_{2}$ ] Craig Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), no. 2, 739–763. MR 694386, https://doi.org/10.1090/S0002-9947-1983-0694386-5
[Hu $_{3}$ ] Craig Huneke, Determinantal ideals of linear type, Arch. Math. (Basel) 47 (1986), no. 4, 324–329. MR 866520, https://doi.org/10.1007/BF01191358
[K] Ernst Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111–115. MR 330158, https://doi.org/10.1016/0021-8693(74)90025-8
[KMU] Andrew R. Kustin, Matthew Miller, and Bernd Ulrich, Linkage theory for algebras with pure resolutions, J. Algebra 102 (1986), no. 1, 199–228. MR 853240, https://doi.org/10.1016/0021-8693(86)90137-7
[Le] Karsten Lebelt, Freie Auflösungen äusserer Potenzen, Manuscripta Math. 21 (1977), no. 4, 341–355 (German, with English summary). MR 450253, https://doi.org/10.1007/BF01167853
[PS] C. Peskine and L. Szpiro, Liaison des variétés algébriques. I, Invent. Math. 26 (1974), 271–302 (French). MR 364271, https://doi.org/10.1007/BF01425554
[SV] A. Simis and W. V. Vasconcelos, The syzygies of the conormal module, Amer. J. Math. 103 (1981), no. 2, 203–224. MR 610474, https://doi.org/10.2307/2374214
[V] Wolmer V. Vasconcelos, Ideals generated by 𝑅-sequences, J. Algebra 6 (1967), 309–316. MR 213345, https://doi.org/10.1016/0021-8693(67)90086-5
[VV] W. V. Vasconcelos and R. Villarreal, On Gorenstein ideals of codimension four, Proc. Amer. Math. Soc. 98 (1986), no. 2, 205–210. MR 854019, https://doi.org/10.1090/S0002-9939-1986-0854019-X
[Vi] R. Villarreal, Koszul homology of Cohen-Macaulay ideals, Ph. D. Thesis, Rutgers University, 1986.
[Wa] Junzo Watanabe, A note on Gorenstein rings of embedding codimension three, Nagoya Math. J. 50 (1973), 227–232. MR 319985
[We] Jerzy Weyman, Resolutions of the exterior and symmetric powers of a module, J. Algebra 58 (1979), no. 2, 333–341. MR 540642, https://doi.org/10.1016/0021-8693(79)90164-9