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, Volume 45, Issue 2, pp 161–191 | Cite as

Generische determinantielle Singularitäten: Homologische Eigenschaften des Derivationenmoduls

  • Udo Vetter


Let K be a field, (X u i ) an (m,n)-matrix of indeterminates over K, and r an integer such that i≤r<m≤n. Let P be the localization of K[Xu/i: 1≤i≤m, 1≤u≤n] at the irrelevant maximal ideal, and S the factorring of P with respect to the ideal generated by all (r+1)-minors of (X u i ). By D=DK(S) we denote the module of Kahler-differentials of S over K and by D* its S-dual Homs(D,S). It will be shown that D*. is a Cohen-Macaulay-module if m<nand has depth equal to dim S−1 if m=n. By this we can describe the syzygetic behaviour of D*. In particular one gets a result of Svanes about the vanishing of the modules Ext S i (D,S) and in some sense a complete description of the rigidity of S.


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  1. [1]
    Auslander, M. and M. Bridger: Stable Module Theory. Mem. Amer. Math. Soc.94 (1969)Google Scholar
  2. [2]
    Bruns, W.: The canonical module of a determinantal ring. In: Sharp, R.Y. (ed.), Commutative Algebra, Durham 1981. London Math. Soc. Lecture Note Series72, Cambridge, 109–120 (1982)Google Scholar
  3. [3]
    Bruns, W.: Generic maps and modules. Compositia Math.47, 171–193 (1982)Google Scholar
  4. [4]
    Buchweitz, R.O.: Déformations de Diagrammes, Déploiements et Singularités très rigides, Liaisions algébriques. These, Paris (1981)Google Scholar
  5. [5]
    De Concini, C., D. Eisenbud and C. Procesi: hodge algebras, astérisque91 (1982)Google Scholar
  6. [6]
    Eisenbud, D.: Introduction to Algebras with Straightening Laws. In: McDonald, B.R. (ed.), Ring Theory and Algebra III. Proc. of the third Oklahoma Conf., M. Dekker, New York and Basel, 243–267 (1980)Google Scholar
  7. [7]
    Eagon, J.A. and D.G. Northcott: Generically acyclic complexes and generically perfect ideals. Proc. Royal Soc. A299, 147–172 (1967)Google Scholar
  8. [8]
    Jähner, U.: Beispiele starrer analytischer Algebren. Dissertation, Clausthal (1974)Google Scholar
  9. [9]
    Matsumura, H.: Commutative Algebra. W.A. Benjamin, New York 1970Google Scholar
  10. [10]
    Svanes, T.: Coherent Cohomology on flag manifolds and rigidity. Ph. D. Thesis, MIT, Cambridge Mass. (1972)Google Scholar
  11. [11]
    Vetter, U,: The depth of the module of differentials of a generic de terminanta1 singularity. Comm. in Algebra11, 1701–1724 (1983)Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Udo Vetter
    • 1
  1. 1.Fachbereich MathematikUniversität Osnabrück Abteilung VechtaVechtaBundesrepublik Deutschland

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