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Cohen-Macaulay modules on quadrics

Research-Articles Singularities

Part of the Lecture Notes in Mathematics book series (LNM,volume 1273)

Abstract

This paper analyzes the graded maximal Cohen-Macaulay modules over rings of the form R=k[x1,...,xr]/Q, when Q is a quadratic form defining a regular projective hypersurface, and k is an arbitrary field (the case when k is algebraically closed of characteristic ≠2 is a special case of the theory developed by Knörrer [1986]). For any nonzero quadratic form Q, regular or not, the graded maximal Cohen-Macaulay R-modules define modules over the even Clifford algebra of Q, and we show that this algebra is semi-simple iff Q is regular (this is classical for char k ≠2). As a result of this and other information about the Clifford algebra, we give a detailed account of the Cohen-Macaulay modules when Q is regular, identifying the number of indecomposables (2 or 3, counting R) their ranks, and the relations of duality and syzygy among them.

Keywords

  • Quadratic Form
  • Spectral Sequence
  • Matrix Factorization
  • Complete Intersection
  • Division Algebra

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

with an appendix by Ragnar-Olaf Buchweitz

Supported by a "Heisenberg-Stipendium", Bu-398/3-1 of the DFG

Partially supported by the NSF

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References

  1. Atiyah, M. F., Bott, R., and Shapiro, A.: Clifford Modules. Topology 3, Suppl. (1964), 3–38.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Auslander, M.: Isolated singularities and existence of almost split sequences. In Representation Theory II, Groups and Orders, Lecture Notes 1178 (1986), 194–241, Springer-Verlag, New York.

    CrossRef  Google Scholar 

  3. Auslander, M. and Reiten, I.: The Cohen-Macaulay Type of Cohen-Macaulay Rings. Preprint (1986).

    Google Scholar 

  4. Buchweitz, R.-O., Greuel, G.-M., and Schreyer, F.-O.: Cohen-Macaulay modules on hypersurface singularities II. Preprint (1986), to appear in Inventiones.

    Google Scholar 

  5. Cohen, P. M.: Algebra, Vol. 2, John Wiley and Sons, (1977).

    Google Scholar 

  6. Eisenbud, D.: Homological Algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260 (1980), 35–64.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Geramita, A. V., and Seberry, J.: Orthogonal designs; quadratic forms and Hadamard matrices. Lecture Notes in Pure and Applied Math. 45 (1979), Marcel Dekker.

    Google Scholar 

  8. Hurwitz, A., Über die Komposition der quadratischen Formen von beliebig vielen Variablen. Nachrichten der k. Gesellschaft der Wissenschaften zu Göttingen (1898), 309–316.

    Google Scholar 

  9. Jacobson, N.: Basic Algebra II, W. H. Freeman and Co., San Francisco, (1980).

    MATH  Google Scholar 

  10. Kneser, M.: Bestimmung des Zentrums der Cliffordschen Algebren einer quadratischen Form über einem Körper der Charakteristik 2. J. Reine u. Angew. Math. 193 (1954), 123–125.

    MathSciNet  MATH  Google Scholar 

  11. Knörrer, H.: Cohen-Macaulay modules on hypersurface singularities I. Preprint (1986), to appear in Inventiones.

    Google Scholar 

  12. Lam, T. Y.: The Algebraic Theory of Quadratic Forms. 2nd printing, with rev. Benjamin, (1980).

    Google Scholar 

  13. Matsumura, H.: Commutative Algebra, 2nd ed., Benjamin Cummings Publ. Co., Reading Mass., (1980).

    MATH  Google Scholar 

  14. Scharlau, W.: Quadratic Forms, Springer-Verlag, New York, (1985).

    MATH  Google Scholar 

  15. Swan, R.: K-theory of quadric hypersurfaces. Annals of Math. 122 (1985), 113–153.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. Vignéras, M.-F.: Arithmétique des Algèbres de Quaternion. Lecture Notes 800 (1980), Springer Verlag, New York.

    CrossRef  MATH  Google Scholar 

References

  1. M. F. Atiyah, R. Bott, A. Shapiro: Clifford Modules, Topology 3 (Suppl.), 3–38, (1964)

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. N. Bourbaki, Algèbre, Masson, Paris 1970 ff

    Google Scholar 

  3. L. Avramov: Local algebra and rational homotopy, Astérisque 113/114, 15–43, (1984)

    MathSciNet  MATH  Google Scholar 

  4. A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Astérisque 100, (1983)

    Google Scholar 

  5. I. N. Bernstein, I. M. Gelfand, S. I. Gelfand: Algebraic Bundles over IPn and Problems of Linear Algebra, Funkt. Anal. 12, No. 3, 66–67, (1978)-engl. translation: 212–214, (1979)-

    MathSciNet  MATH  Google Scholar 

  6. R. Bøgvad, S. Halperin: On a conjecture of Roos, pp. 120–127, in “Algebra, Algebraic Topology and their Interactions”, Proc. Stockholm 1983, ed. by J.-E. Roos, Springer Lect. Notes in Math. 1183, Springer-Verlag Berlin-Heidelberg-New York, 1986

    Google Scholar 

  7. J. Backelin, J.-E. Roos: When is the double Yoneda-Ext-Algebra of a local noetherian ring again noetherian?, pp. 101–120, in the same volume as [B-H]

    Google Scholar 

  8. R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings; preprint.

    Google Scholar 

  9. D. Eisenbud: Homological Algebra on a Complete Intersection, with an Application to Group Representations. Transactions of the AMS 260, 35–64 (1980)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. R. Hartshorne: Residues and Duality, Springer Lecture Notes in Math. 20, Springer-Verlag, Berlin-Heidelberg-New York, 1966

    MATH  Google Scholar 

  11. I. Kaplansky: Commutative Rings, rev. edition, Univ. of Chicago Press, Chicago Ill., 1974

    MATH  Google Scholar 

  12. C. Löfwall: On the subalgebra generated by one-dimensional elements in the Yoneda-Ext-Algebra, pp. 291–339 in the same volume as [B-H]

    Google Scholar 

  13. S. Priddy: Koszul-resolutions, Transactions of the AMS 152, 39–60, (1970)

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. D. Quillen: On the (co-)homology of commutative rings, Proc. Symp. Pure Math. 17, 65–87, AMS, Providence, 1970

    MATH  Google Scholar 

  15. R. Swan: K-theory of quadric hypersurfaces. Annals of Math. 122 (1985), 113–153

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. J.-L. Verdier: Catégories derivées (Etat O), in Sém. de Géom. Algébrique 4 1/2, Springer Lecture Notes in Math. 569 (1977)

    Google Scholar 

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Dedicated to Maurice Auslander on the occasion of his 60th birthday

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Buchweitz, RO., Eisenbud, D., Herzog, J. (1987). Cohen-Macaulay modules on quadrics. In: Greuel, GM., Trautmann, G. (eds) Singularities, Representation of Algebras, and Vector Bundles. Lecture Notes in Mathematics, vol 1273. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078838

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  • DOI: https://doi.org/10.1007/BFb0078838

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