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.
with an appendix by Ragnar-Olaf Buchweitz
Supported by a "Heisenberg-Stipendium", Bu-398/3-1 of the DFG
Partially supported by the NSF
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Atiyah, M. F., Bott, R., and Shapiro, A.: Clifford Modules. Topology 3, Suppl. (1964), 3–38.
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.
Auslander, M. and Reiten, I.: The Cohen-Macaulay Type of Cohen-Macaulay Rings. Preprint (1986).
Buchweitz, R.-O., Greuel, G.-M., and Schreyer, F.-O.: Cohen-Macaulay modules on hypersurface singularities II. Preprint (1986), to appear in Inventiones.
Cohen, P. M.: Algebra, Vol. 2, John Wiley and Sons, (1977).
Eisenbud, D.: Homological Algebra on a complete intersection, with an application to group representations. Trans. Am. Math. Soc. 260 (1980), 35–64.
Geramita, A. V., and Seberry, J.: Orthogonal designs; quadratic forms and Hadamard matrices. Lecture Notes in Pure and Applied Math. 45 (1979), Marcel Dekker.
Hurwitz, A., Über die Komposition der quadratischen Formen von beliebig vielen Variablen. Nachrichten der k. Gesellschaft der Wissenschaften zu Göttingen (1898), 309–316.
Jacobson, N.: Basic Algebra II, W. H. Freeman and Co., San Francisco, (1980).
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.
Knörrer, H.: Cohen-Macaulay modules on hypersurface singularities I. Preprint (1986), to appear in Inventiones.
Lam, T. Y.: The Algebraic Theory of Quadratic Forms. 2nd printing, with rev. Benjamin, (1980).
Matsumura, H.: Commutative Algebra, 2nd ed., Benjamin Cummings Publ. Co., Reading Mass., (1980).
Scharlau, W.: Quadratic Forms, Springer-Verlag, New York, (1985).
Swan, R.: K-theory of quadric hypersurfaces. Annals of Math. 122 (1985), 113–153.
Vignéras, M.-F.: Arithmétique des Algèbres de Quaternion. Lecture Notes 800 (1980), Springer Verlag, New York.
References
M. F. Atiyah, R. Bott, A. Shapiro: Clifford Modules, Topology 3 (Suppl.), 3–38, (1964)
N. Bourbaki, Algèbre, Masson, Paris 1970 ff
L. Avramov: Local algebra and rational homotopy, Astérisque 113/114, 15–43, (1984)
A. A. Beilinson, J. Bernstein, P. Deligne: Faisceaux pervers, Astérisque 100, (1983)
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)-
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
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]
R.-O. Buchweitz: Maximal Cohen-Macaulay modules and Tate-Cohomology over Gorenstein rings; preprint.
D. Eisenbud: Homological Algebra on a Complete Intersection, with an Application to Group Representations. Transactions of the AMS 260, 35–64 (1980)
R. Hartshorne: Residues and Duality, Springer Lecture Notes in Math. 20, Springer-Verlag, Berlin-Heidelberg-New York, 1966
I. Kaplansky: Commutative Rings, rev. edition, Univ. of Chicago Press, Chicago Ill., 1974
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]
S. Priddy: Koszul-resolutions, Transactions of the AMS 152, 39–60, (1970)
D. Quillen: On the (co-)homology of commutative rings, Proc. Symp. Pure Math. 17, 65–87, AMS, Providence, 1970
R. Swan: K-theory of quadric hypersurfaces. Annals of Math. 122 (1985), 113–153
J.-L. Verdier: Catégories derivées (Etat O), in Sém. de Géom. Algébrique 4 1/2, Springer Lecture Notes in Math. 569 (1977)
Author information
Authors and Affiliations
Editor information
Additional information
Dedicated to Maurice Auslander on the occasion of his 60th birthday
Rights and permissions
Copyright information
© 1987 Springer-Verlag
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/BFb0078838
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18263-4
Online ISBN: 978-3-540-47851-5
eBook Packages: Springer Book Archive