A general resolution for grade four Gorenstein ideals

manuscripta mathematica - Tập 35 - Trang 221-269 - 1981
Andrew R. Kustin1, Matthew Miller2
1Department of Mathematics, University of Kansas, Lawrence, USA
2Department of Mathematics, University of Tennessee, Knoxville, USA

Tóm tắt

Gorenstein rings occur in a multitude of different guises: as rings of invariants, as coordinate rings of certain determinantal varieties and symmetric semigroup curves, as representatives of linkage classes, and so on. In an attempt to unify this motley collection of examples (at least for finite projective dimension) one seeks a generic free resolution whose specializations yield all examples of given embedding codimension. The present paper describes a resolution for codimension four, not generic, but general enough to encompass many diverse examples. The structure of this resolution is intimately related to the differential, graded, commutative algebra that it supports, and to the deformation theory of codimension four Gorenstein algebras. These ideas are brought together in the determination of the singular locus of certain codimension four Gorenstein varieties. More generally they suggest a classification of codimension four Gorenstein rings that begins to impose some order on the examples.

Tài liệu tham khảo

ARTIN, M.: Deformations of Singularities, Bombay, Tata Institute for Fundamental Research, 1976 BASS, H.: On the ubiquity of Gorenstein rings. Math. Zeit.82, 8–28 (1963) BEHNKE, K.: Projective resolutions of symmetric Frobenius algebras and Gorenstein rings, preprint BUCHSBAUM, D., EISENBUD, D.: What makes a complex exact? J. Alg.25, 259–268 (1973) —: Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3. Amer. J. Math.99, 447–485 (1977) GULLIKSEN, T., NEGARD, O.: Un complex résolvant pour certains ideaux determinantiels. C. R. Acad. Sc. Paris (A)274, 16–18 (1972) HERZOG, J.: Deformation of certain Gorenstein singularities, Springer Verlag Lecture Notes in Mathematics, v. 740, 1979 —: Deformationen von Cohen-Macaulay Algebren. Journal für die reine und angewandte Mathematik318, 83–105 (1980) —: Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalenmodul und den Differentialmodul. Math. Zeit.163, 149–162 (1978) KLEPPE, H., LAKSOV, D.: The algebraic structure and deformation of pfaffian schemes. J. Alg.64, 167–189 (1980) KUSTIN, A., MILLER, M.: Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four. Math. Zeit.173, 171–184 (1980) -: Algebra structures on minimal resolutions of Gorenstein rings, to appear in the report of the regional conference held at George Mason University in August, 1979 -: Structure theory for a class of grade four Gorenstein ideals. Trans. Amer. Math. Soc. to appear LICHTENBAUM, S., SCHLESSINGER, M.: The cotangent complex of a morphism. Trans. Amer. Math. Soc.128, 41–70 (1967) MATSUMURA, H.: Commutative Algebra. New York, W. A. Benjamin, 1970 PESKINE, C., SZPIRO, L.: Liaison des variétés algébriques I. Invent. Math.26, 271–302 (1974) PINKHAM, H.: Deformations of algebraic varieties with Gm action. Astérisque20, 1–131 (1974) RUGET, G.: Deformations des germes d'espace analytique I. Astérisque16, 63–81 (1974) SCHAFS, M.: Deformations of Cohen-Macaulay schemes of codimension 2 and non-singular deformations of space curves. Amer. J. Math.99, 669–685 (1977) SCHENZEL, P.: Über die freien Auflösungen extremaler Cohen-Macaulay-Ringe. J. Alg.64, 93–101 (1980) STANLEY, R.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc.1, 475–511 (1979) WALDI, R.: Deformation von Gorenstein-Singularitäten der Kodimension 3. Math. Ann.242, 201–208 (1979)