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Numerical invariants of liaison classes. (English) Zbl 0536.13005
The main purpose of this paper is to obtain numerical invariants for liaison classes. If R is a Cohen-Macaulay local ring and \(I\subseteq R\) an ideal such that R/I is unmixed, then there is a polynomial in \({\mathbb{Z}}[t]\), \(P^ R_{R/I}(t)\), such that for any R/J evenly linked to R/I, \(P^ R_{R/I}(t)=P^ R_{R/J}(t).\) If R/I is reduced, \(P^ R_{R/I}(t)=0\) if and only if the Koszul homology modules \(H_ i(I;R)\) are Cohen-Macaulay R/I-modules. Even if R/I is not reduced \(P^ R_{R/I}(t)=0\) if R/I is in the liaison class of a complete intersection. This vanishing result yields interesting consequences for 0-dimensional (or in general non-reduced) algebras in the liaison class of a complete intersection. For instance if R is a formal power series ring over a field k and R/I is Gorenstein and in the liaison class of a complete intersection then \(T^ 2(R/I,k)=0 (T^ i=\cot angent\) functor) - so in particular every deformation is unobstructed.

MSC:
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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References:
[1] Auslander, M., Buchsbaum, D.: Codimension and multiplicity. Ann. of Math.68, 625-657 (1958) · Zbl 0092.03902 · doi:10.2307/1970159
[2] Avramov, L., Herzog, J.: The Koszul algebra of a codimension two embedding. Math. Zeit.175, 249-260 (1980) · Zbl 0461.14014 · doi:10.1007/BF01163026
[3] Buchsbaum, D., Eisenbud, D.: Algebra structures for finite free resolutions, and some structure theorems for ideals of condimension three. Amer. J. Math.99, 447-485 (1977) · Zbl 0373.13006 · doi:10.2307/2373926
[4] Buchsbaum, D., Eisenbud, D.: What makes a complex axact? J. Algebra25, 259-268 (1973) · Zbl 0264.13007 · doi:10.1016/0021-8693(73)90044-6
[5] Buchweitz, R.-O.: Thesis, l’Université Paris VII. 1981
[6] Dutta, S.: Generalized intersection multiplicities of modules. Trans. Amer. Math. Soc. (to appear) · Zbl 0531.13008
[7] Dutta, S., Hochster, M., McLaughlin, J.: Modules of finite projective dimension with negative intersection multiplicities. Preprint 1983 · Zbl 0588.13020
[8] Gulliksen, T.H., Negard, O.G.: Un complexe résolvant pour certains ideaux déterminantiels. C.R. Acad. Sci. Paris Sér. A274 (1972) · Zbl 0238.13015
[9] Herzog, J.: Komplexe. Auflösungen und Dualität in der lokalen Algebra. Habilitationsschrift, Universität Regensburg 1974
[10] Herzog, J.: Homological properties of the module of differentials. Preprint 1983
[11] Hochster, M.: Euler characteristics over unramified regular local rings. Preprint 1982 · Zbl 0562.13019
[12] Hochster, M.: Properties of Noetherian rings stable under general grade reduction. Arch. Math.24, 393-396 (1973) · Zbl 0268.13013 · doi:10.1007/BF01228228
[13] Huneke, C.: Linkage and the Koszul homology of ideals. Amer. J. Math.104, 1043-1062 (1982) · Zbl 0505.13003 · doi:10.2307/2374083
[14] Huneke, C.: Invariants of liaison. Algebraic Geometry Proceeding, Ann Arbor, 1981. Lecture Notes in Mathematics, vol. 1008, pp. 65-74. Berlin-Heidelberg-New York: Springer 1983
[15] Kaplansky, I.: Commutative Rings. Chicago: University of Chicago Press 1974 · Zbl 0296.13001
[16] Kunz, E.: Almost complete intersections are not Gorenstein rings. J. Algebra28, 111-115 (1974) · Zbl 0275.13025 · doi:10.1016/0021-8693(74)90025-8
[17] Kustin, A., Miller, M.: Deformation and linkage of Gorenstein algebras. Preprint 1982 · Zbl 0545.13010
[18] Lichtenbaum, S.: On the vanishing of Tor in regular local rings. Illinois J. Math.10, 220-226 (1966) · Zbl 0139.26601
[19] Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. I.H.E.S. Publ. Math.42, 323-395 (1973) · Zbl 0268.13008
[20] Peshine, C., Szpiro, L.: Liaison des varietés algébriques. Invent. Math.26, 271-302 (1973) · Zbl 0298.14022 · doi:10.1007/BF01425554
[21] Serre, J-P.: Algèbre locale Multiplicités. Lecture Notes in Math. no. 11. Berlin-Heidelberg-New York: Springer 1965
[22] Simis, A., Vasconcelos, W.: The syzygies of the conormal module. Amer. J. Math.103, 203-224 (1981) · Zbl 0467.13009 · doi:10.2307/2374214
[23] Watanabe, J.: A note on Gorenstein rings of embedding codimension three. Nagoya Math. J.50, 227-232 (1973) · Zbl 0242.13019
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