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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.

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
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