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Betti numbers of the conormal module of licci rings. (English) Zbl 1490.13021

Let \(R\) be a commutative noetherian local ring. For an ideal \(I \subset R\), the conormal module is the module \(I/I^2\). It was shown by D. Ferrand [C. R. Acad. Sci., Paris, Sér. A 264, 427–428 (1967; Zbl 0154.03801)] and by W. V. Vasconcelos [J. Algebra 6, 309–316 (1967; Zbl 0147.29301)] that \(I\) is generated by a regular sequence (i.e. \(I\) is a complete intersection) if and only if \(I\) has finite projective dimension and the conormal module is free as an \(R/I\)-module. When \(R\) is a polynomial ring over a field, L. L. Avramov and J. Herzog [Invent. Math. 117, No. 1, 75–88 (1994; Zbl 0813.13024)] showed that if \(I\) is not a complete intersection then the Betti numbers of the conormal module grow exponentially.
In this paper, the authors consider licci ideals, i.e. ideals that are in the linkage class of a complete intersection, in a regular local ring. They find lower bounds for the Betti numbers of the conormal module. In particular, when \(I\) is a licci Gorenstein ideal, they show that the \(i\)-th Betti number of \(I/I^2\) is at least \(5 f_{2i-1}\) for \(i \geq 1\), where \(f_n\) is the \(n\)-th Fibonacci number.

MSC:

13C14 Cohen-Macaulay modules
13C40 Linkage, complete intersections and determinantal ideals
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