Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 42, No. 1, pp. 103-122 (2001)

\font\mf=eufm10 Stretched {\mf m}-primary Ideals

Maria Evelina Rossi; Giuseppe Valla

Dipartimento di Matematica, Universita di Genova, Via Dodecaneso 35, 16146 Genova, Italy e-mail:;

Abstract: \font\mf=eufm10 \def\m{\hbox{\mf m}} The multiplicity of an $\m$-primary ideal $I$ of a Cohen-Macaulay local ring $(A,\m)$ of dimension $d$ can be written as $e(I)=\lambda(I/I^2)-(d-1)\lambda(A/I)+K-1$ for some integer $K\ge 1.$ In the case $K=1,2$, the Hilbert function of $I$ and the depth of the associated graded ring of $A$ with respect to $I$ are very well understood. In this paper we are dealing with the case $K=3$ and we determine the possible Hilbert functions of stretched ideals whose Cohen-Macaulay type is not too big. Our main result extends to a considerable extent a deep result of J. Sally who proved that the associated graded ring of a Gorenstein local ring with embedding dimension equal to $e(\m)+d-3$, is Cohen-Macaulay.

Keywords: Cohen-Macaulay local ring, associated graded ring, stretched ideals, Hilbert series

Classification (MSC2000): 13A30; 13D40, 13H15

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