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Arithmetic degree and associated graded modules

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Abstract.

We prove that the arithmetic degree of a graded or local ring A is bounded above by the arithmetic degree of any of its associated graded rings with respect to ideal I in A. In particular, if Spec(A) is equidimensional and has an embedded component (i.e., A has an embedded associated prime ideal), then the normal cone of Spec(A) along V(I) has an embedded component too. This extends a result of W. M. Ruppert about embedded components of the tangent cone.

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Acknowledgments.

I would like to thank R. Achilles for his warm encouragment and for having introduced me to this subject. Investigation partially supported by the MIUR and the University of Bologna, funds for Selected Research Topics.

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  1. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

    Natale Paolo Vinai

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  1. Natale Paolo Vinai
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Correspondence to Natale Paolo Vinai.

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Mathematics Subject Classification (2000): Primary 13H15, 13A30; Secondary 13D45, 14Q99

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Vinai, N. Arithmetic degree and associated graded modules. manuscripta math. 115, 299–311 (2004). https://doi.org/10.1007/s00229-004-0492-7

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  • DOI: https://doi.org/10.1007/s00229-004-0492-7

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