Multiplicities and Volumes of Filtrations

Steven Dale Cutkosky ()
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Steven Dale Cutkosky: Department of Mathematics, University of Missouri, Columbia, MO 65211, USA

Mathematics, 2025, vol. 13, issue 5, 1-12

Abstract: In this article, we survey some aspects of the theory of multiplicities of m R -primary ideals in a local ring ( R , m R ) and the extension of this theory to multiplicities of graded families of m R -primary ideals. We first discuss the existence of multiplicities as a limit. Then, we focus on a theorem of Rees, characterizing when two m R -primary ideals I ⊂ J have the same multiplicity, and discuss extensions of this theorem to filtrations of m R -primary ideals. In the final sections, we give outlines of the proof of existence of the multiplicity of a graded family of m R -primary ideals as a limit, with mild conditions on R , and the proof of the extension of Rees’ theorem to divisorial filtrations.

Keywords: multiplicity; graded family of ideals; divisorial filtration (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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