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12 pages, 265 KiB

Open AccessArticle

by Steven Dale Cutkosky

Mathematics 2025, 13(5), 694; https://doi.org/10.3390/math13050694 - 21 Feb 2025

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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. [...] Read more.

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 \subset 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. Full article

(This article belongs to the Special Issue Advanced Algebraic Geometry and Applications)

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