Multiplicities and Volumes of Filtrations
In this article, we survey some aspects of the theory of multiplicities of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></ma...
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-02-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/5/694 |
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| Summary: | In this article, we survey some aspects of the theory of multiplicities of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></math></inline-formula>-primary ideals in a local ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>R</mi><mo>,</mo><msub><mi>m</mi><mi>R</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and the extension of this theory to multiplicities of graded families of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></math></inline-formula>-primary ideals. We first discuss the existence of multiplicities as a limit. Then, we focus on a theorem of Rees, characterizing when two <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></math></inline-formula>-primary ideals <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>J</mi></mrow></semantics></math></inline-formula> have the same multiplicity, and discuss extensions of this theorem to filtrations of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></math></inline-formula>-primary ideals. In the final sections, we give outlines of the proof of existence of the multiplicity of a graded family of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mi>R</mi></msub></semantics></math></inline-formula>-primary ideals as a limit, with mild conditions on <i>R</i>, and the proof of the extension of Rees’ theorem to divisorial filtrations. |
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| ISSN: | 2227-7390 |