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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2025-02-01
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| author | Steven Dale Cutkosky |
| author_facet | Steven Dale Cutkosky |
| author_sort | Steven Dale Cutkosky |
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| description | 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. |
| format | Article |
| id | doaj-art-bfd8f35ff4ed4a9c8d9bd7296f42dd90 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-02-01 |
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| spelling | doaj-art-bfd8f35ff4ed4a9c8d9bd7296f42dd902025-08-20T02:04:48ZengMDPI AGMathematics2227-73902025-02-0113569410.3390/math13050694Multiplicities and Volumes of FiltrationsSteven Dale Cutkosky0Department of Mathematics, University of Missouri, Columbia, MO 65211, USAIn 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.https://www.mdpi.com/2227-7390/13/5/694multiplicitygraded family of idealsdivisorial filtration |
| spellingShingle | Steven Dale Cutkosky Multiplicities and Volumes of Filtrations Mathematics multiplicity graded family of ideals divisorial filtration |
| title | Multiplicities and Volumes of Filtrations |
| title_full | Multiplicities and Volumes of Filtrations |
| title_fullStr | Multiplicities and Volumes of Filtrations |
| title_full_unstemmed | Multiplicities and Volumes of Filtrations |
| title_short | Multiplicities and Volumes of Filtrations |
| title_sort | multiplicities and volumes of filtrations |
| topic | multiplicity graded family of ideals divisorial filtration |
| url | https://www.mdpi.com/2227-7390/13/5/694 |
| work_keys_str_mv | AT stevendalecutkosky multiplicitiesandvolumesoffiltrations |