Colength, multiplicity, and ideal closure operations II
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913476940660736 |
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| author | Ma, Linquan Quy, Pham Hung Smirnov, Ilya |
| author_facet | Ma, Linquan Quy, Pham Hung Smirnov, Ilya |
| contents | Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate'' the two multiplicity theories by asking for analogues invariants already established in one of the theories.
On the Hilbert--Samuel side, we prove that the analog of the Stückrad--Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals is often $1$ under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) $\mathfrak{m}$-primary ideals. On the Hilbert--Kunz side, we study several analogs of the Lech--Mumford and Stückrad--Vogel invariants. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_12469 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Colength, multiplicity, and ideal closure operations II Ma, Linquan Quy, Pham Hung Smirnov, Ilya Commutative Algebra Let $(R, \mathfrak{m})$ be a Noetherian local ring. This paper concerns several extremal invariants arising from the study of the relation between colength and (Hilbert--Samuel or Hilbert--Kunz) multiplicity of an $\mathfrak{m}$-primary ideal. We introduce versions of these invariants by restricting to various closures and ``cross-pollinate'' the two multiplicity theories by asking for analogues invariants already established in one of the theories. On the Hilbert--Samuel side, we prove that the analog of the Stückrad--Vogel invariant (that is, the infimum of the ratio between the multiplicity and colength) for integrally closed $\mathfrak{m}$-primary ideals is often $1$ under mild assumptions. We also compute the supremum and infimum of the relative drops of multiplicity for (integrally closed) $\mathfrak{m}$-primary ideals. On the Hilbert--Kunz side, we study several analogs of the Lech--Mumford and Stückrad--Vogel invariants. |
| title | Colength, multiplicity, and ideal closure operations II |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2305.12469 |