Bounds for the Hilbert-Kunz Multiplicity of Singular Rings
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917585937760256 |
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| author | Aberbach, Ian M. Cox-Steib, Nicholas O |
| author_facet | Aberbach, Ian M. Cox-Steib, Nicholas O |
| contents | In this paper we prove that the Watanabe-Yoshida conjecture holds up to dimension $7$. Our primary new tool is a function, $φ_J\left(R; z^t\right),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and various radical extensions, $R_n$. We prove that this function is concave and show that it's rate of growth is related to the size of $e_{HK}\left(R\right)$. We combine several known techniques to get effective lower bounds for $φ,$ which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension $7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds for the Hilbert-Kunz Multiplicity of Singular Rings Aberbach, Ian M. Cox-Steib, Nicholas O Commutative Algebra 13A35, 13D40 (Primary) 13B02, 14B05 (Secondary) In this paper we prove that the Watanabe-Yoshida conjecture holds up to dimension $7$. Our primary new tool is a function, $φ_J\left(R; z^t\right),$ that interpolates between the Hilbert-Kunz multiplicities of a base ring, $R$, and various radical extensions, $R_n$. We prove that this function is concave and show that it's rate of growth is related to the size of $e_{HK}\left(R\right)$. We combine several known techniques to get effective lower bounds for $φ,$ which translate to improved bounds on the size of Hilbert-Kunz multiplicities of singular rings. The improved inequalities are powerful enough to show that the conjecture of Watanabe and Yoshida holds in dimension $7$. |
| title | Bounds for the Hilbert-Kunz Multiplicity of Singular Rings |
| topic | Commutative Algebra 13A35, 13D40 (Primary) 13B02, 14B05 (Secondary) |
| url | https://arxiv.org/abs/2402.05822 |