$h$-function, Hilbert-Kunz density function and Frobenius-Poincaré function
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| Format: | Preprint |
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2023
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| _version_ | 1866910870278242304 |
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| author | Meng, Cheng Mukhopadhyay, Alapan |
| author_facet | Meng, Cheng Mukhopadhyay, Alapan |
| contents | Given ideals $I,J$ of a noetherian local ring $(R, \mathfrak m)$ such that $I+J$ is $\mathfrak m$-primary and a finitely generated $R$-module $M$, we associate an invariant of $(M,R,I,J)$ called the $h$-function. Our results on $h$-functions allow extensions of the theories of Frobenius-Poincaré functions and Hilbert-Kunz density functions from the known graded case to the local case, answering a question of V.Trivedi. When $J$ is $\mathfrak m$-primary, we describe the support of the corresponding density function in terms of other invariants of $(R, I,J)$. We show that the support captures the $F$-threshold: $c^J(I)$, under mild assumptions, extending results of V. Trivedi and Watanabe. The $h$-function encodes Hilbert-Samuel, Hilbert-Kunz multiplicity and $F$-threshold of the ideal pair involved. Using this feature of $h$-functions, we provide an equivalent formulation of a conjecture of Huneke, Mustaţă, Takagi, Watanabe; recover a result of Smirnov and Betancourt; give a new proof of a result answering Watanabe-Yoshida's question comparing Hilbert-Kunz and Hilbert-Samuel multiplicity and establish lower bounds on $F$-thresholds. We also point out that a conjecture of Smirnov-Betancourt as stated is false and suggest a correction which we relate to the conjecture of Huneke et al.
We develop the theory of $h$-functions in a more general setting which yields a density function for $F$-signature. A key to many results on $h$-functions is a `convexity technique' that we introduce, which in particular proves differentiability of Hilbert-Kunz density functions almost everywhere on $(0,\infty)$, thus contributing to another question of Trivedi. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_10270 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $h$-function, Hilbert-Kunz density function and Frobenius-Poincaré function Meng, Cheng Mukhopadhyay, Alapan Commutative Algebra Algebraic Geometry Classical Analysis and ODEs Given ideals $I,J$ of a noetherian local ring $(R, \mathfrak m)$ such that $I+J$ is $\mathfrak m$-primary and a finitely generated $R$-module $M$, we associate an invariant of $(M,R,I,J)$ called the $h$-function. Our results on $h$-functions allow extensions of the theories of Frobenius-Poincaré functions and Hilbert-Kunz density functions from the known graded case to the local case, answering a question of V.Trivedi. When $J$ is $\mathfrak m$-primary, we describe the support of the corresponding density function in terms of other invariants of $(R, I,J)$. We show that the support captures the $F$-threshold: $c^J(I)$, under mild assumptions, extending results of V. Trivedi and Watanabe. The $h$-function encodes Hilbert-Samuel, Hilbert-Kunz multiplicity and $F$-threshold of the ideal pair involved. Using this feature of $h$-functions, we provide an equivalent formulation of a conjecture of Huneke, Mustaţă, Takagi, Watanabe; recover a result of Smirnov and Betancourt; give a new proof of a result answering Watanabe-Yoshida's question comparing Hilbert-Kunz and Hilbert-Samuel multiplicity and establish lower bounds on $F$-thresholds. We also point out that a conjecture of Smirnov-Betancourt as stated is false and suggest a correction which we relate to the conjecture of Huneke et al. We develop the theory of $h$-functions in a more general setting which yields a density function for $F$-signature. A key to many results on $h$-functions is a `convexity technique' that we introduce, which in particular proves differentiability of Hilbert-Kunz density functions almost everywhere on $(0,\infty)$, thus contributing to another question of Trivedi. |
| title | $h$-function, Hilbert-Kunz density function and Frobenius-Poincaré function |
| topic | Commutative Algebra Algebraic Geometry Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2310.10270 |