On the Hilbert depth of the Hilbert function of a finitely generated graded module
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909110045245440 |
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| author | Balanescu, Silviu Cimpoeas, Mircea |
| author_facet | Balanescu, Silviu Cimpoeas, Mircea |
| contents | Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq k} (-1)^{k-j} \binom{d-j}{k-j} h_{M}(j) \geq 0 \text{ for all } k\leq d\}, $$ where $h_M(-)$ is the Hilbert function of $M$, and we prove basic results regard it.
Using the theory of hypergeometric functions, we prove that $\operatorname{hdepth}(h_S)=n$, where $S=K[x_1,\ldots,x_n]$. We show that $\operatorname{hdepth}(h_{S/J})=n$, if $J=(f_1,\ldots,f_d)\subset S$ is a complete intersection monomial ideal with $deg(f_i)\geq 2$ for all $1\leq i\leq d$. Also, we show that $\operatorname{hdepth}(h_{\overline M})\geq \operatorname{hdepth}(h_M)$ for any finitely generated graded $S$-module $M$, where $\overline M=M\otimes_S S[x_{n+1}]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_14031 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Hilbert depth of the Hilbert function of a finitely generated graded module Balanescu, Silviu Cimpoeas, Mircea Commutative Algebra 13C15, 13P10, 13F20, 33B15, 05A18, 05A20 Let $K$ be a field, $A$ a standard graded $K$-algebra and $M$ a finitely generated graded $A$-module. Inspired by our previous works, we study the Hilbert depth of $h_M$, that is $$\operatorname{hdepth}(h_M)=\max\{d\;:\; \sum\limits_{j\leq k} (-1)^{k-j} \binom{d-j}{k-j} h_{M}(j) \geq 0 \text{ for all } k\leq d\}, $$ where $h_M(-)$ is the Hilbert function of $M$, and we prove basic results regard it. Using the theory of hypergeometric functions, we prove that $\operatorname{hdepth}(h_S)=n$, where $S=K[x_1,\ldots,x_n]$. We show that $\operatorname{hdepth}(h_{S/J})=n$, if $J=(f_1,\ldots,f_d)\subset S$ is a complete intersection monomial ideal with $deg(f_i)\geq 2$ for all $1\leq i\leq d$. Also, we show that $\operatorname{hdepth}(h_{\overline M})\geq \operatorname{hdepth}(h_M)$ for any finitely generated graded $S$-module $M$, where $\overline M=M\otimes_S S[x_{n+1}]$. |
| title | On the Hilbert depth of the Hilbert function of a finitely generated graded module |
| topic | Commutative Algebra 13C15, 13P10, 13F20, 33B15, 05A18, 05A20 |
| url | https://arxiv.org/abs/2308.14031 |