Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules
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| Format: | Preprint |
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2024
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| _version_ | 1866929619911835648 |
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| author | Puthenpurakal, Tony J. Sahoo, Samarendra |
| author_facet | Puthenpurakal, Tony J. Sahoo, Samarendra |
| contents | Let $(A,\mathfrak{m})$ be a complete intersection ring of codimension $c\geq 2$ and dimension $d\geq 1$. Let $M$ be a finitely generated maximal Cohen-Macaulay $A$-module. Set $M_i=\text{Syz}^A_{i}(M)$. Let $e^{\mathfrak{m}}_i(M)$ be the $i$-th Hilbert coefficient of $M$ with respect to $\mathfrak{m}$. We prove for all $i\gg0$, the function $i\mapsto e^{\mathfrak{m}}_j(M_i)$ is a quasi-polynomial type with period $2$ and degree $\text{cx}(M)-1$ for $j=0,1$, where $\text{cx}(M)$ is the complexity of $M.$ For $\text{cx}(M)=2,$ we prove
$$\lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_1(M_{2n+j})}{n}\geq \lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_0(M_{2n+j})}{n}-\lim_{n\to \infty}\dfrac{μ(M_{2n+j})}{n}$$ for $j=0,1$. When equality holds, we prove that the Castelnuovo-Mumford regularity of the associated graded ring of $M_i$ with respect to the maximal ideal $\mathfrak{m}$ is bounded for all $i\geq 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05860 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules Puthenpurakal, Tony J. Sahoo, Samarendra Commutative Algebra Primary 13A30, 13C14, 13D40, Secondary 13D02, 13D07, 13D45 Let $(A,\mathfrak{m})$ be a complete intersection ring of codimension $c\geq 2$ and dimension $d\geq 1$. Let $M$ be a finitely generated maximal Cohen-Macaulay $A$-module. Set $M_i=\text{Syz}^A_{i}(M)$. Let $e^{\mathfrak{m}}_i(M)$ be the $i$-th Hilbert coefficient of $M$ with respect to $\mathfrak{m}$. We prove for all $i\gg0$, the function $i\mapsto e^{\mathfrak{m}}_j(M_i)$ is a quasi-polynomial type with period $2$ and degree $\text{cx}(M)-1$ for $j=0,1$, where $\text{cx}(M)$ is the complexity of $M.$ For $\text{cx}(M)=2,$ we prove $$\lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_1(M_{2n+j})}{n}\geq \lim_{n\to \infty}\dfrac{e^{\mathfrak{m}}_0(M_{2n+j})}{n}-\lim_{n\to \infty}\dfrac{μ(M_{2n+j})}{n}$$ for $j=0,1$. When equality holds, we prove that the Castelnuovo-Mumford regularity of the associated graded ring of $M_i$ with respect to the maximal ideal $\mathfrak{m}$ is bounded for all $i\geq 0$. |
| title | Asymptotic behavior of invariants of syzygies of maximal Cohen-Macaulay modules |
| topic | Commutative Algebra Primary 13A30, 13C14, 13D40, Secondary 13D02, 13D07, 13D45 |
| url | https://arxiv.org/abs/2412.05860 |