Derived functors and Hilbert polynomials over hypersurface rings-II
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| Format: | Preprint |
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2025
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| _version_ | 1866909665572421632 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\ell(N/IN)$ finite. We show that there is a integer $r_I \geq -1$ (depending only on $I$ and $N$) such that if $M$ is any non-free maximal \CM \ (= MCM) $A$-module the functions $n \rightarrow \ell(\text{Tor}^A_1(M, N/I^{n+1}N))$, $n \rightarrow \ell(\text{Ext}^1_A(M, N/I^{n+1}N))$ and $n \rightarrow \ell(\text{Ext}^{d+1}(N/I^{n+1}N, M))$ (which are all of polynomial type) has degree $r_I$. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi, see \cite[6.6]{T}). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Derived functors and Hilbert polynomials over hypersurface rings-II Puthenpurakal, Tony J. Commutative Algebra Primary 13D09, 13A30, Secondary 13H10 Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$, $N$ a perfect $A$-module and let $I$ be an ideal in $A$ with $\ell(N/IN)$ finite. We show that there is a integer $r_I \geq -1$ (depending only on $I$ and $N$) such that if $M$ is any non-free maximal \CM \ (= MCM) $A$-module the functions $n \rightarrow \ell(\text{Tor}^A_1(M, N/I^{n+1}N))$, $n \rightarrow \ell(\text{Ext}^1_A(M, N/I^{n+1}N))$ and $n \rightarrow \ell(\text{Ext}^{d+1}(N/I^{n+1}N, M))$ (which are all of polynomial type) has degree $r_I$. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi, see \cite[6.6]{T}). |
| title | Derived functors and Hilbert polynomials over hypersurface rings-II |
| topic | Commutative Algebra Primary 13D09, 13A30, Secondary 13H10 |
| url | https://arxiv.org/abs/2506.23241 |