Derived functors and Hilbert polynomials over hypersurface rings
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915441898684416 |
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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 $d \geq 1$ and let $I$ be an $\mathfrak{m}$-primary ideal. We show that there is a non-negative integer $r_I$ (depending only on $I$) such that if $M$ is any non-free maximal Cohen-Macaulay $A$-module the function $n \rightarrow \ell(Tor^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree $r_I$. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Derived functors and Hilbert polynomials over hypersurface rings Puthenpurakal, Tony J. Commutative Algebra Primary 13D09, 13A30, Secondary 13H10 Let $(A,\mathfrak{m})$ be a hypersurface local ring of dimension $d \geq 1$ and let $I$ be an $\mathfrak{m}$-primary ideal. We show that there is a non-negative integer $r_I$ (depending only on $I$) such that if $M$ is any non-free maximal Cohen-Macaulay $A$-module the function $n \rightarrow \ell(Tor^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree $r_I$. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly a key ingredient is the classification of thick subcategories of the stable category of MCM $A$-modules (obtained by Takahashi). |
| title | Derived functors and Hilbert polynomials over hypersurface rings |
| topic | Commutative Algebra Primary 13D09, 13A30, Secondary 13H10 |
| url | https://arxiv.org/abs/2404.14938 |