Derived functors and Hilbert polynomials over regular local rings
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866929612170199040 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions \[ t^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Ext}_A^i(N, A/I^n)) \] of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$. We prove that $t^I(N) = e^I(N) = \max\{ \dim N, d -1 \}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_16982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Derived functors and Hilbert polynomials over regular local rings Puthenpurakal, Tony J. Commutative Algebra Primary 13D02, 13D07, Secondary 13A30, 13D40, 13D09 Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, $I$ an $\mathfrak{m}$-primary ideal. Let $N$ be a non-zero finitely generated $A$-module. Consider the functions \[ t^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Tor}^A_i(N, A/I^n)) \ \text{and}\ e^I(N, n) = \sum_{i = 0}^{ d}\ell(\text{Ext}_A^i(N, A/I^n)) \] of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$. We prove that $t^I(N) = e^I(N) = \max\{ \dim N, d -1 \}$. |
| title | Derived functors and Hilbert polynomials over regular local rings |
| topic | Commutative Algebra Primary 13D02, 13D07, Secondary 13A30, 13D40, 13D09 |
| url | https://arxiv.org/abs/2312.16982 |