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Derived functors and Hilbert polynomials over regular local rings

Part of: Commutative algebra: Homological methods General commutative ring theory

Published online by Cambridge University Press:  23 October 2024

Tony J. Puthenpurakal Open the ORCID record for Tony J. Puthenpurakal [Opens in a new window]
Tony J. Puthenpurakal*
Affiliation:
Department of Mathematics, IIT Bombay, Powai, Mumbai, India
*
Email: tputhen@gmail.com
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Abstract

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d \geq 1$, I an $\mathfrak{m}$-primary ideal. Let N be a nonzero finitely generated A-module. Consider the functions

\begin{equation*}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))\end{equation*}

of polynomial type and let their degrees be $t^I(N) $ and $e^I(N)$. We prove that $t^I(N) = e^I(N) = \max\{\dim N, d -1 \}$. A crucial ingredient in the proof is that $D^b(A)_f$, the bounded derived category of A with finite length cohomology, has no proper thick subcategories.

Keywords

torsion and extension functorsbounded homotopy category of projectives

MSC classification

Primary: 13D02: Syzygies, resolutions, complexes 13D07: Homological functors on modules (Tor, Ext, etc.)
Secondary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D09: Derived categories

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 4 , November 2024 , pp. 1137 - 1147
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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