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Derived functors and Hilbert polynomials over regular local rings
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 4 / November 2024
- Published online by Cambridge University Press:
- 23 October 2024, pp. 1137-1147
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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.
