Derived functors and Hilbert polynomials over regular local rings

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.