Derived functors and Hilbert polynomials over regular local rings

Abstract

Let $(A,\mathfrak{m})$ be a regular local ring of dimension $d\ge 1$, $I$ 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 $t^I(N)=e^I(N)=\max \{\dim N,d-1\}$.

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