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Derived functors and Hilbert polynomials over regular local rings (2312.16982v1)
Published 28 Dec 2023 in math.AC
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 non-zero finitely generated $A$-module. Consider the functions [ tI(N, n) = \sum_{i = 0}{ d}\ell(\text{Tor}A_i(N, A/In)) \ \text{and}\ eI(N, n) = \sum_{i = 0}{ d}\ell(\text{Ext}_Ai(N, A/In)) ] of polynomial type and let their degrees be $tI(N) $ and $eI(N)$. We prove that $tI(N) = eI(N) = \max{ \dim N, d -1 }$.