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Derived functors and Hilbert polynomials over hypersurface rings
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 02 April 2025, pp. 1-11
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Let
$(A,\mathfrak{m} )$ be a hypersurface local ring of dimension 
$d \geq 1$ and let I be an 
$\mathfrak{m} $-primary ideal. We show that there is a integer rI
$\geq\;-1$ (depending only on I) such that if M is any non-free maximal Cohen–Macaulay (= MCM) A-module the function 
$n \rightarrow \ell(\operatorname{Tor}^A_1(M, A/I^{n+1}))$ (which is of polynomial type) has degree rI. Analogous results hold for Hilbert polynomials associated to Ext-functors. Surprisingly, a key ingredient is the classification of thick subcategories of the stable category of MCM A-modules (obtained by Takahashi, see [11, 6.6]).
