This is a special case of [Satz 1, Faltings-annulators].

Lemma 51.11.5. Let $A$ be a Gorenstein Noetherian local ring. Let $I \subset A$ be an ideal and set $Z = V(I) \subset \mathop{\mathrm{Spec}}(A)$. Let $M$ be a finite $A$-module. Let $s = s_{A, I}(M)$ as in (51.11.1.1). Then $H^ i_ Z(M)$ is finite for $i < s$, but $H^ s_ Z(M)$ is not finite.

Proof. Since a Gorenstein local ring has a dualizing complex, this is a special case of Proposition 51.11.1. It would be helpful to have a short proof of this special case, which will be used in the proof of a general finiteness theorem below. $\square$

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