Lemma 51.9.1. Let A be a Noetherian ring. Let I \subset A be an ideal. Let M be a finite A-module. Let \mathfrak p \in V(I) be a prime ideal. Assume e = \text{depth}_{IA_\mathfrak p}(M_\mathfrak p) < \infty . Then there exists a nonempty open U \subset V(\mathfrak p) such that \text{depth}_{IA_\mathfrak q}(M_\mathfrak q) \geq e for all \mathfrak q \in U.
Proof. By definition of depth we have IM_\mathfrak p \not= M_\mathfrak p and there exists an M_\mathfrak p-regular sequence f_1, \ldots , f_ e \in IA_\mathfrak p. After replacing A by a principal localization we may assume f_1, \ldots , f_ e \in I form an M-regular sequence, see Algebra, Lemma 10.68.6. Consider the module M' = M/IM. Since \mathfrak p \in \text{Supp}(M') and since the support of a finite module is closed, we find V(\mathfrak p) \subset \text{Supp}(M'). Thus for \mathfrak q \in V(\mathfrak p) we get IM_\mathfrak q \not= M_\mathfrak q. Hence, using that localization is exact, we see that \text{depth}_{IA_\mathfrak q}(M_\mathfrak q) \geq e for any \mathfrak q \in V(I) by definition of depth. \square
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