Lemma 52.10.5. Let I \subset \mathfrak a be ideals of a Noetherian ring A. Let M be a finite A-module. Let s and d be integers. If we assume
A has a dualizing complex,
\text{cd}(A, I) \leq d,
if \mathfrak p \not\in V(I) and \mathfrak q \in V(\mathfrak p) \cap V(\mathfrak a) then \text{depth}_{A_\mathfrak p}(M_\mathfrak p) > s or \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > d + s.
Then A, I, V(\mathfrak a), M, s, d are as in Situation 52.10.1.
Proof. We have to show that assumptions (1), (3), (4), and (6) of Situation 52.10.1 hold. It is clear that (a) \Rightarrow (1), (b) \Rightarrow (3), and (c) \Rightarrow (4). To finish the proof in the next paragraph we show (6) holds.
Let \mathfrak q \in V(\mathfrak a). Denote A', I', \mathfrak m', M' the I-adic completions of A_\mathfrak q, I_\mathfrak q, \mathfrak qA_\mathfrak q, M_\mathfrak q. Let \mathfrak p' \subset A' be a nonmaximal prime with V(\mathfrak p') \cap V(I') = \{ \mathfrak m'\} . Observe that this implies \dim (A'/\mathfrak p') \leq d by Local Cohomology, Lemma 51.4.10. Denote \mathfrak p \subset A the image of \mathfrak p'. We have \text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p) and \text{depth}(M'_{\mathfrak p'}) + \dim (A'/\mathfrak p') = \text{depth}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) by Local Cohomology, Lemma 51.11.3. By assumption (c) either we have \text{depth}(M'_{\mathfrak p'}) \geq \text{depth}(M_\mathfrak p) > s and we're done or we have \text{depth}(M'_{\mathfrak p'}) + \dim (A'/\mathfrak p') > s + d which implies \text{depth}(M'_{\mathfrak p'}) > s because of the already shown inequality \dim (A'/\mathfrak p') \leq d. In both cases we obtain what we want. \square
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