The Stacks project

Situation 52.10.1. Here $A$ is a Noetherian ring. We have an ideal $I \subset A$, a finite $A$-module $M$, and a subset $T \subset V(I)$ stable under specialization. We have integers $s$ and $d$. We assume

1. $A$ has a dualizing complex,

2. $\text{cd}(A, I) \leq d$,

3. given primes $\mathfrak p \subset \mathfrak r \subset \mathfrak q$ with $\mathfrak p \not\in V(I)$, $\mathfrak r \in V(I) \setminus T$, $\mathfrak q \in T$ we have

   \[ \text{depth}_{A_\mathfrak p}(M_\mathfrak p) \geq s \quad \text{or}\quad \text{depth}_{A_\mathfrak p}(M_\mathfrak p) + \dim ((A/\mathfrak p)_\mathfrak q) > d + s \]

4. given $\mathfrak q \in T$ denoting $A', \mathfrak m', I', M'$ are the usual $I$-adic completions of $A_\mathfrak q, \mathfrak qA_\mathfrak q, I_\mathfrak q, M_\mathfrak q$ we have

   \[ \text{depth}(M'_{\mathfrak p'}) > s \]

   for all $\mathfrak p' \in \mathop{\mathrm{Spec}}(A') \setminus V(I')$ with $V(\mathfrak p') \cap V(I') = \{ \mathfrak m'\} $.