The Stacks project

Proposition 51.11.1. Let $A$ be a Noetherian ring which has a dualizing complex. Let $T \subset \mathop{\mathrm{Spec}}(A)$ be a subset stable under specialization. Let $s \geq 0$ an integer. Let $M$ be a finite $A$-module. The following are equivalent

1. $H^ i_ T(M)$ is a finite $A$-module for $i \leq s$, and

2. for all $\mathfrak p \not\in T$, $\mathfrak q \in T$ with $\mathfrak p \subset \mathfrak q$ we have

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