The Stacks project

Proposition 51.14.2. 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 $s \in \mathbf{Z}$. Let $K \in D_{\textit{Coh}}^+(A)$. The following are equivalent

  1. $H^ i_ T(K)$ 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

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

Proof. Formal consequence of Proposition 51.13.2 and Lemma 51.14.1. $\square$

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