Remark 52.23.2. In the situation of Proposition 52.23.1 if we assume $A$ has a dualizing complex, then the condition that $H^0(U, \mathcal{F}_1)$ and $H^1(U, \mathcal{F}_1)$ are finite is equivalent to

$\text{depth}(\mathcal{F}_{1, y}) + \dim (\mathcal{O}_{\overline{\{ y\} }, z}) > 2$

for all $y \in U \cap Y$ and $z \in Z \cap \overline{\{ y\} }$. See Local Cohomology, Lemma 51.12.1. This holds for example if $\mathcal{F}_1$ is a finite locally free $\mathcal{O}_{U \cap Y}$-module, $Y$ is $(S_2)$, and $\text{codim}(Z', Y') \geq 3$ for every pair of irreducible components $Y'$ of $Y$, $Z'$ of $Z$ with $Z' \subset Y'$.

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