Lemma 52.19.5. In Situation 52.16.1 let \mathcal{F} be a coherent \mathcal{O}_ U-module and d \geq 1. Assume
A is I-adically complete, has a dualizing complex, and \text{cd}(A, I) \leq d,
the completion \mathcal{F}^\wedge of \mathcal{F} satisfies the strict (1, 1+ d)-inequalities, and
for x \in U with \overline{\{ x\} } \cap Y \subset Z we have \text{depth}(\mathcal{F}_ x) \geq 2.
Then H^0(U, \mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^0(U, \mathcal{F}/I^ n\mathcal{F}) is an isomorphism.
Proof. We will prove this by showing that Lemma 52.12.4 applies. Thus we let x \in \text{Ass}(\mathcal{F}) with x \not\in Y. Set W = \overline{\{ x\} }. By condition (3) we see that W \cap Y \not\subset Z. By Lemma 52.19.4 we see that no irreducible component of W \cap Y is contained in Z. Thus if z \in W \cap Z, then there is an immediate specialization y \leadsto z, y \in W \cap Y, y \not\in Z. For existence of y use Properties, Lemma 28.6.4. Then \delta ^ Y_ Z(y) = 1 and the assumption implies that \dim (\mathcal{O}_{W, y}) > d. Hence \dim (\mathcal{O}_{W, z}) > 1 + d and we win. \square
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