The Stacks project

Lemma 52.19.5. In Situation 52.16.1 let $\mathcal{F}$ be a coherent $\mathcal{O}_ U$-module and $d \geq 1$. Assume

  1. $A$ is $I$-adically complete, has a dualizing complex, and $\text{cd}(A, I) \leq d$,

  2. the completion $\mathcal{F}^\wedge $ of $\mathcal{F}$ satisfies the strict $(1, 1+ d)$-inequalities, and

  3. 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 52.12.3 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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