Lemma 52.21.2. In the situation above assume $X$ locally has a dualizing complex. Let $T' \subset T \subset Y$ be subsets stable under specialization. Let $d \geq 0$ be an integer. Assume

affine locally we have $X = \mathop{\mathrm{Spec}}(A_0)$ and $Y = V(I_0)$ and $\text{cd}(A_0, I_0) \leq d$,

for $y \in T$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) = \{ \mathfrak m_ y^\wedge \} $ we have

\[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) > 0 \]for $y \in T'$ and for a prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $\mathfrak p \not\in V(\mathcal{I}_ y^\wedge )$ and $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) \not= \{ \mathfrak m_ y^\wedge \} $ we have

\[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) \geq 1 \quad \text{or}\quad \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) + \dim (\mathcal{O}_{X, y}^\wedge /\mathfrak p) > 1 + d \]for $y \in T'$ and a nonmaximal prime $\mathfrak p \subset \mathcal{O}_{X, y}^\wedge $ with $V(\mathfrak p) \cap V(\mathcal{I}_ y^\wedge ) = \{ \mathfrak m_ y^\wedge \} $ we have

\[ \text{depth}_{(\mathcal{O}_{X, y})_\mathfrak p} ((\mathcal{F}^\wedge _ y)_\mathfrak p) > 1 \]if $y \leadsto y'$ is an immediate specialization and $y' \in T'$, then $y \in T$.

Then there exists a canonical map $(\mathcal{F}_ n) \to (\mathcal{F}_ n'')$ of inverse systems of coherent $\mathcal{O}_ X$-modules with the following properties

for $y \in T$ we have $\text{depth}(\mathcal{F}''_{n, y}) \geq 1$,

for $y' \in T'$ we have $\text{depth}(\mathcal{F}''_{n, y'}) \geq 2$,

$(\mathcal{F}''_ n)$ is isomorphic as a pro-system to an object $(\mathcal{H}_ n)$ of $\textit{Coh}(X, \mathcal{I})$,

the induced morphism $(\mathcal{F}_ n) \to (\mathcal{H}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ has kernel and cokernel annihilated by a power of $\mathcal{I}$.

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