Lemma 51.15.4. Let $X$ be a Noetherian scheme which locally has a dualizing complex. Let $T' \subset X$ be a subset stable under specialization. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that if $x \leadsto x'$ is an immediate specialization of points in $X$ with $x' \in T'$ and $x \not\in T'$, then $\text{depth}(\mathcal{F}_ x) \geq 1$. Then there exists a unique map $\mathcal{F} \to \mathcal{F}''$ of coherent $\mathcal{O}_ X$-modules such that
$\mathcal{F}_ x \to \mathcal{F}''_ x$ is an isomorphism for $x \not\in T'$,
$\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 2$ for $x \in T'$.
If $f : Y \to X$ is a Cohen-Macaulay morphism with $Y$ Noetherian, then $f^*\mathcal{F} \to f^*\mathcal{F}''$ satisfies the same properties with respect to $f^{-1}(T') \subset Y$.
Proof.
Let $\mathcal{F} \to \mathcal{F}'$ be the quotient of $\mathcal{F}$ constructed in Lemma 51.15.1 using $T'$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $T'$.
Proof of uniqueness. Let $\mathcal{F} \to \mathcal{F}''$ be as in the lemma. We get a factorization $\mathcal{F} \to \mathcal{F}' \to \mathcal{F}''$ since $\text{Ass}(\mathcal{F}'') \cap T' = \emptyset $ by condition (2). Let $U \subset X$ be a maximal open subscheme such that $\mathcal{F}'|_ U \to \mathcal{F}''|_ U$ is an isomorphism. We see that $U$ contains all the points of $T'$. Then by Divisors, Lemma 31.5.11 we conclude that $\mathcal{F}'' = j_*(\mathcal{F}'|_ U)$. In this way we get uniqueness (small detail: if we have two of these $\mathcal{F}''$ then we take the intersection of the opens $U$ we get from either).
Proof of existence. We will define
\[ \mathcal{F}'' = \mathop{\mathrm{colim}}\nolimits j_*(\mathcal{F}'|_ V) \]
where $j : V \to X$ runs over the open subschemes such that $X \setminus V \subset T'$. Observe that the colimit is filtered as $T'$ is stable under specialization. Each of the maps $\mathcal{F}' \to j_*(\mathcal{F}'|_ V)$ is injective as $\text{Ass}(\mathcal{F}')$ is disjoint from $T'$. Thus $\mathcal{F}' \to \mathcal{F}''$ is injective.
Suppose $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{F}$ corresponds to the finite $A$-module $M$. Then $\mathcal{F}'$ corresponds to $M' = M / H^0_{T'}(M)$, see proof of Lemma 51.15.1. Applying Lemmas 51.2.2 and 51.5.3 we see that $\mathcal{F}''$ corresponds to an $A$-module $M''$ which fits into the short exact sequence
\[ 0 \to M' \to M'' \to H^1_{T'}(M') \to 0 \]
By Proposition 51.11.1 and our condition on immediate specializations in the statement of the lemma we see that $M''$ is a finite $A$-module. In this way we see that $\mathcal{F}''$ is coherent.
The final statement follows from the formula for the change in depth along a flat local homomorphism, see Algebra, Lemma 10.163.1 and the assumption on the fibres of $f$ inherent in $f$ being Cohen-Macaulay. Details omitted.
$\square$
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