Lemma 51.15.3. Let $X$ be a Noetherian scheme. Let $Z \subset X$ be a closed subscheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $X$ is universally catenary and the formal fibres of local rings have $(S_1)$. 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 $x \in X \setminus Z$,
$\mathcal{F}_ x \to \mathcal{F}''_ x$ is surjective and $\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) = 1$ for $x \in Z$ such that there exists an immediate specialization $x' \leadsto x$ with $x' \not\in Z$ and $x' \in \text{Ass}(\mathcal{F})$,
$\text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}''_ x) \geq 2$ for the remaining $x \in Z$.
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 $f^{-1}(Z) \subset Y$.
Proof.
Let $\mathcal{F} \to \mathcal{F}'$ be the map constructed in Lemma 51.15.1 for the subset $Z$ of $X$. Recall that $\mathcal{F}'$ is the quotient of $\mathcal{F}$ by the subsheaf of sections supported on $Z$.
We first prove 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 Z = \emptyset $ by conditions (2) and (3). 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 as in (2). 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. Recall that $\text{Ass}(\mathcal{F}') = \{ x_1, \ldots , x_ n\} $ is finite and $x_ i \not\in Z$. Let $Y_ i$ be the closure of $\{ x_ i\} $. Let $Z_{i, j}$ be the irreducible components of $Z \cap Y_ i$. Observe that $\text{Supp}(\mathcal{F}') \cap Z = \bigcup Z_{i, j}$. Let $z_{i, j} \in Z_{i, j}$ be the generic point. Let
\[ d_{i, j} = \dim (\mathcal{O}_{\overline{\{ x_ i\} }, z_{i, j}}) \]
If $d_{i, j} = 1$, then $z_{i, j}$ is one of the points as in (2). Thus we do not need to modify $\mathcal{F}'$ at these points. Furthermore, still assuming $d_{i, j} = 1$, using Lemma 51.9.2 we can find an open neighbourhood $z_{i, j} \in V_{i, j} \subset X$ such that $\text{depth}_{\mathcal{O}_{X, z}}(\mathcal{F}'_ z) \geq 2$ for $z \in Z_{i, j} \cap V_{i, j}$, $z \not= z_{i, j}$. Set
\[ Z' = X \setminus \left( X \setminus Z \cup \bigcup \nolimits _{d_{i, j} = 1} V_{i, j}) \right) \]
Denote $j' : X \setminus Z' \to X$. By our choice of $Z'$ the assumptions of Lemma 51.8.9 are satisfied. We conclude by setting $\mathcal{F}'' = j'_*(\mathcal{F}'|_{X \setminus Z'})$ and applying Lemma 51.15.2.
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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