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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