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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