Lemma 38.43.11 (0F8X)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.43: Blowing up complexes, II
Lemma 38.43.11 (cite)

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Lemma 38.43.11. In Situation 38.43.1. Let $b : X' \to X$ and $D'$ be as in Lemma 38.43.6. Let $Q = L\eta _{\mathcal{I}'}Lb^*M$ be as in Lemma 38.43.7. Let $W \subset X$ be the maximal open where $M$ has locally free cohomology modules. Then there exists a closed immersion $i : T \to D'$ of finite presentation such that

$D' \cap b^{-1}(W) \subset T$ scheme theoretically,
$Li^*Q$ has locally free cohomology sheaves, and
for $t \in T$ mapping to $w \in W$ the rank of $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$ .

Proof. Lemma 38.43.9 tells us that $b$ is an isomorphism over $W$ . Hence $b^{-1}(W) \subset X'$ is contained in the maximal open $W' \subset X'$ where $Lb^*M$ has locally free cohomology sheaves. Then the actual statements in the lemma are an immediate consequence of Lemma 38.43.10 applied to $(X', D', Lb^*M)$ and the other lemmas mentioned in the statement. $\square$

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