Lemma 38.43.9 (0F8W)—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 38.43.9 (cite)

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Lemma 38.43.9. In Situation 38.43.1 let $W \subset X$ be the maximal open subscheme over which the cohomology sheaves of $M$ are locally free. Then the morphism $b : X' \to X$ of Lemma 38.43.6 is an isomorphism over $W$ .

Proof. This is true because for any affine chart $(U, A, f, M^\bullet )$ with $U \subset W$ we have that $I_ i(M^\bullet , f)$ are locally generated by a power of $f$ by More on Algebra, Lemma 15.96.4. Since $f$ is a nonzerodivisor, the blowing up $b^{-1}(U) \to U$ is an isomorphism. $\square$

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