Lemma 31.32.13 (02NS)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.32: Blowing up
Lemma 31.32.13 (cite)

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Lemma 31.32.13. Let $X$ be a scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $b : X' \to X$ be the blowing up of $X$ in the ideal sheaf $\mathcal{I}$. If $\mathcal{I}$ is of finite type, then

$b : X' \to X$ is a projective morphism, and
$\mathcal{O}_{X'}(1)$ is a $b$-relatively ample invertible sheaf.

Proof. The surjection of graded $\mathcal{O}_ X$-algebras

\[ \text{Sym}_{\mathcal{O}_ X}^*(\mathcal{I}) \longrightarrow \bigoplus \nolimits _{d \geq 0} \mathcal{I}^ d \]

defines via Constructions, Lemma 27.18.5 a closed immersion

\[ X' = \underline{\text{Proj}}_ X (\bigoplus \nolimits _{d \geq 0} \mathcal{I}^ d) \longrightarrow \mathbf{P}(\mathcal{I}). \]

Hence $b$ is projective, see Morphisms, Definition 29.43.1. The second statement follows for example from the characterization of relatively ample invertible sheaves in Morphisms, Lemma 29.37.4. Some details omitted. $\square$

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