The Stacks project

Lemma 30.32.4. Let $X$ be a scheme. Let $Z \subset X$ be a closed subscheme. The blowing up $b : X' \to X$ of $Z$ in $X$ has the following properties:

  1. $b|_{b^{-1}(X \setminus Z)} : b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism,

  2. the exceptional divisor $E = b^{-1}(Z)$ is an effective Cartier divisor on $X'$,

  3. there is a canonical isomorphism $\mathcal{O}_{X'}(-1) = \mathcal{O}_{X'}(E)$

Proof. As blowing up commutes with restrictions to open subschemes (Lemma 30.32.3) the first statement just means that $X' = X$ if $Z = \emptyset $. In this case we are blowing up in the ideal sheaf $\mathcal{I} = \mathcal{O}_ X$ and the result follows from Constructions, Example 26.8.14.

The second statement is local on $X$, hence we may assume $X$ affine. Say $X = \mathop{\mathrm{Spec}}(A)$ and $Z = \mathop{\mathrm{Spec}}(A/I)$. By Lemma 30.32.2 we see that $X'$ is covered by the spectra of the affine blowup algebras $A' = A[\frac{I}{a}]$. Then $IA' = aA'$ and $a$ maps to a nonzerodivisor in $A'$ according to Algebra, Lemma 10.69.2. This proves the lemma as the inverse image of $Z$ in $\mathop{\mathrm{Spec}}(A')$ corresponds to $\mathop{\mathrm{Spec}}(A'/IA') \subset \mathop{\mathrm{Spec}}(A')$.

Consider the canonical map $\psi _{univ, 1} : b^*\mathcal{I} \to \mathcal{O}_{X'}(1)$, see discussion following Constructions, Definition 26.16.7. We claim that this factors through an isomorphism $\mathcal{I}_ E \to \mathcal{O}_{X'}(1)$ (which proves the final assertion). Namely, on the affine open corresponding to the blowup algebra $A' = A[\frac{I}{a}]$ mentioned above $\psi _{univ, 1}$ corresponds to the $A'$-module map

\[ I \otimes _ A A' \longrightarrow \left(\Big(\bigoplus \nolimits _{d \geq 0} I^ d\Big)_{a^{(1)}}\right)_1 \]

where $a^{(1)}$ is as in Algebra, Definition 10.69.1. We omit the verification that this is the map $I \otimes _ A A' \to IA' = aA'$. $\square$


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