The Stacks project

Lemma 70.17.12. Let $S$ be a scheme and let $X$ be an algebraic space over $S$. Assume $X$ is quasi-compact and quasi-separated. Let $Z \subset X$ be a closed subspace of finite presentation. Let $b : X' \to X$ be the blowing up with center $Z$. Let $Z' \subset X'$ be a closed subspace of finite presentation. Let $X'' \to X'$ be the blowing up with center $Z'$. There exists a closed subspace $Y \subset X$ of finite presentation, such that

  1. $|Y| = |Z| \cup |b|(|Z'|)$, and

  2. the composition $X'' \to X$ is isomorphic to the blowing up of $X$ in $Y$.

Proof. The condition that $Z \to X$ is of finite presentation means that $Z$ is cut out by a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, see Morphisms of Spaces, Lemma 66.28.12. Write $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$ so that $X' = \underline{\text{Proj}}(\mathcal{A})$. Note that $X \setminus Z$ is a quasi-compact open subspace of $X$ by Limits of Spaces, Lemma 69.14.1. Since $b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism (Lemma 70.17.4) the same result shows that $b^{-1}(X \setminus Z) \setminus Z'$ is quasi-compact open subspace in $X'$. Hence $U = X \setminus (Z \cup b(Z'))$ is quasi-compact open subspace in $X$. By Lemma 70.16.3 there exist a $d > 0$ and a finite type $\mathcal{O}_ X$-submodule $\mathcal{F} \subset \mathcal{I}^ d$ such that $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ and such that the support of $\mathcal{I}^ d/\mathcal{F}$ is contained in $X \setminus U$.

Since $\mathcal{F} \subset \mathcal{I}^ d$ is an $\mathcal{O}_ X$-submodule we may think of $\mathcal{F} \subset \mathcal{I}^ d \subset \mathcal{O}_ X$ as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this $\mathcal{J} \subset \mathcal{O}_ X$ to prevent confusion. Since $\mathcal{I}^ d / \mathcal{J}$ and $\mathcal{O}/\mathcal{I}^ d$ are supported on $|X| \setminus |U|$ we see that $|V(\mathcal{J})|$ is contained in $|X| \setminus |U|$. Conversely, as $\mathcal{J} \subset \mathcal{I}^ d$ we see that $|Z| \subset |V(\mathcal{J})|$. Over $X \setminus Z \cong X' \setminus b^{-1}(Z)$ the sheaf of ideals $\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence $|V(\mathcal{J})|$ equals $|Z| \cup |b|(|Z'|)$. It follows that also $|V(\mathcal{I}\mathcal{J})| = |Z| \cup |b|(|Z'|)$. Moreover, $\mathcal{I}\mathcal{J}$ is an ideal of finite type as a product of two such. We claim that $X'' \to X$ is isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$ which finishes the proof of the lemma by setting $Y = V(\mathcal{I}\mathcal{J})$.

First, recall that the blowup of $X$ in $\mathcal{I}\mathcal{J}$ is the same as the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$, see Lemma 70.17.10. Hence it suffices to show that the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$ agrees with the blowup of $X'$ in $Z'$. We will show that

\[ b^{-1}\mathcal{J} \mathcal{O}_{X'} = \mathcal{I}_ E^ d \mathcal{I}_{Z'} \]

as ideal sheaves on $X''$. This will prove what we want as $\mathcal{I}_ E^ d$ cuts out the effective Cartier divisor $dE$ and we can use Lemmas 70.17.6 and 70.17.10.

To see the displayed equality of the ideals we may work locally. With notation $A$, $I$, $a \in I$ as in Lemma 70.17.2 we see that $\mathcal{F}$ corresponds to an $R$-submodule $M \subset I^ d$ mapping isomorphically to an ideal $J \subset R$. The condition $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ means that $Z' \cap \mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ is cut out by the ideal generated by the elements $m/a^ d$, $m \in M$. Say the element $m \in M$ corresponds to the function $f \in J$. Then in the affine blowup algebra $A' = A[\frac{I}{a}]$ we see that $f = (a^ dm)/a^ d = a^ d (m/a^ d)$. Thus the equality holds. $\square$


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