Separate irreducible components by blowing up.

Lemma 30.34.5. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U, V$ be quasi-compact disjoint open subschemes of $X$. Then there exist a $U \cup V$-admissible blowup $b : X' \to X$ such that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ with $b^{-1}(U) \subset X'_1$ and $b^{-1}(V) \subset X'_2$.

Proof. Choose a finite type quasi-coherent sheaf of ideals $\mathcal{I}$, resp. $\mathcal{J}$ such that $X \setminus U = V(\mathcal{I})$, resp. $X \setminus V = V(\mathcal{J})$, see Properties, Lemma 27.24.1. Then $V(\mathcal{I}\mathcal{J}) = X$ set theoretically, hence $\mathcal{I}\mathcal{J}$ is a locally nilpotent sheaf of ideals. Since $\mathcal{I}$ and $\mathcal{J}$ are of finite type and $X$ is quasi-compact there exists an $n > 0$ such that $\mathcal{I}^ n \mathcal{J}^ n = 0$. We may and do replace $\mathcal{I}$ by $\mathcal{I}^ n$ and $\mathcal{J}$ by $\mathcal{J}^ n$. Whence $\mathcal{I} \mathcal{J} = 0$. Let $b : X' \to X$ be the blowing up in $\mathcal{I} + \mathcal{J}$. This is $U \cup V$-admissible as $V(\mathcal{I} + \mathcal{J}) = X \setminus U \cup V$. We will show that $X'$ is a disjoint union of open subschemes $X' = X'_1 \amalg X'_2$ such that $b^{-1}\mathcal{I}|_{X'_2} = 0$ and $b^{-1}\mathcal{J}|_{X'_1} = 0$ which will prove the lemma.

We will use the description of the blowing up in Lemma 30.32.2. Suppose that $U = \mathop{\mathrm{Spec}}(A) \subset X$ is an affine open such that $\mathcal{I}|_ U$, resp. $\mathcal{J}|_ U$ corresponds to the finitely generated ideal $I \subset A$, resp. $J \subset A$. Then

$b^{-1}(U) = \text{Proj}(A \oplus (I + J) \oplus (I + J)^2 \oplus \ldots )$

This is covered by the affine open subsets $A[\frac{I + J}{x}]$ and $A[\frac{I + J}{y}]$ with $x \in I$ and $y \in J$. Since $x \in I$ is a nonzerodivisor in $A[\frac{I + J}{x}]$ and $IJ = 0$ we see that $J A[\frac{I + J}{x}] = 0$. Since $y \in J$ is a nonzerodivisor in $A[\frac{I + J}{y}]$ and $IJ = 0$ we see that $I A[\frac{I + J}{y}] = 0$. Moreover,

$\mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{x}}]) \cap \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{y}}]) = \mathop{\mathrm{Spec}}(A[\textstyle {\frac{I + J}{xy}}]) = \emptyset$

because $xy$ is both a nonzerodivisor and zero. Thus $b^{-1}(U)$ is the disjoint union of the open subscheme $U_1$ defined as the union of the standard opens $\mathop{\mathrm{Spec}}(A[\frac{I + J}{x}])$ for $x \in I$ and the open subscheme $U_2$ which is the union of the affine opens $\mathop{\mathrm{Spec}}(A[\frac{I + J}{y}])$ for $y \in J$. We have seen that $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_2$ and $b^{-1}\mathcal{I}\mathcal{O}_{X'}$ restricts to zero on $U_1$. We omit the verification that these open subschemes glue to global open subschemes $X'_1$ and $X'_2$. $\square$

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