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The Stacks project

Separate irreducible components by blowing up.

Lemma 31.34.5. 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 28.24.1. 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 31.32.2. 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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