The Stacks project

Proof. Immediate from the more precise Lemma 31.32.14. $\square$

Proof. Let $\mathcal{I} \subset \mathcal{O}_ V$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I})$ is disjoint from $U \cap V$ and such that $V'$ is isomorphic to the blowup of $V$ in $\mathcal{I}$. Let $\mathcal{I}' \subset \mathcal{O}_{U \cup V}$ be the quasi-coherent sheaf of ideals whose restriction to $U$ is $\mathcal{O}_ U$ and whose restriction to $V$ is $\mathcal{I}$ (see Sheaves, Section 6.33). By Properties, Lemma 28.22.2 there exists a finite type quasi-coherent sheaf of ideals $\mathcal{J} \subset \mathcal{O}_ X$ whose restriction to $U \cup V$ is $\mathcal{I}'$. The lemma follows. $\square$

Proof. Let $\mathcal{I}_ i \subset \mathcal{O}_ X$ be the finite type quasi-coherent sheaf of ideals such that $V(\mathcal{I}_ i)$ is disjoint from $U$ and such that $X_ i$ is isomorphic to the blowup of $X$ in $\mathcal{I}_ i$. Set $\mathcal{I} = \mathcal{I}_1 \cdot \ldots \cdot \mathcal{I}_ n$ and let $X'$ be the blowup of $X$ in $\mathcal{I}$. Then $X' \to X$ factors through $b_ i$ by Lemma 31.32.12. $\square$

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

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,

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 $X'_2$. $\square$

Proof. From the definition of the ideal of denominators in Definition 31.23.10 we immediately see that $b$ is a $U$-admissible blowup. For the notation $1_ D$, $1_ E$, and $\mathcal{O}_{X'}(D - E)$ please see Definition 31.14.1. The pullback $b^*s$ is defined by Lemmas 31.32.11 and 31.23.8. Thus the statement of the lemma makes sense. We can reinterpret the final assertion as saying that $b^*s$ is a global regular section of $b^*\mathcal{L}(E)$ whose zero scheme is $D$. This uniquely defines $D$ hence to prove the lemma we may work affine locally on $X$ and $X'$. Assume $X = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{L} = \mathcal{O}_ X$. Then $s$ is a regular meromorphic function and shrinking further we may assume $s = a'/a$ with $a', a \in A$ nonzerodivisors. Then the ideal of denominators of $s$ corresponds to the ideal $I = \{ x \in A \mid xa' \in aA\} $. Recall that $X'$ is covered by spectra of affine blowup algebras $A' = A[\frac{I}{x}]$ with $x \in I$ (Lemma 31.32.2). Fix $x \in I$ and write $xa' = a a''$ for some $a'' \in A$. The divisor $E \subset X'$ is cut out by $x \in A'$ over the spectrum of $A'$ and hence $1/x$ is a generator of $\mathcal{O}_{X'}(E)$ over $\mathop{\mathrm{Spec}}(A')$. Finally, in the total quotient ring of $A'$ we have $a'/a = a''/x$. Hence $b^*s = a'/a$ restricts to a regular section of $\mathcal{O}_{X'}(E)$ which is over $\mathop{\mathrm{Spec}}(A')$ given by $a''/x$. This finishes the proof. (The divisor $D \cap \mathop{\mathrm{Spec}}(A')$ is cut out by the image of $a''$ in $A'$.) $\square$