The Stacks project

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 $A'$.) $\square$