# stacks/stacks-project

Split out a lemma

This commit removes exactly the same number of lines as it adds.
 @@ -10466,63 +10466,13 @@ \subsection{Blowing up lemmas} \begin{proof} Let $\mathcal{I} \subset \mathcal{O}_X$ be the quasi-coherent ideal sheaf of denominators of $s$. Namely, we declare a local section $f$ of $\mathcal{O}_X$ to be a local section of $\mathcal{I}$ if and only if $fs$ is a local section of $\mathcal{L}$. On any affine open $U = \Spec(A)$ of $X$ write $\mathcal{L}|_U = \widetilde{L}$ for some invertible $A$-module $L$. Then $A$ is a Noetherian domain with fraction field $K = R(X)$ and we may think of $s|_U$ as an element of $L \otimes_A K$ (see Divisors, Lemma \ref{divisors-lemma-locally-Noetherian-K}). Let $I = \{x \in A \mid xs \in L\}$. Then we see that $\mathcal{I}|_U = \widetilde{I}$ (details omitted) and hence $\mathcal{I}$ is quasi-coherent. \medskip\noindent Consider the closed subscheme $Z \subset X$ defined by $\mathcal{I}$. It is clear that $U = X \setminus Z$. This suggests we should blow up $Z$. Let $$\pi : X' = \underline{\text{Proj}}_X \left(\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n\right) \longrightarrow X$$ be the blowing up of $X$ along $Z$. The quasi-coherent sheaf of $\mathcal{O}_X$-algebras $\bigoplus\nolimits_{n \geq 0} \mathcal{I}^n$ is generated in degree $1$ over $\mathcal{O}_X$. Moreover, the degree $1$ part is a coherent $\mathcal{O}_X$-module, in particular of finite type. Hence we see that $\pi$ is projective and $\mathcal{O}_{X'}(1)$ is relatively very ample. \medskip\noindent of denominators of $s$, see Divisors, Definition \ref{divisors-definition-regular-meromorphic-ideal-denominators}. By Divisors, Lemma \ref{divisors-lemma-blowing-up-denominators} we get (2), (3), and (4). By Divisors, Lemma \ref{divisors-lemma-blow-up-integral-scheme} we have $X'$ is integral. By Divisors, Lemma \ref{divisors-lemma-blowing-up-gives-effective-Cartier-divisor} there exists an effective Cartier divisor $E \subset X'$ such that $\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{X'} = \mathcal{I}_E$. Also, by the same lemma we see that $\pi^{-1}(U) \cong U$. \medskip\noindent Denote $s'$ the pullback of the meromorphic section $s$ to a meromorphic section of $\mathcal{L}' = \pi^*\mathcal{L}$ over $X'$. It follows from the fact that $\mathcal{I}s \subset \mathcal{L}$ that $\mathcal{I}_Es' \subset \mathcal{L}'$. In other words, $s'$ gives rise to an $\mathcal{O}_{X'}$-linear map $\mathcal{I}_E \to \mathcal{L}'$, or in other words a section $t \in \mathcal{L}' \otimes \mathcal{O}_{X'}(E)$. By Divisors, Lemma \ref{divisors-lemma-characterize-OD} we obtain a unique effective Cartier divisor $D \subset X'$ such that $\mathcal{L}' \otimes \mathcal{O}_{X'}(E) \cong \mathcal{O}_{X'}(D)$ with $t$ corresponding to $1_D$. Reversing this procedure we conclude that $\mathcal{L}' = \mathcal{O}_{X'}(-E) \cong \mathcal{O}_{X'}(D)$ with $s'$ corresponding to $1_D \otimes 1_E^{-1}$ as in (4). \medskip\noindent we get (1). By Divisors, Lemma \ref{divisors-lemma-blowing-up-projective} the morphism $\pi$ is projective. We still have to prove (5). By Lemma \ref{lemma-equal-c1-as-cycles} we have $$ @@ -8105,9 +8105,12 @@ \section{Blowing up} \begin{lemma} \label{lemma-blow-up-pullback-effective-Cartier} Let X be a scheme. Let b : X' \to X be a blowup of X in a closed subscheme. For any effective Cartier divisor D on X the pullback b^{-1}D is defined (see Definition \ref{definition-pullback-effective-Cartier-divisor}). subscheme. The pullback b^{-1}D is defined for all effective Cartier divisors D \subset X and pullbacks of meromorphic functions are defined for b (Definitions \ref{definition-pullback-effective-Cartier-divisor} and \ref{definition-pullback-meromorphic-sections}). \end{lemma} \begin{proof} @@ -8799,7 +8802,54 @@ \section{Admissible blowups} open subschemes X'_1 and X'_2. \end{proof} \begin{lemma} \label{lemma-blowing-up-denominators} Let X be a locally Noetherian scheme. Let \mathcal{L} be an invertible \mathcal{O}_X-module. Let s be a regular meromorphic section of \mathcal{L}. Let U \subset X be the maximal open subscheme such that s corresponds to a section of \mathcal{L} over U. The blowup b : X' \to X in the ideal of denominators of s is U-admissible. There exists an effective Cartier divisor D \subset X' and an isomorphism$$ b^*\mathcal{L} = \mathcal{O}_{X'}(D - E),  where $E \subset X'$ is the exceptional divisor such that the meromorphic section $b^*s$ corresponds, via the isomorphism, to the meromorphic section $1_D \otimes (1_E)^{-1}$. \end{lemma} \begin{proof} From the definition of the ideal of denominators in Definition \ref{definition-regular-meromorphic-ideal-denominators} 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 \ref{definition-invertible-sheaf-effective-Cartier-divisor}. Finally, note that $b^*s$ is defined by Lemma \ref{lemma-blow-up-pullback-effective-Cartier}. 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 $X = \Spec(A)$ is affine and $\mathcal{L} = \mathcal{O}_X$. Shrinking further we may assume $s = a/b$ with $a, b \in A$ nonzerodivisors. Then the ideal of denominators of $s$ corresponds to the ideal $I = \{x \in A \mid xa \in bA\}$. Recall that $X'$ is covered by spectra of affine blowup algebras $A' = A[\frac{I}{x}]$ with $x \in I$ (Lemma \ref{lemma-blowing-up-affine}). In $A'$ we have $b = x (b/x)$ as $b \in I$ and $E$ is cut out by $x$. Thus if we let $D' \cap \Spec(A')$ be the effective Cartier divisor cut out by the nonzerodivisor $ab/x$ of $A'$, then the lemma holds over the open $\Spec(A') \subset X'$ as desired. \end{proof}