The Stacks project

Proof. Note that the restriction $\mathcal{I}|_ U$ is equal to the pullback of $\mathcal{I}$ via the morphism $U \to X$, see Properties of Spaces, Section 66.26. Thus the lemma follows on combining Lemma 71.11.2 with Divisors, Lemma 31.32.2. $\square$

Proof. Let $\mathcal{I}_2$ be the ideal sheaf of $Z_2$ in $X_2$. Denote $g : X_1 \to X_2$ the given morphism. Then the ideal sheaf $\mathcal{I}_1$ of $Z_1$ is the image of $g^*\mathcal{I}_2 \to \mathcal{O}_{X_1}$ (see Morphisms of Spaces, Definition 67.13.2 and discussion following the definition). By Lemma 71.11.5 we see that $X_1 \times _{X_2} X_2'$ is the relative Proj of $\bigoplus _{n \geq 0} g^*\mathcal{I}_2^ n$. Because $g$ is flat the map $g^*\mathcal{I}_2^ n \to \mathcal{O}_{X_1}$ is injective with image $\mathcal{I}_1^ n$. Thus we see that $X_1 \times _{X_2} X_2' = X_1'$. $\square$

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 71.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.4. $\square$

Proof. We see that $b : X' \to X$ is an object of $\mathcal{C}$ according to Lemma 71.17.4. Let $f : Y \to X$ be an object of $\mathcal{C}$. We have to show there exists a unique morphism $Y \to X'$ over $X$. Let $D = f^{-1}(Z)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal sheaf of $Z$ and let $\mathcal{I}_ D$ be the ideal sheaf of $D$. Then $f^*\mathcal{I} \to \mathcal{I}_ D$ is a surjection to an invertible $\mathcal{O}_ Y$-module. This extends to a map $\psi : \bigoplus f^*\mathcal{I}^ d \to \bigoplus \mathcal{I}_ D^ d$ of graded $\mathcal{O}_ Y$-algebras. (We observe that $\mathcal{I}_ D^ d = \mathcal{I}_ D^{\otimes d}$ as $D$ is an effective Cartier divisor.) By Lemma 71.11.11. the triple $(f : Y \to X, \mathcal{I}_ D, \psi )$ defines a morphism $Y \to X'$ over $X$. The restriction

is unique. The open $Y \setminus D$ is scheme theoretically dense in $Y$ according to Lemma 71.6.4. Thus the morphism $Y \to X'$ is unique by Morphisms of Spaces, Lemma 67.17.8 (also $b$ is separated by Lemma 71.11.6). $\square$

Proof. Immediate from the universal property of blowups (Lemma 71.17.5). $\square$

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 71.17.3) we can prove each of these statements after base change to $U$. This reduces us to the case of schemes. In this case the result is Divisors, Lemma 31.32.8. $\square$

Proof. Follows immediately from the lemma cited in the statement, the étale local description of blowing ups in Lemma 71.17.2, and Divisors, Lemma 31.32.10. $\square$

Proof. By Lemmas 71.17.2 and 71.6.2 this reduces to the following algebra fact: Let $A$ be a ring, $I \subset A$ an ideal, $a \in I$, and $x \in A$ a nonzerodivisor. Then the image of $x$ in $A[\frac{I}{a}]$ is a nonzerodivisor. Namely, suppose that $x (y/a^ n) = 0$ in $A[\frac{I}{a}]$. Then $a^ mxy = 0$ in $A$ for some $m$. Hence $a^ my = 0$ as $x$ is a nonzerodivisor. Whence $y/a^ n$ is zero in $A[\frac{I}{a}]$ as desired. $\square$

Proof. Let $E \subset X'$ be the exceptional divisor of $b$ which is an effective Cartier divisor by Lemma 71.17.4. Then $(b')^{-1}E$ is an effective Cartier divisor on $X''$ by Lemma 71.17.9. Let $E' \subset X''$ be the exceptional divisor of $b'$ (also an effective Cartier divisor). Consider the effective Cartier divisor $E'' = E' + (b')^{-1}E$. By construction the ideal of $E''$ is $(b \circ b')^{-1}\mathcal{I} (b \circ b')^{-1}\mathcal{J} \mathcal{O}_{X''}$. Hence according to Lemma 71.17.5 there is a canonical morphism from $X''$ to the blowup $c : Y \to X$ of $X$ in $\mathcal{I}\mathcal{J}$. Conversely, as $\mathcal{I}\mathcal{J}$ pulls back to an invertible ideal we see that $c^{-1}\mathcal{I}\mathcal{O}_ Y$ defines an effective Cartier divisor, see Lemma 71.6.8. Thus a morphism $c' : Y \to X'$ over $X$ by Lemma 71.17.5. Then $(c')^{-1}b^{-1}\mathcal{J}\mathcal{O}_ Y = c^{-1}\mathcal{J}\mathcal{O}_ Y$ which also defines an effective Cartier divisor. Thus a morphism $c'' : Y \to X''$ over $X'$. We omit the verification that this morphism is inverse to the morphism $X'' \to Y$ constructed earlier. $\square$

Proof. Let $U$ be a scheme and let $U \to X$ be a surjective étale morphism. As blowing up commutes with flat base change (Lemma 71.17.3) we can prove each of these statements after base change to $U$ (see Morphisms of Spaces, Lemma 67.40.2). This reduces us to the case of schemes. In this case the morphism $b$ is projective by Divisors, Lemma 31.32.13 hence proper by Morphisms, Lemma 29.43.5. $\square$

Proof. The condition that $Z \to X$ is of finite presentation means that $Z$ is cut out by a finite type quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, see Morphisms of Spaces, Lemma 67.28.12. Write $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$ so that $X' = \underline{\text{Proj}}(\mathcal{A})$. Note that $X \setminus Z$ is a quasi-compact open subspace of $X$ by Limits of Spaces, Lemma 70.14.1. Since $b^{-1}(X \setminus Z) \to X \setminus Z$ is an isomorphism (Lemma 71.17.4) the same result shows that $b^{-1}(X \setminus Z) \setminus Z'$ is quasi-compact open subspace in $X'$. Hence $U = X \setminus (Z \cup b(Z'))$ is quasi-compact open subspace in $X$. By Lemma 71.16.3 there exist a $d > 0$ and a finite type $\mathcal{O}_ X$-submodule $\mathcal{F} \subset \mathcal{I}^ d$ such that $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ and such that the support of $\mathcal{I}^ d/\mathcal{F}$ is contained in $X \setminus U$.

Since $\mathcal{F} \subset \mathcal{I}^ d$ is an $\mathcal{O}_ X$-submodule we may think of $\mathcal{F} \subset \mathcal{I}^ d \subset \mathcal{O}_ X$ as a finite type quasi-coherent sheaf of ideals on $X$. Let's denote this $\mathcal{J} \subset \mathcal{O}_ X$ to prevent confusion. Since $\mathcal{I}^ d / \mathcal{J}$ and $\mathcal{O}/\mathcal{I}^ d$ are supported on $|X| \setminus |U|$ we see that $|V(\mathcal{J})|$ is contained in $|X| \setminus |U|$. Conversely, as $\mathcal{J} \subset \mathcal{I}^ d$ we see that $|Z| \subset |V(\mathcal{J})|$. Over $X \setminus Z \cong X' \setminus b^{-1}(Z)$ the sheaf of ideals $\mathcal{J}$ cuts out $Z'$ (see displayed formula below). Hence $|V(\mathcal{J})|$ equals $|Z| \cup |b|(|Z'|)$. It follows that also $|V(\mathcal{I}\mathcal{J})| = |Z| \cup |b|(|Z'|)$. Moreover, $\mathcal{I}\mathcal{J}$ is an ideal of finite type as a product of two such. We claim that $X'' \to X$ is isomorphic to the blowing up of $X$ in $\mathcal{I}\mathcal{J}$ which finishes the proof of the lemma by setting $Y = V(\mathcal{I}\mathcal{J})$.

First, recall that the blowup of $X$ in $\mathcal{I}\mathcal{J}$ is the same as the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$, see Lemma 71.17.10. Hence it suffices to show that the blowup of $X'$ in $b^{-1}\mathcal{J} \mathcal{O}_{X'}$ agrees with the blowup of $X'$ in $Z'$. We will show that

as ideal sheaves on $X''$. This will prove what we want as $\mathcal{I}_ E^ d$ cuts out the effective Cartier divisor $dE$ and we can use Lemmas 71.17.6 and 71.17.10.

To see the displayed equality of the ideals we may work locally. With notation $A$, $I$, $a \in I$ as in Lemma 71.17.2 we see that $\mathcal{F}$ corresponds to an $R$-submodule $M \subset I^ d$ mapping isomorphically to an ideal $J \subset R$. The condition $Z' = \underline{\text{Proj}}(\mathcal{A}/\mathcal{F}\mathcal{A})$ means that $Z' \cap \mathop{\mathrm{Spec}}(A[\frac{I}{a}])$ is cut out by the ideal generated by the elements $m/a^ d$, $m \in M$. Say the element $m \in M$ corresponds to the function $f \in J$. Then in the affine blowup algebra $A' = A[\frac{I}{a}]$ we see that $f = (a^ dm)/a^ d = a^ d (m/a^ d)$. Thus the equality holds. $\square$