The Stacks project

65.48 Relative normalization of algebraic spaces

This section is the analogue of Morphisms, Section 29.53.

Lemma 65.48.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. There exists a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras $\mathcal{A}' \subset \mathcal{A}$ such that for any affine object $U$ of $X_{\acute{e}tale}$ the ring $\mathcal{A}'(U) \subset \mathcal{A}(U)$ is the integral closure of $\mathcal{O}_ X(U)$ in $\mathcal{A}(U)$.

Proof. By Properties of Spaces, Lemma 64.18.5 it suffices to prove that the rule given above defines a quasi-coherent module on $X_{affine, {\acute{e}tale}}$. To see this it suffices to show the following: Let $U_1 \to U_2$ be a morphism of affine objects of $X_{\acute{e}tale}$. Say $U_ i = \mathop{\mathrm{Spec}}(R_ i)$. Say $\mathcal{A}|_{(U_1)_{\acute{e}tale}}$ is the quasi-coherent sheaf associated to the $R_2$-algebra $A$. Let $A' \subset A$ be the integral closure of $R_2$ in $A$. Then $A' \otimes _{R_2} R_1$ is the integral closure of $R_1$ in $A \otimes _{R_2} R_1$. This is Algebra, Lemma 10.147.2. $\square$

Definition 65.48.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras. The integral closure of $\mathcal{O}_ X$ in $\mathcal{A}$ is the quasi-coherent $\mathcal{O}_ X$-subalgebra $\mathcal{A}' \subset \mathcal{A}$ constructed in Lemma 65.48.1 above.

We will apply this in particular when $\mathcal{A} = f_*\mathcal{O}_ Y$ for a quasi-compact and quasi-separated morphism of algebraic spaces $f : Y \to X$ (see Lemma 65.11.2). We can then take the relative spectrum of the quasi-coherent $\mathcal{O}_ X$-algebra (Lemma 65.20.7) to obtain the normalization of $X$ in $Y$.

Definition 65.48.3. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $\mathcal{O}'$ be the integral closure of $\mathcal{O}_ X$ in $f_*\mathcal{O}_ Y$. The normalization of $X$ in $Y$ is the morphism of algebraic spaces

\[ \nu : X' = \underline{\mathop{\mathrm{Spec}}}_ X(\mathcal{O}') \to X \]

over $S$. It comes equipped with a natural factorization

\[ Y \xrightarrow {f'} X' \xrightarrow {\nu } X \]

of the initial morphism $f$.

To get the factorization, use Remark 65.20.9 and functoriality of the $\underline{\mathop{\mathrm{Spec}}}$ construction.

Lemma 65.48.4. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $Y \to X' \to X$ be the normalization of $X$ in $Y$.

  1. If $W \to X$ is an étale morphism of algebraic spaces over $S$, then $W \times _ X X'$ is the normalization of $W$ in $W \times _ X Y$.

  2. If $Y$ and $X$ are representable, then $Y'$ is representable and is canonically isomorphic to the normalization of the scheme $X$ in the scheme $Y$ as constructed in Morphisms, Section 29.54.

Proof. It is immediate from the construction that the formation of the normalization of $X$ in $Y$ commutes with étale base change, i.e., part (1) holds. On the other hand, if $X$ and $Y$ are schemes, then for $U \subset X$ affine open, $f_*\mathcal{O}_ Y(U) = \mathcal{O}_ Y(f^{-1}(U))$ and hence $\nu ^{-1}(U)$ is the spectrum of exactly the same ring as we get in the corresponding construction for schemes. $\square$

Here is a characterization of this construction.

Lemma 65.48.5. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. The factorization $f = \nu \circ f'$, where $\nu : X' \to X$ is the normalization of $X$ in $Y$ is characterized by the following two properties:

  1. the morphism $\nu $ is integral, and

  2. for any factorization $f = \pi \circ g$, with $\pi : Z \to X$ integral, there exists a commutative diagram

    \[ \xymatrix{ Y \ar[d]_{f'} \ar[r]_ g & Z \ar[d]^\pi \\ X' \ar[ru]^ h \ar[r]^\nu & X } \]

    for a unique morphism $h : X' \to Z$.

Moreover, in (2) the morphism $h : X' \to Z$ is the normalization of $Z$ in $Y$.

Proof. Let $\mathcal{O}' \subset f_*\mathcal{O}_ Y$ be the integral closure of $\mathcal{O}_ X$ as in Definition 65.48.3. The morphism $\nu $ is integral by construction, which proves (1). Assume given a factorization $f = \pi \circ g$ with $\pi : Z \to X$ integral as in (2). By Definition 65.45.2 $\pi $ is affine, and hence $Z$ is the relative spectrum of a quasi-coherent sheaf of $\mathcal{O}_ X$-algebras $\mathcal{B}$. The morphism $g : X \to Z$ corresponds to a map of $\mathcal{O}_ X$-algebras $\chi : \mathcal{B} \to f_*\mathcal{O}_ Y$. Since $\mathcal{B}(U)$ is integral over $\mathcal{O}_ X(U)$ for every affine $U$ étale over $X$ (by Definition 65.45.2) we see from Lemma 65.48.1 that $\chi (\mathcal{B}) \subset \mathcal{O}'$. By the functoriality of the relative spectrum Lemma 65.20.7 this provides us with a unique morphism $h : X' \to Z$. We omit the verification that the diagram commutes.

It is clear that (1) and (2) characterize the factorization $f = \nu \circ f'$ since it characterizes it as an initial object in a category. The morphism $h$ in (2) is integral by Lemma 65.45.12. Given a factorization $g = \pi ' \circ g'$ with $\pi ' : Z' \to Z$ integral, we get a factorization $f = (\pi \circ \pi ') \circ g'$ and we get a morphism $h' : X' \to Z'$. Uniqueness implies that $\pi ' \circ h' = h$. Hence the characterization (1), (2) applies to the morphism $h : X' \to Z$ which gives the last statement of the lemma. $\square$

Lemma 65.48.6. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Let $X' \to X$ be the normalization of $X$ in $Y$. If $Y$ is reduced, so is $X'$.

Proof. This follows from the fact that a subring of a reduced ring is reduced. Some details omitted. $\square$

Lemma 65.48.7. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of schemes. Let $X' \to X$ be the normalization of $X$ in $Y$. If $x' \in |X'|$ is a point of codimension $0$ (Properties of Spaces, Definition 64.10.2), then $x'$ is the image of some $y \in |Y|$ of codimension $0$.

Proof. By Lemma 65.48.4 and the definitions, we may assume that $X = \mathop{\mathrm{Spec}}(A)$ is affine. Then $X' = \mathop{\mathrm{Spec}}(A')$ where $A'$ is the integral closure of $A$ in $\Gamma (Y, \mathcal{O}_ Y)$ and $x'$ corresponds to a minimal prime of $A'$. Choose a surjective étale morphism $V \to Y$ where $V = \mathop{\mathrm{Spec}}(B)$ is affine. Then $A' \to B$ is injective, hence every minimal prime of $A'$ is the image of a minimal prime of $B$, see Algebra, Lemma 10.30.5. The lemma follows. $\square$

Lemma 65.48.8. Let $S$ be a scheme. Let $f : Y \to X$ be a quasi-compact and quasi-separated morphism of algebraic spaces over $S$. Suppose that $Y = Y_1 \amalg Y_2$ is a disjoint union of two algebraic spaces. Write $f_ i = f|_{Y_ i}$. Let $X_ i'$ be the normalization of $X$ in $Y_ i$. Then $X_1' \amalg X_2'$ is the normalization of $X$ in $Y$.

Proof. Omitted. $\square$

Lemma 65.48.9. Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact, quasi-separated and universally closed morphisms of algebraic spaces over $S$. Then $f_*\mathcal{O}_ X$ is integral over $\mathcal{O}_ Y$. In other words, the normalization of $Y$ in $X$ is equal to the factorization

\[ X \longrightarrow \underline{\mathop{\mathrm{Spec}}}_ Y(f_*\mathcal{O}_ X) \longrightarrow Y \]

of Remark 65.20.9.

Proof. The question is étale local on $Y$, hence we may reduce to the case where $Y = \mathop{\mathrm{Spec}}(R)$ is affine. Let $h \in \Gamma (X, \mathcal{O}_ X)$. We have to show that $h$ satisfies a monic equation over $R$. Think of $h$ as a morphism as in the following commutative diagram

\[ \xymatrix{ X \ar[rr]_ h \ar[rd]_ f & & \mathbf{A}^1_ Y \ar[ld] \\ & Y & } \]

Let $Z \subset \mathbf{A}^1_ Y$ be the scheme theoretic image of $h$, see Definition 65.16.2. The morphism $h$ is quasi-compact as $f$ is quasi-compact and $\mathbf{A}^1_ Y \to Y$ is separated, see Lemma 65.8.9. By Lemma 65.16.3 the morphism $X \to Z$ has dense image on underlying topological spaces. By Lemma 65.40.6 the morphism $X \to Z$ is closed. Hence $h(X) = Z$ (set theoretically). Thus we can use Lemma 65.40.7 to conclude that $Z \to Y$ is universally closed (and even proper). Since $Z \subset \mathbf{A}^1_ Y$, we see that $Z \to Y$ is affine and proper, hence integral by Lemma 65.45.7. Writing $\mathbf{A}^1_ Y = \mathop{\mathrm{Spec}}(R[T])$ we conclude that the ideal $I \subset R[T]$ of $Z$ contains a monic polynomial $P(T) \in R[T]$. Hence $P(h) = 0$ and we win. $\square$

Lemma 65.48.10. Let $S$ be a scheme. Let $f : Y \to X$ be an integral morphism of algebraic spaces over $S$. Then the integral closure of $X$ in $Y$ is equal to $Y$.

Proof. By Lemma 65.45.7 this is a special case of Lemma 65.48.9. $\square$

Lemma 65.48.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that

  1. $Y$ is Nagata,

  2. $f$ is quasi-separated of finite type,

  3. $X$ is reduced.

Then the normalization $\nu : Y' \to Y$ of $Y$ in $X$ is finite.

Proof. The question is étale local on $Y$, see Lemma 65.48.4. Thus we may assume $Y = \mathop{\mathrm{Spec}}(R)$ is affine. Then $R$ is a Noetherian Nagata ring and we have to show that the integral closure of $R$ in $\Gamma (X, \mathcal{O}_ X)$ is finite over $R$. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Choose an affine scheme $U$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 64.6.3). Then $\Gamma (X, \mathcal{O}_ X) \subset \Gamma (U, \mathcal{O}_ X)$. Since $R$ is Noetherian it suffices to show that the integral closure of $R$ in $\Gamma (U, \mathcal{O}_ U)$ is finite over $R$. As $U \to Y$ is of finite type this follows from Morphisms, Lemma 29.53.15. $\square$


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