The Stacks project

87.6 Base change to the completion

The following simple lemma will turn out to be a useful tool in what follows.

Lemma 87.6.1. Let $(A, \mathfrak m, \kappa )$ be a local ring with finitely generated maximal ideal $\mathfrak m$. Let $X$ be a decent algebraic space over $A$. Let $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ where $A^\wedge $ is the $\mathfrak m$-adic completion of $A$. For a point $q \in |Y|$ with image $p \in |X|$ lying over the closed point of $\mathop{\mathrm{Spec}}(A)$ the map $\mathcal{O}_{X, p}^ h \to \mathcal{O}_{Y, q}^ h$ of henselian local rings induces an isomorphism on completions.

Proof. This follows immediately from the case of schemes by choosing an elementary étale neighbourhood $(U, u) \to (X, p)$ as in Decent Spaces, Lemma 66.11.4, setting $V = U \times _ X Y = U \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ and $v = (u, q)$. The case of schemes is Resolution of Surfaces, Lemma 54.11.1. $\square$

Lemma 87.6.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $X \to \mathop{\mathrm{Spec}}(A)$ be a morphism which is locally of finite type with $X$ a decent algebraic space. Set $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$. Let $y \in |Y|$ with image $x \in |X|$. Then

  1. if $\mathcal{O}_{Y, y}^ h$ is regular, then $\mathcal{O}_{X, x}^ h$ is regular,

  2. if $y$ is in the closed fibre, then $\mathcal{O}_{Y, y}^ h$ is regular $\Leftrightarrow \mathcal{O}_{X, x}^ h$ is regular, and

  3. If $X$ is proper over $A$, then $X$ is regular if and only if $Y$ is regular.

Proof. By étale localization the first two statements follow immediately from the counter part to this lemma for schemes, see Resolution of Surfaces, Lemma 54.11.2. For part (3), since $Y \to X$ is surjective (as $A \to A^\wedge $ is faithfully flat) we see that $Y$ regular implies $X$ regular by part (1). Conversely, if $X$ is regular, then the henselian local rings of $Y$ are regular for all points of the special fibre. Let $y \in |Y|$ be a general point. Since $|Y| \to |\mathop{\mathrm{Spec}}(A^\wedge )|$ is closed in the proper case, we can find a specialization $y \leadsto y_0$ with $y_0$ in the closed fibre. Choose an elementary étale neighbourhood $(V, v_0) \to (Y, y_0)$ as in Decent Spaces, Lemma 66.11.4. Since $Y$ is decent we can lift $y \leadsto y_0$ to a specialization $v \leadsto v_0$ in $V$ (Decent Spaces, Lemma 66.12.2). Then we conclude that $\mathcal{O}_{V, v}$ is a localization of $\mathcal{O}_{V, v_0}$ hence regular and the proof is complete. $\square$

Lemma 87.6.3. Let $(A, \mathfrak m)$ be a local Noetherian ring. Let $X$ be an algebraic space over $A$. Assume

  1. $A$ is analytically unramified (Algebra, Definition 10.160.9),

  2. $X$ is locally of finite type over $A$,

  3. $X \to \mathop{\mathrm{Spec}}(A)$ is étale at every point of codimension $0$ in $X$.

Then the normalization of $X$ is finite over $X$.

Proof. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Then $U \to \mathop{\mathrm{Spec}}(A)$ satisfies the assumptions and hence the conclusions of Resolution of Surfaces, Lemma 54.11.5. $\square$

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