The Stacks project

Lemma 89.6.2. 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 68.11.4. Since $Y$ is decent we can lift $y \leadsto y_0$ to a specialization $v \leadsto v_0$ in $V$ (Decent Spaces, Lemma 68.12.2). Then we conclude that $\mathcal{O}_{V, v}$ is a localization of $\mathcal{O}_{V, v_0}$ hence regular and the proof is complete. $\square$


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