The Stacks project

Lemma 54.11.2. 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. 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 $\mathcal{O}_{Y, y}$ is regular, then $\mathcal{O}_{X, x}$ is regular,

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

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

Proof. Since $A \to A^\wedge $ is faithfully flat (Algebra, Lemma 10.97.3), we see that $Y \to X$ is flat. Hence (1) by Algebra, Lemma 10.164.4. Lemma 54.11.1 shows the morphism $Y \to X$ induces an isomorphism on complete local rings at points of the special fibres. Thus (2) by More on Algebra, Lemma 15.43.4. If $X$ is proper over $A$, then $Y$ is proper over $A^\wedge $ (Morphisms, Lemma 29.41.5) and we see every closed point of $X$ and $Y$ lies in the closed fibre. Thus we see that $Y$ is a regular scheme if and only if $X$ is so by Properties, Lemma 28.9.2. $\square$

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