Lemma 37.26.6 (0C0D)—The Stacks project

Lemma 37.26.6. Let $X \to \mathop{\mathrm{Spec}}(R)$ be a proper flat morphism where $R$ is a discrete valuation ring. If the special fibre is reduced, then both $X$ and the generic fibre $X_\eta $ are reduced.

Proof. Assume the special fibre $X_ s$ is reduced. Let $x \in X$ be any point, and let us show that $\mathcal{O}_{X, x}$ is reduced; this will prove that $X$ and $X_\eta $ are reduced. Let $x \leadsto x'$ be a specialization with $x'$ in the special fibre; such a specialization exists as a proper morphism is closed. Consider the local ring $A = \mathcal{O}_{X, x'}$. Then $\mathcal{O}_{X, x}$ is a localization of $A$, so it suffices to show that $A$ is reduced. Let $\pi \in R$ be a uniformizer. If $a \in A$ then there exists an $n \geq 0$ and an element $a' \in A$ such that $a = \pi ^ n a'$ and $a' \not\in \pi A$. This follows from Krull intersection theorem (Algebra, Lemma 10.51.4). If $a$ is nilpotent, so is $a'$, because $\pi $ is a nonzerodivisor by flatness of $A$ over $R$. But $a'$ maps to a nonzero element of the reduced ring $A/\pi A = \mathcal{O}_{X_ s, x'}$. This is a contradiction unless $A$ is reduced, which is what we wanted to show. $\square$

Comments (4)

Comment #8076 by Laurent Moret-Bailly on January 22, 2023 at 09:22

First line of proof: $\mathcal{O}_{X_\eta}$ should be $\mathcal{O}_{X_\eta,x}$ . But in fact it is not very clear whether one is proving that $X$ is reduced or that $X_\eta$ is. I propose to change the first 3 lines as follows:

Assume the special fibre is reduced. Let $x \in X$ be any point, and let us show that $\mathcal{O}_{X,x}$
is reduced. (This will prove that $X$ and $X_\eta$ are reduced). Let $x \leadsto x'$ be a specialization with $x'$ in the special fibre; such a specialization exists as a proper morphism is closed. Consider the local ring $A = \mathcal{O}_{X, x'}$ . Then $\mathcal{O}_{X,x}$ is a localization of $A$ , so it suffices to show that $A$ is reduced. Let $\pi \in R$ be a uniformizer. (The rest is unchanged)

Comment #8088 by Laurent Moret-Bailly on January 24, 2023 at 12:12

Not sure this is useful, but the lemma works for any valuation ring $R$ : instead of Krull's intersection theorem, use the fact that $a$ has a content ideal (Lemma 0ASX) which is principal by Comment 8087. (I do realize that Lemma 0ASX comes a bit later).

Comment #8201 by Aise Johan de Jong on March 03, 2023 at 09:53

@#8076 Thanks and fixed here.

Comment #8205 by Stacks Project on March 03, 2023 at 11:48

@#8088 OK, I added this fun curiosity in a later chapter. See this commit.

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