Lemma 37.26.2 (0576)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.26: Reduced fibres
Lemma 37.26.2 (cite)

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Lemma 37.26.2. Let $f : X \to Y$ be a morphism of schemes. Let $g : Y' \to Y$ be any morphism, and denote $f' : X' \to Y'$ the base change of $f$. Then

\begin{align*} \{ y' \in Y' \mid X'_{y'}\text{ is geometrically reduced}\} \\ = g^{-1}(\{ y \in Y \mid X_ y\text{ is geometrically reduced}\} ). \end{align*}

Proof. This comes down to the statement that for $y' \in Y'$ with image $y \in Y$ the fibre $X'_{y'} = X_ y \times _ y y'$ is geometrically reduced over $\kappa (y')$ if and only if $X_ y$ is geometrically reduced over $\kappa (y)$. This follows from Varieties, Lemma 33.6.6. $\square$

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