Lemma 31.11.7 (0BCM)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.11: Torsion free modules
Lemma 31.11.7 (cite)

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Lemma 31.11.7. Let $f : X \to Y$ be a flat morphism of schemes. If $Y$ is integral and the generic fibre of $f$ is integral, then $X$ is integral.

Proof. The algebraic analogue is this: let $A$ be a domain with fraction field $K$ and let $B$ be a flat $A$ -algebra such that $B \otimes _ A K$ is a domain. Then $B$ is a domain. This is true because $B$ is torsion free by More on Algebra, Lemma 15.22.9 and hence $B \subset B \otimes _ A K$ . $\square$

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