Lemma 4

Let $A$ be a discrete valuation ring, $B$ a noetherian local ring, and $\rho : A \to B$ a local and flat homomorphism. If the ring $\kappa_A \otimes_A B$ is reduced, then $B$ is reduced.

Suppose there exists a nonzero nilpotent element $x$ of $B$, and let $\pi$ be a uniformizer of $A$. Since one has $\pi B \subset m_B$, the ring $B$ is separated for the $\pi B$-adic topology. There therefore exist $n \in \mathbf{N}$ and $y \in B$ with $x = \pi^n y$ and $y \notin \pi B$. Since $B$ is flat over $A$, multiplication by $\pi$ is injective in $B$. The class of $y$ in $B/\pi B$ is therefore a nonzero nilpotent element, which contradicts the assumption.