= Integrality of the generic fibre of an affine dominant morphism
For an injective <ring homomorphism> between nonzero <integral domains> $A\to B$, the <scheme-theoretic fibre> over the <generic point> of $\operatorname{Spec}A$ is $\operatorname{Spec}(S^{-1}B)$ with $S=A\setminus\{0\}$. Localization at these nonzero elements is a nonzero integral domain, so the generic fibre is a nonempty <integral scheme>. A surjective morphism between these affine schemes necessarily gives an injective ring map: a prime above $(0)$ contains its kernel.
Back to article page
