Lemma 1

Let $\rho : A \to B$ be a homomorphism of rings and p a prime ideal of A. The continuous mapping $^a h : \mathrm{Spec}(B \otimes_A \kappa(p)) \to \mathrm{Spec}(B)$, associated with the canonical homomorphism $h : B \to B \otimes_A \kappa(p)$, induces a homeomorphism of $\mathrm{Spec}(B \otimes_A \kappa(p))$ onto the subspace $(\alpha\rho)^{-1}(p)$ of $\mathrm{Spec}(B)$ formed by the prime ideals of $B$ lying above $p$.

The homomorphism $h$ is composed of the homomorphism of passing to the quotient of $B$ into $B/\rho(p)$ $B$ and of the canonical homomorphism of $B/\rho(p)$ $B$ into its ring of fractions $(\rho(A - p))^{-1}(B/\rho(p) \ B)$. According to the remark and the corollary to prop. 13 of II, § 4, no 3, $^a h$ therefore induces a homeomorphism of $\mathrm{Spec}(B \otimes_A \kappa(p))$ onto the subspace of $\mathrm{Spec}(B)$ formed by the prime ideals $q$ of $B$ which contain $\rho(p)$ and are disjoint from $\rho(A - p)$, that is to say which lie above $p$.