Exercise 11, § 6

Let $\rho : A \to B$ be a ring homomorphism, $^a\rho : \mathrm{Spec}(B) \to \mathrm{Spec}(A)$ the corresponding mapping. a) Prove that for every $q \in \mathrm{Spec}(A)$, $\mathrm{Spec}(B \otimes_A \kappa(q))$ is identified with $^a\rho^{-1}(q)$. b) Suppose that $B$ is an $A$-algebra of finite type. Prove that $p$ is an isolated point in its fibre $^a\rho^{-1}(^a\rho(p))$ if and only if $\dim_p(B \otimes_A \kappa(^a\rho(p)))$ is zero. c) Deduce from Exerc. 10 that under the hypotheses of b), the set of points of $\mathrm{Spec}(B)$ which are isolated in their fibre, is an open subset of $\mathrm{Spec}(B)$.