Remark 1

\* In algebraic geometry, Prop. 7 implies that the dimension of the fibres of a projective morphism of algebraic varieties is upper semi-continuous. \* 2) We shall see later that if $\rho : A \to B$ is a ring homomorphism which makes B into a finitely generated A-algebra, the function $q \mapsto \dim_q(B \otimes_A \kappa(\rho^{-1}(q)))$ is upper semi-continuous on $\mathrm{Spec}(B)$ (cf. p. 101, Exercise 10).