Proposition 19

Let $A, B$ be two rings, $\phi : A \to B$ a homomorphism and

the continuous mapping associated with $\phi$ (Proposition 13). For every $A$-module $M$, Let $q$ be a prime ideal of $B$ and $p = \phi^{-1}(q)$. Suppose that $q$ belongs to $\operatorname{Supp}(M_{(B)})$; then $M_{(B)} \otimes_B B_q = (M \otimes_A B) \otimes_B B_q = M \otimes_A B_q = (M \otimes_A A_p) \otimes_A B_q$, since the homomorphism $A \to B \to B_q$ factors into $A \to A_p \to B_q$ (§ 2, no. 1, Proposition 2); the hypothesis $M_{(B)} \otimes_B B_q \neq 0$ implies therefore $M \otimes_A A_q \neq 0$, whence the first assertion. As the homomorphism $\phi_q : A_p \to B_q$ is local, the second assertion follows from the following lemma: