Lemma 4

Let $A, B$ be two local rings, $p : A \to B$ a local homomorphism and $E$ a finitely generated $A$-module. If $E \neq 0$, then $E_{(B)} \neq 0$.

Let $m$ be the maximal ideal of $A$ and $k = A/m$ the residue field; the hypothesis implies that $B \otimes_A k = B/mB \neq 0$; since the tensor product is associative, $(E \otimes_A B) \otimes_A k$ is isomorphic to $E \otimes_A (B \otimes_A k)$, hence also to $E \otimes_A (k \otimes_k (B \otimes_A k))$ and finally to $(E \otimes_A k) \otimes_k (B \otimes_A k)$; by § 3, no. 2, Proposition 4, $E \otimes_A k \neq 0$, hence $(E \otimes_A B) \otimes_A k \neq 0$ (Algebra, Chapter II, § 3, no. 7) and a fortiori $E \otimes_A B \neq 0$.