Remark

Let $\rho : A \to B$ be a homomorphism of rings, and let $p \in \mathrm{Spec}(A)$. Let us denote by $\overline{B}$ the ring $\kappa(p) \otimes_A B$. It may be identified with $S^{-1}B/p(S^{-1}B)$, where $S$ is the multiplicative subset $\rho(A - p)$ of $B$; the prime ideals of $\overline{B}$ are therefore the ideals $q\overline{B}$, where $q$ is a prime ideal of $B$ containing $pB$ and not meeting $S$, in other words a prime ideal of $B$ lying over $p$. For such an ideal $q$, one has $S \subset B - q$, hence the local ring of $\overline{B}$ at $q\overline{B}$ may be identified with $B_q/pB_q$, that is to say again with $\kappa(p) \otimes_A B_q$.

Analogously, if $N$ is a $B$-module, the $\overline{B}_{q\overline{B}}$-module $(\kappa(p) \otimes_A N)_{q\overline{B}}$ may be identified with $\kappa(p) \otimes_A N_q$. Suppose moreover that the $B$-module $N$ is finitely generated; by Nakayama's lemma, the condition $\kappa(p) \otimes_A N_q = 0$ is equivalent to $N_q = 0$. Thus the support of the $\overline{B}$-module $\kappa(p) \otimes_A N$ consists of the ideals $q\overline{B}$, where $q$ runs through the prime ideals of $\mathrm{Supp}_B(N)$ lying over $p$. In particular, in order that the module $\kappa(p) \otimes_A N$ be nonzero, it is necessary and sufficient that there exist a prime ideal of $\mathrm{Supp}_B(N)$ lying over $p$.