Lemma 10.39.6. Let $R \to R'$ be a ring map and let $M$ be a finite $R$-module. Then $\text{Supp}(M \otimes _ R R')$ is the inverse image of $\text{Supp}(M)$.

**Proof.**
Let $\mathfrak p \in \text{Supp}(M)$. By Nakayama's lemma (Lemma 10.19.1) we see that

is a nonzero $\kappa (\mathfrak p)$ vector space. Hence for every prime $\mathfrak p' \subset R'$ lying over $\mathfrak p$ we see that

is nonzero. This implies $\mathfrak p' \in \text{Supp}(M \otimes _ R R')$. For the converse, if $\mathfrak p' \subset R'$ is a prime lying over an arbitrary prime $\mathfrak p \subset R$, then

Hence if $\mathfrak p' \in \text{Supp}(M \otimes _ R R')$ lies over the prime $\mathfrak p \subset R$, then $\mathfrak p \in \text{Supp}(M)$. $\square$

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