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- Jun 20, 2014

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Given commutative rings with unity $R$ and $S$, let $\phi : R \to S$ be a morphism of rings. It induces a morphism $\phi^* : \operatorname{Spec}(S) \to \operatorname{Spec}(R)$ of prime spectra such that $\phi^*(\mathfrak{q}) = \phi^{-1}(\mathfrak{q})$ for all $\mathfrak{q}\in \operatorname{Spec}(S)$. Show that if $X$ is a finitely generated $R$-module, the support of $S\otimes_R X$ is the inverse image of the support of $X$ under the induced map $\phi^*$.

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