Remark 10.17.8. A fundamental commutative diagram associated to a ring map $\varphi : R \to S$, a prime $\mathfrak q \subset S$ and the corresponding prime $\mathfrak p = \varphi ^{-1}(\mathfrak q)$ of $R$ is the following

In this diagram the arrows in the outer left and outer right columns are identical. The horizontal maps induce on the associated spectra always a homeomorphism onto the image. The lower two rows of the diagram make sense without assuming $\mathfrak q$ exists. The lower squares induce fibre squares of topological spaces. This diagram shows that $\mathfrak p$ is in the image of the map on Spec if and only if $S \otimes _ R \kappa (\mathfrak p)$ is not the zero ring.

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