Remark 10.18.5 (00E6)—The Stacks project

Remark 10.18.5. A fundamental commutative diagram associated to a ring map $\varphi : R \to S$ and a prime $\mathfrak p \subset R$ is the following

$\xymatrix{ \kappa (\mathfrak p) \otimes _ R S = S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} & S_{\mathfrak p} \ar[l] & S \ar[r] \ar[l] & S/\mathfrak pS \ar[r] & (R \setminus \mathfrak p)^{-1}S/\mathfrak pS \\ \kappa (\mathfrak p) = R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u] & R_{\mathfrak p} \ar[u] \ar[l] & R \ar[u] \ar[r] \ar[l] & R/\mathfrak p \ar[u] \ar[r] & \kappa (\mathfrak p) \ar[u] }$

In this diagram the outer left and outer right columns are identical. On spectra the horizontal maps induce homeomorphisms onto their images and the squares induce fibre squares of topological spaces (see Lemmas 10.17.5 and 10.17.7). This 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. If there does exist a prime $\mathfrak q \subset S$ lying over $\mathfrak p$ , i.e., with $\mathfrak p = \varphi ^{-1}(\mathfrak q)$ then we can extend the diagram to the following diagram

$\xymatrix{ \kappa (\mathfrak q) = S_{\mathfrak q}/{\mathfrak q}S_{\mathfrak q} & S_{\mathfrak q} \ar[l] & S \ar[r] \ar[l] & S/\mathfrak q \ar[r] & \kappa (\mathfrak q) \\ \kappa (\mathfrak p) \otimes _ R S = S_{\mathfrak p}/{\mathfrak p}S_{\mathfrak p} \ar[u] & S_{\mathfrak p} \ar[u] \ar[l] & S \ar[u] \ar[r] \ar[l] & S/\mathfrak pS \ar[u] \ar[r] & (R \setminus \mathfrak p)^{-1}S/\mathfrak pS \ar[u] \\ \kappa (\mathfrak p) = R_{\mathfrak p}/{\mathfrak p}R_{\mathfrak p} \ar[u] & R_{\mathfrak p} \ar[u] \ar[l] & R \ar[u] \ar[r] \ar[l] & R/\mathfrak p \ar[u] \ar[r] & \kappa (\mathfrak p) \ar[u] }$

In this diagram it is still the case that the outer left and outer right columns are identical and that on spectra the horizontal maps induce homeomorphisms onto their image.

The Stacks project

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