Lemma 10.156.1. Let R \to S be a ring map. Let \mathfrak q be a prime of S lying over \mathfrak p in R. Assume R \to S is quasi-finite at \mathfrak q. The commutative diagram
\xymatrix{ R_{\mathfrak p}^ h \ar[r] & S_{\mathfrak q}^ h \\ R_{\mathfrak p} \ar[u] \ar[r] & S_{\mathfrak q} \ar[u] }
of Lemma 10.155.6 identifies S_{\mathfrak q}^ h with the localization of R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q} at the prime generated by \mathfrak q. Moreover, the ring map R_{\mathfrak p}^ h \to S_{\mathfrak q}^ h is finite.
Proof.
Note that R_{\mathfrak p}^ h \otimes _ R S is quasi-finite over R_{\mathfrak p}^ h at the prime ideal corresponding to \mathfrak q, see Lemma 10.122.6. Hence the localization S' of R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q} is henselian and finite over R_{\mathfrak p}^ h, see Lemma 10.153.4. As a localization S' is a filtered colimit of étale R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}-algebras. By Lemma 10.155.8 we see that S_\mathfrak q^ h is the henselization of R_{\mathfrak p}^ h \otimes _{R_{\mathfrak p}} S_{\mathfrak q}. Thus S' = S_\mathfrak q^ h by the uniqueness result of Lemma 10.154.7.
\square
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