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The Stacks project

Lemma 10.126.7. Let R be a ring. Let S, S' be of finite presentation over R. Let \mathfrak q \subset S and \mathfrak q' \subset S' be primes. If S_{\mathfrak q} \cong S'_{\mathfrak q'} as R-algebras, then there exist g \in S, g \not\in \mathfrak q and g' \in S', g' \not\in \mathfrak q' such that S_ g \cong S'_{g'} as R-algebras.

Proof. Let \psi : S_{\mathfrak q} \to S'_{\mathfrak q'} be the isomorphism of the hypothesis of the lemma. Write S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ r) and S' = R[y_1, \ldots , y_ m]/J. For each i = 1, \ldots , n choose a fraction h_ i/g_ i with h_ i, g_ i \in R[y_1, \ldots , y_ m] and g_ i \bmod J not in \mathfrak q' which represents the image of x_ i under \psi . After replacing S' by S'_{g_1 \ldots g_ n} and R[y_1, \ldots , y_ m, y_{m + 1}] (mapping y_{m + 1} to 1/(g_1\ldots g_ n)) we may assume that \psi (x_ i) is the image of some h_ i \in R[y_1, \ldots , y_ m]. Consider the elements f_ j(h_1, \ldots , h_ n) \in R[y_1, \ldots , y_ m]. Since \psi kills each f_ j we see that there exists a g \in R[y_1, \ldots , y_ m], g \bmod J \not\in \mathfrak q' such that g f_ j(h_1, \ldots , h_ n) \in J for each j = 1, \ldots , r. After replacing S' by S'_ g and R[y_1, \ldots , y_ m, y_{m + 1}] as before we may assume that f_ j(h_1, \ldots , h_ n) \in J. Thus we obtain a ring map S \to S', x_ i \mapsto h_ i which induces \psi on local rings. By Lemma 10.6.2 the map S \to S' is of finite presentation. By Lemma 10.126.6 we may assume that S' = S \times C. Thus localizing S' at the idempotent corresponding to the factor C we obtain the result. \square


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