The Stacks project

Lemma 32.5.2. Suppose given a cartesian diagram of rings

\[ \xymatrix{ B \ar[r]_ s & R \\ B'\ar[u] \ar[r] & R' \ar[u]_ t } \]

Let $W' \subset \mathop{\mathrm{Spec}}(R')$ be an open of the form $W' = D(f_1) \cup \ldots \cup D(f_ n)$ such that $t(f_ i) = s(g_ i)$ for some $g_ i \in B$ and $B_{g_ i} \cong R_{s(g_ i)}$. Then $B' \to R'$ induces an open immersion of $W'$ into $\mathop{\mathrm{Spec}}(B')$.

Proof. Set $h_ i = (g_ i, f_ i) \in B'$. More on Algebra, Lemma 15.5.3 shows that $(B')_{h_ i} \cong (R')_{f_ i}$ as desired. $\square$

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