The Stacks project

Remark 15.116.4. The construction in Lemma 15.116.3 satisfies the following “functoriality”. Suppose we have a commutative diagram

\[ \xymatrix{ A'_2 \ar[r] & A_2 \\ A'_1 \ar[r] \ar[u] & A_1 \ar[u] } \]

with injective horizontal arrows. Suppose given an element $f \in A'_1$ such that $(A'_1 \subset A_1, f)$ and $(A'_2 \subset A_2, f)$ satisfy properties (a), (b), (c) of Lemma 15.116.3. Let $n_{0, 1}$ and $n_{0, 2}$ be the integers found in the lemma for these two situations. Finally, let $B'_1 \to B'_2$ be a ring map, let $g \in B'_1$ be a nonzerodivisor on $B_1$ and $B_2$, let $n \geq \max (n_{0, 1}, n_{0, 2})$, and let a commutative diagram

\[ \xymatrix{ A'_2/f^ nA'_2 \ar[r]_{\varphi '_2} & B'_2/g^ nB'_2 \\ A'_1/f^ nA'_1 \ar[r]^{\varphi '_1} \ar[u] & B'_2/g^ nB'_2 \ar[u] } \]

be given whose horizontal arrows are isomorphisms and where $\varphi '_1(f) \equiv g$. Then we obtain commutative diagrams

\[ \vcenter { \xymatrix{ B'_2 \ar[r] & B_2 \\ B'_1 \ar[r] \ar[u] & B_1 \ar[u] } } \quad \text{and}\quad \vcenter { \xymatrix{ A_2/fA_2 \ar[r]_{\varphi _2} & B_2/gB_2 \\ A_1/fA_1 \ar[r]^{\varphi _1} \ar[u] & B_2/gB_2 \ar[u] } } \]

where $(B'_1 \subset B_1, \varphi _1)$ and $(B'_2 \subset B_2, \varphi _2)$ are constructed as in the proof of Lemma 15.116.3. We omit the detailed verification.

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