Lemma 10.168.3. Let $A = \mathop{\mathrm{colim}}\nolimits _{i \in I} A_ i$ be a directed colimit of rings. Let $0 \in I$ and $\varphi _0 : B_0 \to C_0$ a map of $A_0$-algebras. Assume

$A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0$ is finite,

$C_0$ is of finite type over $B_0$.

Then there exists an $i \geq 0$ such that the map $A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0$ is finite.

**Proof.**
Let $x_1, \ldots , x_ m$ be generators for $C_0$ over $B_0$. Pick monic polynomials $P_ j \in A \otimes _{A_0} B_0[T]$ such that $P_ j(1 \otimes x_ j) = 0$ in $A \otimes _{A_0} C_0$. For some $i \geq 0$ we can find $P_{j, i} \in A_ i \otimes _{A_0} B_0[T]$ mapping to $P_ j$. Since $\otimes $ commutes with colimits we see that $P_{j, i}(1 \otimes x_ j)$ is zero in $A_ i \otimes _{A_0} C_0$ after possibly increasing $i$. Then this $i$ works.
$\square$

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