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
Comments (0)