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The Stacks project

Lemma 10.168.9. 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

  1. A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0 is syntomic (resp. a relative global complete intersection),

  2. C_0 is of finite presentation 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 syntomic (resp. a relative global complete intersection).

Proof. Assume A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0 is a relative global complete intersection. By Lemma 10.136.11 there exists a finite type \mathbf{Z}-algebra R, a ring map R \to A \otimes _{A_0} B_0, a relative global complete intersection R \to S, and an isomorphism

(A \otimes _{A_0} B_0) \otimes _ R S \longrightarrow A \otimes _{A_0} C_0

Because R is of finite type (and hence finite presentation) over \mathbf{Z}, there exists an i and a map R \to A_ i \otimes _{A_0} B_0 lifting the map R \to A \otimes _{A_0} B_0, see Lemma 10.127.3. Using the same lemma, there exists an i' \geq i such that (A_ i \otimes _{A_0} B_0) \otimes _ R S \to A \otimes _{A_0} C_0 comes from a map (A_ i \otimes _{A_0} B_0) \otimes _ R S \to A_{i'} \otimes _{A_0} C_0. Thus we may assume, after replacing i by i', that the displayed map comes from an A_ i \otimes _{A_0} B_0-algebra map

(A_ i \otimes _{A_0} B_0) \otimes _ R S \longrightarrow A_ i \otimes _{A_0} C_0

By Lemma 10.168.6 after increasing i this map is an isomorphism. This finishes the proof in this case because the base change of a relative global complete intersection is a relative global complete intersection by Lemma 10.136.9.

Assume A \otimes _{A_0} B_0 \to A \otimes _{A_0} C_0 is syntomic. Then there exist elements g_1, \ldots , g_ m in A \otimes _{A_0} C_0 generating the unit ideal such that A \otimes _{A_0} B_0 \to (A \otimes _{A_0} C_0)_{g_ j} is a relative global complete intersection, see Lemma 10.136.15. We can find an i and elements g_{i, j} \in A_ i \otimes _{A_0} C_0 mapping to g_ j. After increasing i we may assume g_{i, 1}, \ldots , g_{i, m} generate the unit ideal of A_ i \otimes _{A_0} C_0. The result of the previous paragraph implies that, after increasing i, we may assume the maps A_ i \otimes _{A_0} B_0 \to (A_ i \otimes _{A_0} C_0)_{g_{i, j}} are relative global complete intersections. Then A_ i \otimes _{A_0} B_0 \to A_ i \otimes _{A_0} C_0 is syntomic by Lemma 10.136.4 (and the already used Lemma 10.136.15). \square


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