Lemma 10.127.7. Let $A$ be a ring and let $B, C$ be $A$-algebras. Suppose that $R = \mathop{\mathrm{colim}}\nolimits _{i \in I} R_ i$ is a directed colimit of $A$-algebras.

If $B$ is a finite type $A$-algebra, and $u, u' : B \to C$ are $A$-algebra maps such that $u \otimes 1 = u' \otimes 1 : B \otimes _ A R \to C \otimes _ A R$ then for some $i$ we have $u \otimes 1 = u' \otimes 1 : B \otimes _ A R_ i \to C \otimes _ A R_ i$.

If $C$ is a finite type $A$-algebra and $u : B \to C$ is an $A$-algebra map such that $u \otimes 1 : B \otimes _ A R \to C \otimes _ A R$ is surjective, then for some $i$ the map $u \otimes 1 : B \otimes _ A R_ i \to C \otimes _ A R_ i$ is surjective.

If $C$ is of finite presentation over $A$ and $v : C \otimes _ A R \to B \otimes _ A R$ is an $R$-algebra map, then there exists an $i$ and an $R_ i$-algebra map $v_ i : C \otimes _ A R_ i \to B \otimes _ A R_ i$ such that $v = v_ i \otimes 1$.

If $B$ is a finite type $A$-algebra, $C$ is a finitely presented $A$-algebra, and $u \otimes 1 : B \otimes _ A R \to C \otimes _ A R$ is an isomorphism, then for some $i$ the map $u \otimes 1 : B \otimes _ A R_ i \to C \otimes _ A R_ i$ is an isomorphism.

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