Corollary

Tag 05YV, permanent. Cite it as [Bourbaki, Tag 05YV].

Algebra, chapter III, § 1. Algebras

Let $E, E'$ be two $A$ -algebras; for every $A$ -homomorphism of algebras $u : E \to E', u \otimes 1_B$ is the unique $B$ -homomorphism of algebras $v : E \otimes_A B \to E' \otimes_A B$ rendering commutative the diagram

\begin{array}{ccc} E & \xrightarrow{\phi_E} & E \otimes_A B \\ u \downarrow & & \downarrow v \\ E' & \xrightarrow{\phi_{E'}} & E' \otimes_A B \end{array}

Let $C$ be a third commutative ring and $\sigma : B \to C$ a ring homomorphism; it is immediate that the canonical $C$ -homomorphism

\sigma^*(\rho^*(E)) \to (\sigma \circ \rho)^*(E)

mapping $(x \otimes 1) \otimes 1$ to $x \otimes 1$ for all $x \in E$ (II, § 5, no. 1, Proposition 2) is an algebra isomorphism.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

II, § 5, no. 1, Proposition 2