Proposition 3

Tag 060E, permanent. Cite it as [Bourbaki, Tag 060E].

Algebra, chapter III, § 4. Tensor products of algebras

Let $A, B$ be two commutative rings, $\varphi : A \to B$ a ring homomorphism and $E, F$ two $A$ -algebras. Then the canonical $B$ -module isomorphism

\varphi^*(E) \otimes_B \varphi^*(F) \to \varphi^*(E \otimes_A F)

(II, § 5, no. 1, Proposition 3) is a $B$ -algebra isomorphism.

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

Cited by

§ 15. Brauer Groups

Cites

II, § 5, no. 1, Proposition 3