Proposition 7

Tag 062G, permanent. Cite it as [Bourbaki, Tag 062G].

Let $A, B$ be two commutative rings, $\rho : A \to B$ a ring homomorphism and $M$ an $A$ -module. The canonical extension

\psi : S_B(B \otimes_A M) \to B \otimes_A S_A(M)

of the $B$ -linear mapping $l_B \otimes \phi_M' : B \otimes_A M \to B \otimes_A S_A(M)$ is a graded $B$ -algebra isomorphism.

The proof is derived from that of § 5, no. 3, Proposition 5 replacing $T$ by $S$ and $\phi_M$ by $\phi_M'$ .

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

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