Remark

Let $M$ be an $A$-module and $\rho : A \to B$ a commutative ring homomorphism. Then there is a commutative diagram of graded $B$-algebra homomorphisms

where the first row is a homomorphism composed of the homomorphism $\theta_T \otimes 1_B : T(M^*) \otimes_A B \to T(M)^{*gr}$ and the canonical isomorphism

§ 5, no. 3, Proposition 5). It is immediately verified, using formula (28) and the definition of the homomorphism $v_E$ (II, § 5, no. 4), that this diagram is commutative. When M is a finitely generated projective A-module, $M_{(B)}$ is a finitely generated projective B-module (II, § 5, no. 1, Corollary to Proposition 4) and all the homomorphisms of the above diagram are bijective (Proposition 7 and II, § 5, no. 4, Proposition 8). There are analogous commutative diagrams with T replaced by S or $\Lambda$; the diagram for $\Lambda$ also consists of bijective homomorphisms when M is projective and finitely generated (Proposition 7); if further A is an algebra over $\mathbf{Q}$, the diagram for S also consists of bijective homomorphisms (Proposition 8).