Proposition 8

Tag 0684, permanent. Cite it as [Bourbaki, Tag 0684].

Algebra, chapter III, § 11. Cogebras, products of multilinear forms, inner products and duality

Suppose that the ring $A$ is an algebra over the field $\mathbf{Q}$ of rational numbers; then, for every finitely generated projective $A$ -module $M$ , the homomorphism

\theta_S : S(M^*) \to S(M)^{*gr}

is bijective.

It amounts to proving this when $M$ is finitely generated and free; we pass from this to the general case using Lemma 2 as in the proof of Proposition 7.

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

Cited by

Remark, § 11
§ 8. Symmetric invariants

Cites

Lemma 2
Proposition 7