Definition 3

Tag 06DD, permanent. Cite it as [Bourbaki, Tag 06DD].

Algebra, chapter IV, § 5. Symmetric tensors and polynomial mappings

— Let M and N be A-modules and q an integer $\geq 0$ . Suppose that M is free, and denote by $\mathrm{Pol}_A^q(M, N)$ or simply $\mathrm{Pol}^q(M, N)$ the set of mappings of M into N satisfying the conditions of Prop. 13. The elements of $\mathrm{Pol}^q(M, N)$ are called homogeneous polynomial mappings of degree q of M into N.

Prop. 13 (i) defines a homomorphism of A-modules :

\mathscr{L}_q(M, ..., M; N) \to \mathrm{Pol}^q(M, N).

Prop. 13 (ii) defines a homomorphism of A-modules

\mathrm{Hom}_A(\mathbf{TS}^q(M), N) \to \mathrm{Pol}^q(M, N).

These homomorphisms are called canonical. They are surjective.

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

Cited by

Corollary, § 18

Cites

Prop. 13