Proposition 21

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

— The canonical homomorphism of A-modules (IV, p. 55)

u : \mathbf{TS}(M)^{*gr} \to \mathrm{Pol}_A(M, A)

is an algebra homomorphism.

Let $a \in \mathbf{TS}^q(M)^*, b \in TS^r(M)^*, x \in M$ ; we have

u(ab)(x) = \langle ab, \gamma_{q+r}(x) \rangle = \langle a \otimes b, c(\gamma_{q+r}(x)) \rangle = = \langle a \otimes b, \gamma_q(x) \otimes \gamma_r(x) \rangle = \langle a, \gamma_q(x) \rangle \langle b, \gamma_r(x) \rangle = u(a)(x) \cdot u(b)(x),

whence the result.

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

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