Bourbaki

Remark

Tag 067U, permanent. Cite it as [Bourbaki, Tag 067U].

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

Let C be a counital coassociative A-coalgebra, B a unital associative A-algebra and M a left B-module. The A-bilinear mapping (b,m)↦bm(b, m) \mapsto bm of B×MB \times M into M defines an A-bilinear mapping

Hom⁡A(C,B)×Hom⁡A(C,M)→Hom⁡A(C,M) \operatorname{Hom}_A(C, B) \times \operatorname{Hom}_A(C, M) \to \operatorname{Hom}_A(C, M)
by the general procedure described at the beginning of this no. It is immediately verified that this mapping defines on Hom⁡A(C,M)\operatorname{Hom}_A(C, M) a left module structure over the ring Hom⁡A(C,B)\operatorname{Hom}_A(C, B).

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

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Cites

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Transcribed from the printed edition, 1998, Springer. Not translated.