Proposition 1

Tag 006R, permanent. Cite it as [Bourbaki, Tag 006R].

Algebra, VIII, p. 97, § 6. Morita Equivalence of Modules and Algebras

Suppose that the mappings $b\mapsto b_P$ from B to End $_A(P)$ and $a\mapsto a_P$ from A to End $_B(P)$ are bijective. Then $\sigma$ and $\widetilde{\sigma}$ are inverse isomorphisms, and we have the relations

\Lambda =\widetilde{\Theta}\circ (1_P\otimes \sigma ) \tag{13}

\widetilde{\Lambda} = \Theta \circ (\widetilde{\sigma}\otimes 1_P) \tag{14}

For $x\in P,x^*\in P^*$ , and $y\in P$ , by relations (4), (8), (10), and (11), we have

(15)\langle x, x^*\rangle y=x\Theta (x^*\otimes y) =x\langle \sigma (x^*), y\rangle =\widetilde{\Theta}(x\otimes \sigma (x^*))y=\langle x,\widetilde{\sigma}(\sigma (x^*))\rangle y

Likewise, for $x\in P,\widetilde{y}\in \widetilde{P}$ , and $y\in P$ , we have

x\langle \widetilde{y}, y\rangle =\widetilde{\Theta}(x\otimes \widetilde{y})y=\langle x,\widetilde{\sigma}(\widetilde{y})\rangle y=x\Theta (\widetilde{\sigma}(\widetilde{y})\otimes y) =x\langle \sigma (\widetilde{\sigma}(\widetilde{y})), y\rangle \tag{16}

Because of the assumptions, P is faithful as an A-module and as a B-module. From relations (15) and (16), respectively, we deduce $\widetilde{\sigma}\circ \sigma = 1_{P^*}$ and $\sigma \circ \widetilde{\sigma}=$ $1_{\widetilde{P}}$ . Relations (13) and (14) then follow from (12) and (9), respectively.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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