Proposition 4

Let A and B be $k$-algebras and P an $(A,B)_k$-bimodule. Suppose that the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective.

a) If the A-module P is generating, then the right B-module P is projective and finitely generated.

b) If the A-module P is projective and finitely generated, then the right B-module P is generating.

Suppose that the A-module P is generating. It is then faithful and balanced (VIII, p. 82, Theorem 2), and consequently the mapping $a\mapsto a_P$ from A to End$_B(P)$ is bijective (VIII, p. 97, Remark 1). Moreover, the mapping $\Lambda : P\otimes_BP^*\rightarrow_sA_d$ is bijective (VIII, p. 98, Proposition 3). We view P as a $(B^o,A^o)_k$-bimodule. The mapping Λ induces a bijective mapping $P^*\otimes_{B^o}P\rightarrow A^o$; by Proposition 2 of VIII, p. 98, (iv) $\Rightarrow$ (i), the right B-module P is projective and finitely generated.

Now, suppose that the A-module P is projective and finitely generated. Then the mapping $\Theta : P^*\otimes_AP\rightarrow_sB_d$ is bijective (loc. cit., (i) $\Rightarrow$ (ii)). By the implication (iii) $\Rightarrow$ (i) of Proposition 3 above applied to the $(B^o,A^o)_k$-bimodule P, the right B-module P is generating.