Proposition 2

Let A and B be algebras over $k$, and let P be an $(A,B)_k$-bimodule. Suppose that the mapping $b\mapsto b_P$ from B to End$_A(P)$ is bijective. The following assertions are equivalent:

(i) The A-module P is projective and finitely generated.

(ii) The mapping Θ (VIII, p. 96) is an isomorphism of $(B,B)_k$-bimodules from $P^*\otimes_AP$ to $_sB_d$.

(iii) The image of Θ contains the unit element of B.

If, moreover, the mapping $a\mapsto a_P$ from A to End$_B(P)$ is bijective, then the assertions above are equivalent to the following statement:

(iv) There exist a $(B,A)_k$-bimodule Q and a surjective homomorphism of $(B,B)_k$-bimodules from $Q\otimes_AP$ to $_sB_d$.

By the corollary of II, §4, No. 2, p. 271, assertion (i) implies (ii). Moreover, (iii) follows from (ii). Let $t\in B^*\otimes_AB$ be such that $\Theta (t) = 1$. Let $n$ be an integer, and let $(x_1, . . . , x_n)\in P^n$ and $(x^*_1, . . . , x^*_n)\in P^{*n}$ be such that $t=\sum^n_{i=1}x^*_i\otimes x_i$. For every $x$ belonging to P, the relation $x\Theta (t) =x$ can be written as

It follows that the A-module P is finitely generated by the family $(x_i)_{1\leqslant i\leqslant n}$, and we conclude the proof of the implication (iii) $\Rightarrow$ (i) using Proposition 12 of II, §2, No. 6, p. 238.

We obviously have (ii) $\Rightarrow$ (iv). Let Q be a $(B,A)_k$-bimodule and $\theta$ a surjective homomorphism of $(B,B)_k$-bimodules from $Q\otimes_AP$ to $_sB_d$. In the notation of the previous subsection, there exists a homomorphism of $(B,A)_k$-bimodules $\zeta : Q\rightarrow \widetilde{P}$ such that $\theta (y\otimes x) =\langle \zeta (y), x\rangle$ for $x\in P$ and $y\in Q$, and we have $\theta =\widetilde{\Lambda}\circ (\zeta \otimes 1_P)$. Since $\theta$ is surjective, the same holds for $\widetilde{\Lambda}$. If the mapping $a\mapsto a_P$ from A to End$_B(P)$ is bijective, then Θ is surjective by relation (14) of Proposition 1 of VIII, p. 97; the implication (iv) $\Rightarrow$ (iii) follows.