Example 3

Let V be a left B-module. Then the correspondence given above sends the annihilator of the B-module V to the annihilator of the A-module $P\otimes_BV$. Indeed, denote the annihilator of the A-module $P\otimes_BV$ by $\mathfrak{a}$ and that of the B-module V by $\mathfrak{b}$. Let W be the $(A,B)_k$-sub-bimodule of P consisting of the elements such that $p\otimes v= 0$ for every element $v$ of V. We have the inclusion $P\mathfrak{b}\subset W$; conversely, for every $p\in W$ and $q\in Q$, we have that $\theta (q\otimes p)$ belongs to $\mathfrak{b}$. Hence, the element $\mathfrak{b}$ of $D(B_s)$ corresponds to the element W of D(P). Likewise, $\mathfrak{a}\in D(A_d)$ corresponds to W.