Morita duality for Grothendieck categories and its application (Q1895587)

Let \((_R \Gamma_S,{_S U_R},{_R V_T},{_T \Delta_R})\) be such that 1. \((_R \Gamma_S,{_S U_R})\) is a Morita context with the trace ideals \(I\) in \(R\) and \(J\) in \(S\) and \(UI = U\), \(u \in Ju\) for every \(u\) in \(U\); 2. \((_R V_T,{_T \Delta_R})\) is a Morita context with the trace ideals \(K\) in \(R\) and \(L\) in \(T\), \(KV = V\) and \(v \in vL\) for every \(v\) in \(V\); 3. \(U_R\), \(_RV\) are \(\text{QF-}3'\) modules with \(E(U)\)-dom.dim \(U_R \geq 2\), \(E(V)\)-dom.dim \(_R V \geq 2\); 4. \(\text{ann}_U K = 0 = \text{ann}_V I\). Then \(_S U \otimes_R V_T\) is a \(\text{QF-}3''\) module both as a left \(S\)- and as a right \(T\)-module. Let \(({\mathcal T}, {\mathcal F})\) and \(({\mathcal T}', {\mathcal F}')\) be hereditary torsion theories generated by \(U \otimes V\) in \(\text{Gen}(_S U)\) and \(\text{Gen} (V_T)\) with the corresponding torsion radicals \(\tau_1\) and \(\tau_2\), respectively. For each \(X\) in \(\text{Gen} (_S U)\), \(Y\) in \(\text{Gen} (V_T)\) put \(D_1(X) = \tau_2 (X^*)\), \(D_2(Y) = \tau_2(Y^*)\), where \(X^* = \text{Hom}_S (X, U\otimes V)\), \(Y^* = \text{Hom}_T (Y, U \otimes V)\). Then \(D_1\) and \(D_2\) are adjoint functors with the adjunctions \(\eta : 1_{\text{Gen} (_S U)} \to D_2D_1\) \(\eta' : 1_{\text{Gen}(V_T)} \to D_1 D_2\). Let \(\mathcal L\) and \(\mathcal L'\) be full subcategories of those \(X\) and \(Y\) such that \(\eta_X\) and \(\eta_Y'\) are isomorphisms, respectively. Then \(\mathcal L\) and \(\mathcal L'\) are Grothendieck categories, \(D_1\) and \(D_2\) induce a strong Morita duality between them. Some applications and examples are also given in the final section of the paper.