Separable functors in group coring. (Q2348675)

Let \(k\) be a field, \(\pi\) a group with identity \(e\) and \(A\) a \(k\)-algebra. A \(\pi\)-\(A\)-coring \(\mathcal C\) is a family \(\{\mathcal C_\alpha\}_{\alpha\in\pi}\) of \(A\)-bimodules together with a family of \(A\)-bimodule maps \(\Delta_{\alpha,\beta}\colon\mathcal C_{\alpha\beta}\to\mathcal C_\alpha\otimes_A\mathcal C_\beta\), \(\varepsilon\colon\mathcal C_e\to A\) such that \((\Delta_{\alpha,\beta}\otimes_A\text{id}_{\mathcal C_\gamma})\circ\Delta_{\alpha\beta,\gamma}=(\text{id}_{\mathcal C_\alpha}\otimes\Delta_{\beta,\gamma})\circ\Delta_{\alpha,\beta\gamma}\), and \((\text{id}_{\mathcal C_\alpha}\otimes_A\varepsilon)\circ\Delta_{\alpha,e}=\text{id}_{\mathcal C_\alpha}=(\varepsilon\otimes_A\text{id}_{\mathcal C_\alpha})\circ\Delta_{e,\alpha}\), for all \(\alpha,\beta,\gamma\in\pi\), where \(\Delta_{\alpha,\beta}(c)=c_{(1,\alpha)}\otimes_Ac_{(2,\beta)}\). A right \((\pi\)-\(\mathcal C)\)-comodule \(M\) is a family of right \(A\)-modules \(\{M_\alpha\}_{\alpha\in\pi}\) together with a family of right \(A\)-linear maps \(\rho^M=\{\rho_{\alpha,\beta}^M\}_{\alpha,\beta\in\pi}\) where \(\rho_{\alpha,\beta}^M\colon M_{\alpha\beta}\to M_\alpha\otimes_A\mathcal C_\beta\) such that \((\text{id}_{M_\alpha}\otimes_A\Delta_{\beta,\gamma})\circ\rho_{\alpha,\beta\gamma}^M=(\rho_{\alpha,\beta}^M\otimes_A\text{id}_{\mathcal C_\gamma})\circ\rho_{\alpha\beta,\gamma}^M\) and \((\text{id}_{M_\alpha}\otimes_A\varepsilon)\circ\rho_{\alpha,e}^M=\text{id}_{M_\alpha}\) for all \(\alpha,\beta,\gamma\in\pi\). A morphism between right \((\pi\)-\(\mathcal C)\)-comodules \(\{M_\alpha\}\) and \(\{N_\alpha\}\) is a family of right \(A\)-linear maps \(f=\{f_\alpha\}_{\alpha\in\pi}\), \(f_\alpha\colon M_\alpha\to N_\alpha\) such that \((f_\alpha\otimes_A\text{id}_{\mathcal C_\beta})\circ\rho_{\alpha,\beta}^M=\rho_{\alpha,\beta}^N\circ f_{\alpha\beta}\). Denote the category of right \((\pi\)-\(\mathcal C)\)-comodules by \(\mathcal M^{\pi,\mathcal C}\). Let \(\mathcal C\) and \(\mathcal D\) be two categories, \(F\colon\mathcal C\to\mathcal D\) a covariant functor, and the natural transformation induced by \(F\), \(\mathcal F\colon\Hom_{\mathcal C}(\cdot,\cdot)\to\Hom_{\mathcal D}(F(\cdot),F(\cdot))\). Then \(F\) is called a separable functor if \(\mathcal F\) splits. Let \((F,G)\) be adjoint functors between \(\mathcal M^{\pi,\mathcal C}\) and \(\mathcal M_A\) where \(F\colon\mathcal M^{\pi,\mathcal C}\to\mathcal M_A\) by \(F(M)=M_e\), \(F(f)=f_e\) and \(G(N)=N\otimes_A\mathcal C_\alpha=\{N\otimes_A\mathcal C_\alpha\}_{\alpha\in\pi}\), for \(M\in\mathcal M^{\pi,\mathcal C}\), \(N\in\mathcal M_A\). It is shown that for a \(\pi\)-\(A\)-coring \(\mathcal C\), the forgetful functor \(F\colon\mathcal M^{\pi,\mathcal C}\to\mathcal M_A\) is separable if and only if there exists a family of \(A\)-bimodules \(\theta=\{\theta^{(\alpha)}\colon\mathcal C_{\alpha^{-1}}\otimes_A\mathcal C_\alpha\to A\}_{\alpha\in\pi}\) such that \(\theta^{(\alpha)}(c'_{(1,\alpha^{-1})}\otimes_Ac'_{(2,\alpha)})=\varepsilon (c')\), and \(c_{(1,\beta)}\theta^{(\alpha\beta)}(c_{(2,\beta^{-1}\alpha^{-1})}\otimes_Ad)=\theta^{(\alpha)}(c\otimes_Ad_{(1,\alpha)})d_{(2,\beta)}\) for all \(c'\in\mathcal C_e\), \(c\in\mathcal C_{\alpha^{-1}}\), \(d\in\mathcal C_{\alpha\beta}\). The result is applied to group entwined modules, Doi-Hopf group modules and relative Hopf modules.