Modified induction and a torsion-theoretic equivalence for relative BiHom-Hopf modules

Abstract

We develop an induction theory for relative BiHom-Hopf modules, the special BiHom-Doi--Hopf case associated with the datum $(H,A,H)$ in which the final copy of $H$ carries its regular right $H$-module coalgebra structure. Let $H$ be a monoidal BiHom-Hopf algebra, let $A$ be a right $H$-BiHom-comodule algebra, and let $B=A^{\mathrm{co}H}$. We show that the balanced tensor product defines an induction functor $-{\otimes }_{B}A$ left adjoint to the coinvariant functor. If $H$ has a fixed Haar integral, untwisting yields a Haar identity for the BiHom setting and a canonical projection onto coinvariants. These constructions define a hereditary torsion theory with radical $\kappa$ and a torsion-free reflector $Q(M)=M/\kappa (M)$. The modified induction functor $Q(-{\otimes }_{B}A)$ then gives an equivalence between right $B$-BiHom-modules and torsion-free relative BiHom-Hopf modules generated by their coinvariants. Equal structure maps recover the corresponding Hom result, while identity structure maps recover the classical relative Hopf-module setting.