The Hopf modules over bimonoids are modules in the category of comodules or viceversa. This notion has many generalizations and variants. Relative Hopf modules are an algebraic and possibly noncommutative analogue of a notion of an equivariant sheaf.
Given a -bialgebra , a left-right Hopf module of is a -module with the structure of left -module and right -comodule, where the action and right -coaction are compatible in the sense that the coaction is a morphism of left modules. In this, the structure of left module on is the standard tensor product of modules over Hopf algebras, with the action given by as -linear map where is the standard flip of tensor factors in the symmetric monoidal category of -modules.
An immediate generalization of Hopf modules is for the case where is a right -comodule algebra (a monoid in the category of -comodules); then one can define the category of left - right - relative Hopf modules (less precisely, -relative Hopf modules, or simply (relative) Hopf modules), which are left -modules that are right -comodules with a natural compatibility condition. In Sweedler notation for comodules. where , , the compatibility condition for the left-right relative Hopf modules is for all and .
There are further generalizations where instead of a bialgebra and a -comodule algebra one replaces by an arbitrary algebra , and by a coalgebra and introduces a compatibility in the sense of a mixed distributive law or entwining (structure). Then the relative Hopf modules become a special case of so-called entwined modules, see the monograph [BW 2003].
Geometrically, relative Hopf modules are instances of equivariant objects (equivariant quasicoherent sheaves) in noncommutative algebraic geometry, the statement of which can be made precise, cf. [Škoda 2008].
Furthermore, in the context of relative Hopf modules there is an analogue of the faithfully flat descent along torsors from commutative algebraic geometry, and the Galois descent theorems in algebra. Its main instance is Schneider's theorem, asserting that if is a Hopf algebra and a faithfully flat -Hopf-Galois extension then the natural adjunction between the categories of relative -Hopf modules and left -modules is an equivalence of categories. This corresponds to the classical theorem saying that the category of equivariant quasicoherent sheaves over the total space of a torsor is equivalent to the category of the quasicoherent sheaves over the base of the torsor.
One can also consider Hopf bimodules, and similar categories. A Hopf -bimodule is left and right -comodule and left and right -bimodule, where all four structure are compatible in standard way.
The category of Hopf bimodules, is monoidally equivalent to the category of Yetter-Drinfeld modules.
If is a Hopf algebra over a field , then the category of the ordinary Hopf modules is equivalent to the category of -vector spaces. See e.g. Montgomery’s book.