nLab
Hopf module

Given a $k$-bialgebra $(H,m_H,\eta ,\Delta ,ϵ)$, a left-right Hopf module of $H$ is a $k$-module $M$ with the structure of left $H$-module and right $H$-comodule, where the action $\nu :H\otimes M\to M$ and right $H$-coaction $\rho :M\to M\otimes H$ are compatible in the sense that the coaction is a morphism of left modules (the structure of left module on $M\otimes H$ is the standard tensor product of modules over Hopf algebras, with action given in this case by $(\nu \otimes m_H)\circ (H\otimes \tau \otimes H)\circ (\Delta \otimes M\otimes H)$ as $k$-linear map $H\otimes (M\otimes H)\to M\otimes H$ where $\tau ={\tau }_{H,M}:H\otimes M\to M\otimes H$ is the standard flip of tensor factors in the symmetric monoidal category of $k$-modules).

An immediate generalization of Hopf modules is for the case where $E$ is a right $H$-comodule algebra (a monoid in the category of $H$-comodules); then one can define the category of left $E$- right $H$- relative Hopf modules (less precisely, $(E,H)$-relative Hopf modules, or simply (relative) Hopf modules), which are left $E$-modules that are right $H$-comodules with a natural compatibility condition. There are further generalizations where instead of a bialgebra $H$ and a $H$-comodule algebra $E$ one replaces $E$ by an arbitrary algebra $A$, and $H$ by a coalgebra $C$ 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].

The entwined modules are first introduced under the name “bialgebras” by van Osdol ([van Osdol 1973]) in the more general case of monads and comonads instead of $k$-algebras and $k$-coalgebras.

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 $H$ is a Hopf algebra and $U\hookrightarrow E$ a faithfully flat $H$-Hopf-Galois extension then the natural adjunction between the categories of relative $(E,H)$-Hopf modules and left $U$-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.

References

• BW2003: T. Brzezinski, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
• Škoda 2008: Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry arXiv:0811.4770
• van Osdol 1973: D. H. van Osdol, Bicohomology theory, Trans. Amer. Math. Soc. 183 (1973), 449–476.