symmetric monoidal (∞,1)-category of spectra
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 commutative ring $k$ and a $k$-bialgebra $(H,m_H,\eta,\Delta,\epsilon)$, 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. In this, the structure of left module on $M\otimes H$ is the standard tensor product of modules over Hopf algebras, with the action given 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. morphism of relative Hopf modules is a morphism of modules which is also a morphism of comodules.
An immediate generalization of Hopf modules is for the case where $(E,\rho_E)$ is a right $H$-comodule algebra (a monoid in the category of $H$-comodules); then one can define the category ${}_E\mathcal{M}^H$ 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. In Sweedler notation for comodules. where $\rho(m) = \sum m_{(0)}\otimes m_{(1)}$, $\rho_E(e) = \sum e_{(0)}\otimes e_{(1)}$, the compatibility condition for the left-right relative Hopf modules is $\rho (e m) = \sum e_{(0)} m_{(0)} \otimes e_{(1)} m_{(1)}$ for all $m\in M$ and $e\in E$. A morphism of relative Hopf modules is a morphism of modules which is also a morphism of comodules. If the category of $k$-modules is denoted by $\mathcal{M}$ then the left $E$- right $H$-relative Hopf modules and their morphisms form a category in Hopf algebraic literature denoted by ${}_E\mathcal{M}^H$.
Given a left $H$-comodule algebras $E$ and $F$, one considers also the category ${}_E\mathcal{M}_F^H$ of two-sided relative Hopf modules (sometimes also called the category of relative Hopf bimodules. It is equivalent to the category ${}_{E\otimes F^{op}}\mathcal{M}^H$. See Caen. et al 2007.
One can also consider Hopf bimodules, and similar categories. A Hopf $H$-bimodule is left and right $H$-comodule and left and right $H$-bimodule, where all four structure are compatible in standard way.
The category of Hopf bimodules, ${}_H^H\mathcal{M}^H_H$ is monoidally equivalent to the category of Yetter-Drinfeld modules.
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 Brzezinski, Wisbauer 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. For another approach/candidate for noncommutative equivariant objects, see d’Andrea, Paris 2019 and its MR review by Gabriella Böhm for an excellent comparison.
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.
If $H$ is a Hopf algebra over a field $k$, then the category of the ordinary Hopf modules ${}_H^H\mathcal{M}$ is equivalent to the category of $k$-vector spaces. See Section 1 of Montgomery 1993 for more.
The equivalence may be seen as follows. Any vector space $V$ can be endowed with a (left-) Hopf module structure, for $H$ a Hopf algebra, simply by tensoring with $H$. The action of $H$ is given as
and the coaction as
for $\Delta:H\to H\otimes H$ the comultiplication. This is known as a trivial Hopf module.
The fundamental theorem of Hopf modules states that any Hopf module $M$ arises precisely in this way, as one shows that
where $M^{\text{co} H}:= \{m\in M \vert \sigma(m)= 1_H \otimes m\}$ is the space of coinvariant of $M$ under the coaction $\sigma$ of $H$. In fact, the operations
come as functors realizing an equivalence of categories between vector spaces, and $H$-Hopf modules (see Vercruysse 2012 for more on this).
Related entries include comodule algebra, Schneider's descent theorem, Yetter-Drinfeld module, entwined module
Last revised on May 16, 2024 at 15:39:51. See the history of this page for a list of all contributions to it.