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\section*{Hopf module}

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a $k$-[[bialgebra|bialgebra]] $(H,m_H,\eta,\Delta,\epsilon)$, a left-right \textbf{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.

An immediate generalization of Hopf modules is for the case where $(E,\eho_E)$ 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 $H$- \textbf{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 $e\in E$.

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 module]]s, see the monograph [BW 2003].

Geometrically, relative Hopf modules are instances of [[equivariant object]]s (equivariant quasicoherent sheaves) in [[noncommutative algebraic geometry]], the statement of which can be made precise, cf. [\v{S}koda 2008].

Furthermore, in the context of relative Hopf modules there is an analogue of the faithfully flat descent along [[torsor]]s 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.

\hypertarget{hopf_bimodules}{}\subsection*{{Hopf bimodules}}\label{hopf_bimodules}

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.\newline The category of Hopf bimodules, ${}_H^H\mathcal{M}^H_H$ is monoidally equivalent to the category of [[Yetter-Drinfeld module]]s.

\hypertarget{fundamental_theorem_on_hopf_modules}{}\subsection*{{Fundamental theorem on Hopf modules}}\label{fundamental_theorem_on_hopf_modules}

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 e.g. Montgomery's book.

\hypertarget{references}{}\subsection*{{References}}\label{references}

Related entries include [[comodule algebra]], [[Schneider's descent theorem]], [[Yetter-Drinfeld module]], [[entwined module]]

\begin{itemize}%
\item BW2003: [[T. Brzeziński]], R. Wisbauer, \textbf{Corings and comodules}, London Math. Soc. Lec. Note Series 309, Cambridge 2003.
\item \v{S}koda 2008: [[Z. Škoda]], \emph{Some equivariant constructions in noncommutative algebraic geometry}, Georgian Mathematical Journal \textbf{16} (2009), No. 1, 183--202, \href{http://arxiv.org/abs/0811.4770}{arXiv:0811.4770} \href{http://www.ams.org/mathscinet-getitem?mr=2527623}{MR2011b:14004}
\item Susan Montgomery, \emph{Hopf algebras and their actions on rings}, CBMS Lecture Notes \textbf{82}, AMS 1993, 240p.
\item Peter Schauenburg, \emph{Hopf modules and Yetter - Drinfel$\prime$d modules}, J. Algebra \textbf{169}:3 (1994) 874-890 \href{http://dx.doi.org/10.1006/jabr.1994.1314}{doi}; \emph{Hopf modules and the double of a quasi-Hopf algebra}, Trans. Amer. Math. Soc. 354 (2002), 3349-3378 \href{http://dx.doi.org/10.1090/S0002-9947-02-02980-X}{doi} \href{http://www.ams.org/journals/tran/2002-354-08/S0002-9947-02-02980-X/S0002-9947-02-02980-X.pdf}{pdf}; \emph{Actions of monoidal categories, and generalized Hopf smash products}, Journal of Algebra \textbf{270} (2003) 521-563,  \href{http://www.mathematik.uni-muenchen.de/%7Eschauen/papers/amcghsp.ps}{ps}
\item A. Borowiec, G. A. Vazquez Coutino, \emph{Hopf modules and their duals}, \href{http://arxiv.org/abs/math/0007151}{math.QA/0007151}
\item H-J. Schneider, \emph{Principal homogeneous spaces for arbitrary Hopf algebras}, Israel J. Math. \textbf{72} (1990), no. 1-2, 167--195 \href{http://www.ams.org/mathscinet-getitem?mr=1098988}{MR92a:16047} \href{http://dx.doi.org/10.1007/BF02764619}{doi}
\item Francesco d'Andrea, Alessandro de Paris, \emph{On noncommutative equivariant bundles}, \href{http://arxiv.org/abs/1606.09130}{arxiv/1606.09130}

\end{itemize}
category: algebra [[!redirects Hopf modules]] [[!redirects Hopf bimodule]] [[!redirects Hopf bimodules]]



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