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\section*{separable functor}

\hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation}

The original motivation was the functorial [[Maschke's theorem]] over rings and its various cousins: namely the classical Maschke's theorem about finite group rings over fields, generalizes to the statements that when the order of the group is invertible in the ground ring then the splitting of an exact sequence of $kG$-module can be obtained functorially from its splitting as an exact sequence of $k$-modules. Functors similar to the forgetful functor ${}_{kG} Mod\to {}_{k}Mod$ in the sense of having such a functorial Maschke's property are abstracted into the notion of a separable functor.

Similar phenomena appear in the study of ring extensions $f: R\to S$: such a ring extension is separable iff restriction of scalars $\operatorname{Res}_R^S: {}_{S}Mod \to {}_{R}Mod$ is a separable functor.

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

Let $F:C\to D$ be a functor. There is a corresponding (bi)natural transformation with components

\begin{displaymath}
\mathcal{F}_{x,y} = C(x,y)\to D(Fx,Fy),\,\,\,\,(x\stackrel{f}\to y)\mapsto (Fx\stackrel{Ff}\to Fy).
\end{displaymath}
We way that $F$ is a separable functor if $\mathcal{F}$ splits (i.e. $\mathcal{F}_{x,y}$ has a section natural in $x$ and $y$).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\textbf{Rafael's theorem.} Let $F\dashv G$ be a pair of adjoint functors. Then $F$ is separable iff the unit $\eta:1\to GF$ has a section (= a natural transformation $\nu$ which is its right inverse, $\nu\circ\eta = 1$). $G$ is separable iff the counit $\epsilon:FG\to 1$ has a retraction (= a natural transformation $\zeta$ which is its left inverse, $\eta\circ\zeta =1$).

\hypertarget{generalization_for_categories}{}\subsection*{{Generalization for $S$-categories}}\label{generalization_for_categories}

See \hyperlink{Brzezinski}{T. Brzeziski 2008}.

\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Separable functors were defined in

\begin{itemize}%
\item C. Nstsescu, M. van den Bergh, [[F. van Oystaeyen]], \emph{Separable functors applied to graded rings}, J. Algebra \textbf{123} (1989), 397-413, .

\end{itemize}
Now there is a monograph available:

\begin{itemize}%
\item S. Caenepeel, G. Militaru, S. Zhu, \emph{Frobenius and separable functors for generalized module categories and nonlinear equations}, Lec. Notes in Math. \textbf{1787}, Springer 2002. xiv+354 pp.

\end{itemize}
Other references

\begin{itemize}%
\item M. D. Rafael, \emph{Separable functors revisited}, Comm. in Algebra \textbf{18} (1990), 1445-1459.


\item S. Caenepeel, B. Ion, G. Militaru, \emph{The structure of Frobenius algebras and separable algebras}, $K$-Theory \textbf{19} (2000), no. 4, 365--402.


\item J. G\'o{}mez-Torrecillas, \emph{Separable functors in [[corings]]}, Int. J. of Math. and Math. Sci. \textbf{30} (2002), 4, Pages 203-225, \href{http://dx.doi.org/10.1155/S016117120201270X}{doi}, \href{http://www.emis.de/journals/HOA/IJMMS/Volume30_4/225.pdf}{pdf}


\item A. Ardizzoni, \emph{Separable functors and [[formal smoothness]]}, J. K-Theory 1 (2008), no. 3, 535--582, \href{http://arxiv.org/abs/math.QA/0407095}{math.QA/0407095}, \href{http://dx.doi.org/10.1017/is007011015jkt012}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=2009k:16069}{MR2009k:16069}


\item A. Ardizzoni, [[G. Böhm]], C. Menini, \emph{A Schneider type theorem for Hopf algebroids}, J. Algebra \textbf{318} (2007), no. 1, 225--269 \href{http://www.ams.org/mathscinet-getitem?mr=2363132}{MR2008j:16103} \href{http://dx.doi.org/10.1016/j.jalgebra.2007.05.017}{doi} \href{http://arxiv.org/abs/math/0612633}{math.QA/0612633} (arXiv version is unified, corrected); \emph{Corrigendum}, J. Algebra \textbf{321}:6 (2009) 1786-1796 \href{http://www.ams.org/mathscinet-getitem?mr=2498268}{MR2010b:16060} \href{http://dx.doi.org/10.1016/j.jalgebra.2008.11.040}{doi}



\end{itemize}
The following article studies formal smoothness and generalizes the separable functors to the context of the so-called [[S-category|S-categories]] which are introduced therein:

\begin{itemize}%
\item [[T. Brzeziński]], \emph{Notes on formal smoothness}, \emph{in}: Modules and Comodules (series \emph{Trends in Mathematics}). T Brzeziski, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkh\"a{}user, Basel, 2008, pp. 113-124 (\href{http://dx.doi.org/10.1007/978-3-7643-8742-6}{doi}, \href{http://arxiv.org/abs/0710.5527}{arXiv:0710.5527})

\end{itemize}
[[!redirects separable functors]]



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