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.
Let $F:C\to D$ be a functor. There is a corresponding (bi)natural transformation with components
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 $y$).
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, $\eta\circ\zeta =1$).
See T. Brzeziński 2008.
Separable functors were defined in
Now there is a monograph available:
Other references
M. D. Rafael, Separable functors revisited, Comm. in Algebra 18 (1990), 1445-1459.
S. Caenepeel, B. Ion, G. Militaru, The structure of Frobenius algebras and separable algebras, $K$-Theory 19 (2000), no. 4, 365–402.
J. Gómez-Torrecillas, Separable functors in corings, Int. J. of Math. and Math. Sci. 30 (2002), 4, Pages 203-225, doi, pdf
A. Ardizzoni, Separable functors and formal smoothness, J. K-Theory 1 (2008), no. 3, 535–582, math.QA/0407095, doi, MR2009k:16069
A. Ardizzoni, G. Böhm, C. Menini, A Schneider type theorem for Hopf algebroids, J. Algebra 318 (2007), no. 1, 225–269 MR2008j:16103 doi math.QA/0612633 (arXiv version is unified, corrected); Corrigendum, J. Algebra 321:6 (2009) 1786-1796 MR2010b:16060 doi
The following article studies formal smoothness and generalizes the separable functors to the context of the so-called S-categories which are introduced therein:
Last revised on July 27, 2017 at 19:09:52. See the history of this page for a list of all contributions to it.