separable functor


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 kGkG-module can be obtained functorially from its splitting as an exact sequence of kk-modules. Functors similar to the forgetful functor kGMod kMod{}_{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:RSf: R\to S: such a ring extension is separable iff restriction of scalars Res R S: SMod RMod\operatorname{Res}_R^S: {}_{S}Mod \to {}_{R}Mod is a separable functor.


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

x,y=C(x,y)D(Fx,Fy),(xfy)(FxFfFy). \mathcal{F}_{x,y} = C(x,y)\to D(Fx,Fy),\,\,\,\,(x\stackrel{f}\to y)\mapsto (Fx\stackrel{Ff}\to Fy).

We way that FF is a separable functor if \mathcal{F} splits (i.e. x,y\mathcal{F}_{x,y} has a section natural in xx and yy).


Rafael’s theorem. Let FGF\dashv G be a pair of adjoint functors. Then FF is separable iff the unit η:1GF\eta:1\to GF has a section (= a natural transformation ν\nu which is its right inverse, νη=1\nu\circ\eta = 1). GG is separable iff the counit ϵ:FG1\epsilon:FG\to 1 has a retraction (= a natural transformation ζ\zeta which is its left inverse, ηζ=1\eta\circ\zeta =1).

Generalization for SS-categories

See T. Brzeziński 2008.


Separable functors were defined in

  • C. Năstăsescu, M. van den Bergh, F. van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989), 397-413, [doi].

Now there is a monograph available:

  • S. Caenepeel, G. Militaru, S. Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Lec. Notes in Math. 1787, Springer 2002. xiv+354 pp.

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, KK-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:

  • T. Brzeziński, Notes on formal smoothness, in: Modules and Comodules (series Trends in Mathematics). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 (doi, arXiv:0710.5527)

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