nLab separable functor

Motivation

The original motivation for the notion of separable functors [Năstăsescu, van den Bergh & van Oystaeyen (1989)] was the functorial form of 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 $k G$-modules can be obtained functorially from its splitting as an exact sequence of underlying $k$-modules.

This “Maschke property” of the forgetful functor ${}_{k G} Mod\to {}_{k}Mod$ is what is generalized by the notion of separable functors.

A further example appears in the study of ring extensions $f \colon R\to S$: Such is separable iff the corresponding restriction of scalars-functor $\operatorname{Res}_R^S \colon {}_{S}Mod \to {}_{R}Mod$ is a separable functor.

Definition

Let $F \colon C\to D$ be a functor with hom-set components

$$\array{ F_{x,y} \colon & C(x,y)&\longrightarrow& D(F x,F y) \\ & (x\stackrel{f}\to y)&\mapsto& (F x\stackrel{F f}\to F y) \mathrlap{\,.} }$$

Then $F$ is called separable [Năstăsescu, van den Bergh & van Oystaeyen (1989), p. 398] if each $F_{x,y}$ has a section which is “natural” in $x$ and $y$, in a suitable sense.

Properties

Theorem

(Rafael’s theorem) Let $F\dashv G$ be a pair of adjoint functors. Then

• $F$ is separable iff the adjunction unit $\eta \con id \to G F$ has a section (= a natural transformation $\nu$ which is its right inverse, $\eta\circ\nu = 1$),

• $G$ is separable iff the adjunction counit $\epsilon \colon F G \to id$ has a retraction (i.e. a natural transformation $\zeta$ that is its left inverse, $\zeta\circ\epsilon =1$).

The latter condition is reminiscent of one of the many equivalent definitions of a separable algebra $A$ over a field, namely one for which multiplication, viewed as an $(A,A)$-bimodule map $A \otimes A^{\mathrm{op}} \to A$, has a right inverse.

Literature

Separable functors were defined in

• Constantin Năstăsescu, Michel van den Bergh, F. van Oystaeyen, Separable functors applied to graded rings, J. Algebra 123 (1989) 397-413 [doi:10.1016/0021-8693(89)90053-7]

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, $K$-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:

• 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)