nLab
Frobenius functor

Contents

Definition

Definition

An adjoint pair of functors

(FG):DC (F\dashv G) : D \to C

is a Frobenius pair if FF is not only a left adjoint to GG but also a right adjoint to GG, hence if we have an adjoint triple

(FGF):DC. (F \dashv G \dashv F) : D \to C \,.

In this case one often says that FF (or GG) is a Frobenius functor.

Note

There is no relation to the notion of Frobenius monoidal functor.

Properties

  • K. Morita proved that the extension of scalars functor for a morphism of rings f:RSf:R\to S is Frobenius iff the morphism ff itself is Frobenius in the sense of (Kasch), that is: RS{}_R S is finitely generated projective and SS RHom R( RS, RR){}_S S_R\cong Hom_R({}_R S, {}_R R) as RSR-S-bimodules.

    This is in the spirit of the finite-dimensional duality coded e.g. in the notion of Frobenius algebra.

References

  • Stefaan Caenepeel, Gigel Militaru, Shenglin Zhu, Frobenius and separable functors for generalized module categories and nonlinear equations, Springer Lec. Notes in Math. 1787 (2002) xiv+354 pp, gBooks

  • F. Kasch, Projektive Frobenius-Erweiterungen, Sitzungsber, Heidelberger Akad. Wiss., Math.- Naturw. Kl. 1960/61 (1961), 89–109.

Revised on April 13, 2011 11:02:50 by Urs Schreiber (188.201.208.83)