nLab
Frobenius functor

Definition

Definition

An adjoint pair of functors

$$(F⊣G):D\to C$$(F\dashv G) : D \to C

is a Frobenius pair if $F$ is not only a left adjoint to $G$ but also a right adjoint to $G$, hence if we have an adjoint triple

$$(F⊣G⊣F):D\to C.$$(F \dashv G \dashv F) : D \to C \,.

In this case one often says that $F$ (or $G$) 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 $f:R\to S$ is Frobenius iff the morphism $f$ itself is Frobenius in the sense of (Kasch), that is: ${}_{R}S$ is finitely generated projective and ${}_S{S}_R\cong {\mathrm{Hom}}_R({}_{R}S,{}_{R}R)$ as $R-S$-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.