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Let ${\mathbb F}:({\mathcal A},{\mathcal E}_{\mathcal A})\to({\mathcal B},{\mathcal E}_{\mathcal B})$ be an exact functor between exact categories, and suppose ${\mathbb F}$ has both a left adjoint ${\mathbb F}_\lambda$ and a right adjoint ${\mathbb F}_\rho$. Then the class

${\mathcal E} := \{X\to Y\to Z\in{\mathcal E}_{\mathcal A}\ |\ {\mathbb F}X\to{\mathbb F}Y\to{\mathbb F}Z\in{\mathcal E}_{\mathcal B}\}$

defines another exact structure on ${\mathcal A}$.

It is interesting to ask for criteria to decide when this exact structure is Frobenius. One such criterion is the following:

Suppose ${\mathcal E}_{\mathcal B}$ is the split exact structure, and that for each $X\in{\mathcal A}$ the unit $\eta_X: X\to{\mathbb F}_\rho{\mathbb F}X$ is an ${\mathcal E}_{\mathcal A}$-monomorphism, while the counit ${\mathbb F}_\lambda{\mathbb F}X\to X$ is an ${\mathcal E}_{\mathcal A}$-epimorphism. Then $({\mathcal A},{\mathcal E})$ has enough projectives and injectives, and the classes ${\mathcal P}$/${\mathcal I}$ of projective/injective objects in $({\mathcal A},{\mathcal E})$ are given by

${\mathcal P} = \{X\in{\mathcal A}\ |\ X\text{ is a summand of some }{\mathbb F}_\lambda , Y\in{\mathcal B}\}$ and

${\mathcal I} = \{X\in{\mathcal A}\ |\ X\text{ is a summand of some }{\mathbb F}_\rho Y, Y\in{\mathcal B}\}$,

respectively. Consequently, if ${\mathbb F}_\lambda$ and ${\mathbb F}_\rho$ have the same image, then $({\mathcal A},{\mathcal E})$ is Frobenius.

This seems very restrictive, but in fact there are at least two cases I know where it can be applied:

(1) If ${\mathcal A}$ is a dg-category, then the forgetful functor ${\mathbb F}: \text{dg-mod}({\mathcal A})\to\text{gr-mod}({\mathcal A})$ fulfills the requirements of the criterion above and thus can be used to construct a Frobenius structure on $\text{dg-mod}({\mathcal A})$ (for pretriangulated dg-categories ${\mathcal A}$, this structure can in turn be restricted to ${\mathcal A}$ itself).

(2) If $G$ is a finite group, $H$ is a subgroup, then the fortgetful functor $G\text{-mod}\to H\text{-mod}$ has left adjoint $\text{Ind}^G_H$ and right adjoint $\text{Coind}^G_H$, and these two functors coincide for $(G:H)<\infty$. In this case, the above criterion therefore applies to provide $G\text{-mod}$ with a Frobenius structure "relative to $H$".

Question

Do you know more criteria for constructing Frobenius structures and situations where they can be applied?

For example, I would be interested in a criterion which can be applied to show that the category of maximal Cohen-Macaulay modules over a Gorenstein ring is Frobenius.

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