Twisted Frobenius extensions of graded superrings

TL;DR

This paper introduces the concept of twisted Frobenius extensions for graded superrings, providing multiple equivalent definitions and exploring their role in categorification and module adjointness.

Contribution

It formalizes twisted Frobenius extensions in graded superrings, linking them to adjoint functors in categorification, and offers a broad class of examples involving Frobenius graded superalgebras.

Findings

Twisted Frobenius extensions characterized by bimodule isomorphisms, trace maps, bilinear forms, and generators.

Induction functor is twisted shifted right adjoint to restriction in these extensions.

Many Frobenius graded superalgebras form examples of twisted Frobenius extensions.

Abstract

We define twisted Frobenius extensions of graded superrings. We develop equivalent definitions in terms of bimodule isomorphisms, trace maps, bilinear forms, and dual sets of generators. The motivation for our study comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors. We show that $A$ is a twisted Frobenius extension of $B$ if and only if induction of $B$-modules to $A$-modules is twisted shifted right adjoint to restriction of $A$-modules to $B$-modules. A large (non-exhaustive) class of examples is given by the fact that any time $A$ is a Frobenius graded superalgebra, $B$ is a graded subalgebra that is also a Frobenius graded superalgebra, and $A$ is projective as a left $B$-module, then $A$ is a twisted Frobenius extension of $B$.