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Twisted Frobenius extensions of graded superrings

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 $B$.