Twisted separability for adjoint functors

TL;DR

This paper introduces twisted separable functors, extending classical separability, and explores their properties and applications, including a version of Rafael's Theorem and implications for Hochschild cohomological dimensions in Hopf-Galois contexts.

Contribution

It generalizes separable functors to twisted separable functors, establishes criteria for when an adjoint functor is twisted separable, and applies these results to Hochschild cohomology in Hopf-Galois theory.

Findings

Twisted separable functors generalize classical separable functors.

A version of Rafael's Theorem is established for twisted separability.

Hochschild cohomological dimensions coincide under certain conditions for Hopf-Galois objects.

Abstract

Twisted separable functors generalize the separable functors of Nastasescu, Van den Bergh and Van Oystaeyen, and provide a convenient tool to compare various projective dimensions. We discuss when an adjoint functor is twisted separable, obtaining a version of Rafael's Theorem in the twisted case. As an application, we show that if $R$ is Hopf-Galois object over a Hopf algebra $A$, then their Hochschild cohomological dimension coincide, provided that the cohomological dimension of $A$ is finite and that $R$ has a unital twisted trace with respect to a semi-colinear automorphism.