Hopfological invariants for tame subextensions

TL;DR

This paper investigates conditions under which H-extensions are tame by analyzing Hopfological and Hopf-cyclic homology, providing criteria based on trace maps and cyclic modules for Hopf-Galois extensions.

Contribution

It introduces a method to determine tameness of H-extensions using Hopfological homology and cyclic modules, expanding understanding of Hopf-Galois extensions.

Findings

Tame H-extensions characterized by surjectivity of trace maps.

Hopf-cyclic homology behavior linked to cyclic modules in Galois extensions.

Criteria for tameness in terms of Hopfological homology established.

Abstract

Let H be a finite dimensional Hopf algebra over a field K. In this paper, we study when an H-extension becomes a tame H-extension by calculating Hopfological homology and Hopf-cyclic homology. In the (derived) category of H'-comodules for a Hopf algebra H', we take Hopf subalgebra H of H' and a certain order A of H. We see the behavior of Hopfological homology for a tame A-subextension S/R in terms of the surjectivity of trace map and of cyclic modules, which induce Hopf-cyclic homology, for Hopf-Galois extensions with H in terms of relative Hopf modules.