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Hopfological invariants for tame subextensions

Mariko Ohara

TL;DR

The paper investigates criteria for a finite $H$-extension to be tame by exploiting Hopfological homology and Hopf-cyclic homology in the Hopf-module setting. It establishes precise links between vanishing Hopfological homology, trace surjectivity, and Hopf-Galois/tame equivalence under rank and integrality conditions, and derives degree-shift relations for bar constructions in the relative Hopf-module context. The results yield computable criteria for tameness via homological invariants and clarify the role of relative Hopf modules and anti-Yetter-Drinfeld modules in these equivalences. The findings advance understanding of module structure over associated orders and provide a homological framework for tameness and Hopf-Galois extensions with potential applications in arithmetic and noncommutative geometry.

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.

Hopfological invariants for tame subextensions

TL;DR

The paper investigates criteria for a finite -extension to be tame by exploiting Hopfological homology and Hopf-cyclic homology in the Hopf-module setting. It establishes precise links between vanishing Hopfological homology, trace surjectivity, and Hopf-Galois/tame equivalence under rank and integrality conditions, and derives degree-shift relations for bar constructions in the relative Hopf-module context. The results yield computable criteria for tameness via homological invariants and clarify the role of relative Hopf modules and anti-Yetter-Drinfeld modules in these equivalences. The findings advance understanding of module structure over associated orders and provide a homological framework for tameness and Hopf-Galois extensions with potential applications in arithmetic and noncommutative geometry.

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.

Paper Structure

This paper contains 3 sections, 10 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Hopfological homology and Hopf-cyclic homology

Key Result

Theorem 1.1

Let $R$ be a commutative local ring. Let $\mathcal{A}$ be a finite $R$-Hopf algebra and $S$ a faithful $\mathcal{A}$-module algebra which is a finite commutative $R$-algebra of $rank_R(S)=rank_R(\mathcal{A})$. Assume that $S^\mathcal{A}=R$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Hopfological homology
  • Remark 2.2
  • Definition 2.3: cf. Childs1, Definition 2.7, Definition 13.1
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • ...and 19 more