Report on $AS$-Gorenstein Hopf algebras

TL;DR

This paper reviews progress on whether noetherian Hopf algebras possess finite injective dimension and explores related homological properties, highlighting key results, open questions, and implications in the field.

Contribution

It summarizes recent advances and open problems concerning the homological dimensions of noetherian Hopf algebras, including specific classes with positive results.

Findings

Positive answers for certain classes of Hopf algebras

Descriptions of consequences of finite injective dimension

Listing of numerous open questions in the area

Abstract

This is a review of progress on the question whether noetherian Hopf algebras always have finite injective dimension and related good homological properties. As well as discussing in detail the main results giving positive answers for particular classes of Hopf algebras, some consequences of such positive answers are also described. Full definitions and references are included, also sketches of some proofs. A considerable number of open questions are listed, additional to the original question, which itself remains open after 30 years.