A Survey on Han's Conjecture

TL;DR

This survey reviews the development and current understanding of Han's conjecture, exploring the relationship between Hochschild cohomology vanishing and finite global dimension in finite-dimensional algebras.

Contribution

It compiles and discusses existing results, examples, and extensions related to Han's conjecture, providing a comprehensive overview of the research progress.

Findings

Examples of algebras satisfying Han's conjecture

Extensions that preserve the conjecture

Progress towards understanding the conjecture's validity

Abstract

In 1989, D. Happel pointed out for a possible connection between the global dimension of a finite-dimensional algebra and its Hochschild cohomology: is it true that the vanishing of Hochschild cohomology higher groups is sufficient to deduce that the global dimension is finite? After the discovery of a counterexample, Y. Han proposed, in 2006, to reformulate this question to homology. In this survey, after introducing the concepts and results involved, I present the efforts made until now towards the comprehension of Han's conjecture; which includes: examples of algebras that have been proven to satisfy it and extensions that preserve it.