Han's conjecture for bounded extensions

TL;DR

This paper proves that for bounded extensions of finite dimensional algebras, Han's conjecture holds for the subalgebra if and only if it holds for the larger algebra, using the Jacobi-Zariski sequence.

Contribution

It establishes an equivalence of Han's conjecture validity between bounded extensions of finite dimensional algebras, regardless of splitting, and provides structural conditions for such extensions.

Findings

Han's conjecture equivalence for bounded extensions

Conditions for extensions by arrows and relations to be bounded

Examples of non-split bounded extensions

Abstract

Let $B\subset A$ be a left or right bounded extension of finite dimensional algebras. We use the Jacobi-Zariski long nearly exact sequence to show that $B$ satisfies Han's conjecture if and only if $A$ does, regardless if the extension splits or not. We provide conditions ensuring that an extension by arrows and relations is left or right bounded. Finally we give a structure result for extensions of an algebra given by a quiver and admissible relations, and examples of non split left or right bounded extensions.