Strongly stratifying ideals, Morita contexts and Hochschild homology

TL;DR

This paper investigates the relationship between strongly stratifying ideals, Morita contexts, and Hochschild homology, showing how Han's conjecture for certain algebras can be reduced to simpler cases using the Jacobi-Zariski sequence.

Contribution

It establishes that Han's conjecture holds for algebras with strongly stratifying chains if it holds for their diagonal subalgebras, linking stratification with homological conjectures.

Findings

Han's conjecture holds for strongly stratifying Morita contexts.

The conjecture reduces to diagonal subalgebras in these contexts.

It holds for algebras with primitive strongly stratifying chains if it holds for local algebras.

Abstract

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for it.