# Han's conjecture and Hochschild homology for null-square projective algebras

## TL;DR

This paper investigates classes of algebras related to Han's conjecture, establishing inductive steps for proving the conjecture for certain triangular and matrix-structured algebras, advancing understanding of Hochschild homology.

## Contribution

It provides new inductive methods to verify Han's conjecture for specific classes of algebras, including triangular and null-square matrix algebras.

## Key findings

- Triangular algebras with components in $\\mathcal H$ are in $\\mathcal H$.
- Null-square matrix algebras with diagonal components in $\\mathcal H$ are in $\\mathcal H$.
- Results support inductive approaches to Han's conjecture.

## Abstract

Let $\mathcal H$ be the class of algebras verifying Han's conjecture. In this paper we analyse two types of algebras with the aim of providing an inductive step towards the proof of this conjecture. Firstly we show that if an algebra $\Lambda$ is triangular with respect to a system of non necessarily primitive idempotents, and if the algebras at the idempotents belong to $\mathcal H$, then $\Lambda$ is in $\mathcal H$. Secondly we consider a $2\times 2$ matrix algebra, with two algebras on the diagonal, two projective bimodules in the corners, and zero corner products. They are not triangular with respect to the system of the two diagonal idempotents. However, the analogous result holds, namely if both algebras on the diagonal belong to $\mathcal H$, then the algebra itself is in $\mathcal H$.