A recollement approach to Han's conjecture

TL;DR

This paper investigates Han's conjecture on Hochschild homology and finite global dimension using recollements of derived categories, reducing the problem to derived 2-simple rings and confirming it for specific algebra classes.

Contribution

It introduces a recollement-based approach to Han's conjecture, reducing it to derived 2-simple rings and verifying it for several algebra classes.

Findings

Han's conjecture holds for rings in certain recollement configurations.

Reduction of Han's conjecture to derived 2-simple rings.

Verification of Han's conjecture for skew-gentle, EI category, and Geiss-Leclerc-Schröer algebras.

Abstract

A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We investigate this conjecture from the viewpoint of recollements of derived categories. It is shown that for a recollement of unbounded derived categories of rings which extends downwards (or upwards) one step, Han's conjecture holds for the ring in the middle if and only if it holds for the two rings on the two sides and hence Han's conjecture is reduced to derived $2$-simple rings. Furthermore, this reduction result is applied to Han's conjecture for Morita contexts rings and exact contexts. Finally it is proved that Han's conjecture holds for skew-gentle algebras, category algebras of finite EI categories and Geiss-Leclerc-Schr\"{o}er algebras associated to Cartan triples.