Speaker: Guodong Zhou (East China Normal Univeristy)
Abstract: A conjecture due to Y. Han asks whether that Hochschild homology groups of a finite dimensional algebra defined over an algebraically closed field vanish for sufficiently large degrees would imply that the algebra is of finite global dimension. We present two approached for this conjecture, one via recollements and ladders of derived categories, one via algebra extension. In the first approach, 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. As applications, it is proved that Han’s conjecture holds for skew-gentle algebras, category algebras of finite EI categories and Geiss-Leclerc-Schroeer algebras associated to Cartan triples. In the second approach, following Cibils, Lanzilotta, Marcos, and Solotar, we study the so-called bounded extensions of algebras. It is shown that for a bounded extension, the two algebras are singularly equivalent of Morita type with level. Additionally, under mild conditions, their stable categories of Gorenstein projective modules and Gorenstein defect categories are equivalent, respectively. Some homological conjectures are also investigated for bounded extensions, including Han’s conjecture, Auslander-Reiten conjecture, finitistic dimension conjecture, Fg condition, and Keller’s conjecture. Applications to trivial extensions and triangular matrix algebras are given. In course of the talk, we will also present reduction methods for singularity categories and Hochschild homology.
Time: 16:00-19:00 (with tea break), Monday, Sept. 23, 2024
Location: Room 1410, Block A, No. 657 Songhu Road, Yangpu District, Shanghai
English 
简体中文