A recollement approach to Han's conjecture

Ren Wang, Xiaoxiao Xu, Jinbi Zhang, Guodong Zhou
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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.
韩氏猜想的重补方法
Y. Han 提出的一个猜想是,无限维代数的霍希契尔德同调群在足够大的度数下消失是否意味着该代数是有限全维的。我们从派生类的重组的角度研究了这一猜想。结果表明,对于向下(或向上)延伸一步的无界派生类环的重组,如果且只有当韩氏猜想对两边的两个环成立时,韩氏猜想才对中间的环成立,因此韩氏猜想被还原为派生的 2 美元简单环。此外,这一还原结果也适用于莫里塔上下文环和精确上下文的韩氏猜想。最后,证明了韩氏猜想对于与卡坦三元组相关联的斜温和代数、有限EI范畴的范畴代数和Geiss-Leclerc-Schr"{o}er代数是成立的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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