Homological invariants of the arrow removal operation

单项式箭向移除与有限维数猜想

Monomial arrow removal and the finitistic dimension conjecture

针对有限维数有限性判定问题，提出单项式箭向移除这一新约化技术，借助阿贝尔范畴裂扩张理论与Green意义下的非交换Gröbner基建立关键关联，并用实例验证理论。

有限维数猜想的箭头约化

Arrow reductions for the finitistic dimension conjecture

本文提出有界箭图代数中移除箭头的新技术，将有限维数猜想约化到更小代数，并引入可移除乘法双模和平凡单箭头扩张等构造，同时保持有限维数与整体维数有限性。

Eventually homological isomorphisms and Gorenstein projective modules

We prove that a certain eventually homological isomorphism between module categories induces a triangle equivalence between their singularity categories, Gorenstein defect categories and the stable categories of Gorenstein projective modules. Further, we show that Auslander-Reiten conjecture and Gorenstein symmetry conjecture can be reduced by eventually homological isomorphisms. Applying the results to arrow removal and vertex removal, we describe the Gorenstein projective modules over some non-monomial algebras, and we verify the Auslander-Reiten conjecture for certain algebras.

三条同调不变量在裂扩展下的行为

Three homological invariants under cleft extensions

本文研究了三种同调不变量在abelian范畴裂扩展下的行为，并将其应用于多种环结构。

Reduction techniques for the finitistic dimension

In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.