无限箭图的表示及其Auslander-Reiten理论

The Auslander-Reiten theory is one of the cornerstones of modern representation theory of algebras. The representation theory of infinite quivers is an important research topic. It is closely related to covering theory and derived categories. This project intends to study the homological properties of interval-finite infinite quivers, and the Auslander-Reiten theory of the category of finitely presented representations. More precisely, we will study the Noetherian property of representations, and provide a necessary and sufficient condition for the category of representations being locally Noetherian. We will study the projective objects and injective objects in the category of finitely presented representations and the one of locally finite dimensional representations, and study the projectively and injectively stable categories. We will characterize the generalized Auslander-Reiten duality, and the morphisms determined by objects in the category of finitely presented representations. Moreover, we will find the formula for the minimal determiner. These studies will help us to understand the irreducible morphisms in the category of finitely presented representations, and moreover enrich the Auslander-Reiten theory.

Auslander-Reiten理论是现代代数表示论的基石之一，无限箭图的表示是重要的研究课题，与覆盖理论以及导出范畴等密切相关。本项目拟研究区间有限的无限箭图上表示范畴的同调性质，以及有限表现表示范畴的Auslander-Reiten理论。具体地，我们将研究区间有限的无限箭图表示的Noether性质，并给出箭图的表示范畴满足局部Noether性质的充要条件；研究局部有限表示范畴与有限表现表示范畴的投射对象与内射对象，以及其投射稳定范畴与内射稳定范畴；刻画有限表现表示范畴的广义Auslander-Reiten对偶，以及由对象决定的态射，并探求其极小决定子公式。这些研究有助于理解有限表现表示范畴中的不可约态射，进而丰富其Auslander-Reiten理论。

Auslander-Reiten理论是现代代数表示论的基石之一，无限箭图的表示是重要的研究课题，与覆盖理论以及导出范畴等密切相关。本项目研究了区间有限的无限箭图上表示范畴的同调性质，以及有限表现表示范畴的Auslander-Reiten理论。具体地，项目研究了有限表现表示范畴的内射对象，以及其内射稳定范畴；刻画了有限表现表示范畴的广义Auslander-Reiten对偶，以及由对象决定的态射。这些研究有助于理解有限表现表示范畴中的不可约态射，进而丰富其Auslander-Reiten理论。