AI 中文总结
本文针对$(n+2)$-角范畴，为偶数$n$的Auslander-Reiten (n+2)-角与局部有限性的蕴含关系建立充分条件，给出该开放问题的部分肯定答案。
AI 中文摘要
设$\boldsymbol{\textit{C}}$为一个$(n+2)$-角范畴。Zhou证明了当$n$为奇数时，若Auslander-Reiten (n+2)-角生成$\boldsymbol{\textit{C}}$的Grothendieck群的关系，则$\boldsymbol{\textit{C}}$是局部有限的。对于偶数$n$，该结论是否成立仍是开放问题。本文通过建立一个充分条件，给出该问题的部分肯定答案，使得相同蕴含关系对偶数$n$成立。进一步证明该充分条件被大量示例满足，表明结果可推广至孤立情形之外。
英文摘要
Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.
Comments14 pages
