We discuss the finiteness of (two-term) silting objects. First, we investigate new triangulated categories without silting object. Second, we study two classes of $\tau$-tilting-finite algebras and give the numbers of their two-term silting objects. Finally, we explore when $\tau$-tilting-finiteness implies representation-finiteness and obtain several classes of algebras in which a $\tau$-tilting-finite algebra is representation-finite.

中文翻译：

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淤积物体的有限性报告

我们讨论（两项）淤积物体的有限性。首先，我们调查没有淤积对象的新三角分类。其次，我们研究两类$\tau$-倾斜有限代数并给出它们的两项淤积对象的数量。最后，我们探讨何时$\tau$-tilting-finiteness 意味着表示有限性并获得几类代数，其中$\tau$-倾斜有限代数是表示有限的。