AI 中文总结
本文研究带紧 silting 生成子的三角范畴的有限维数，统一相关概念并关联阿贝尔心范畴，在 recollement 下建立维数不等式，推广环论定理并应用于多类代数结构。
AI 中文摘要
本文研究具有紧 silting 生成子的三角范畴中的有限维数。我们统一了文献中出现的若干（大）有限维数概念，证明它们本质上等价，并当生成子为 tilting 时将其与阿贝尔心范畴关联，为有限维数猜想提供了新的范畴论视角。此外，我们在 recollement 下建立了（大）有限维数与整体维数的显式不等式，证明中间范畴的有限性等价于外范畴的有限性。这些结果推广了经典的环论定理，并适用于三角矩阵环、正合上下文及平凡扩张。
英文摘要
This paper studies finitistic dimensions in triangulated categories with a compact silting generator. We unify several notions of (big) finitistic dimension appearing in the literature, show that they are essentially equivalent, and relate them to the abelian heart when the generator is tilting. This yields a new categorical perspective on the finitistic dimension conjecture. Furthermore, we establish explicit inequalities for (big) finitistic and global dimensions under recollements, demonstrating that finiteness in the middle category is equivalent to finiteness in the outer categories. These results generalize classical ring-theoretic theorems and apply to triangular matrix rings, exact contexts and trivial extensions.
