标题： 箭头约简用于有限维猜想 显示英文标题

标题： Arrow reductions for the finitistic dimension conjecture

摘要： 我们提出了新的方法来消除有界箭图代数中的箭头，从而将给定代数的Finitistic Dimension Conjecture $\mathsf{(FDC)}$降低到一个较小的代数。 与Green-Psaroudakis-Solberg的经典箭头消除操作不同，我们的方法允许在箭头出现在代数定义可接受理想的每个生成集时进行消除。 我们的第一个主要结果在理想 $K$的特定同调和结构条件下，在左阿廷环的更广泛背景下，建立了环 $\Lambda$和其商 $\Lambda/K$的有限性（以及整体）维数之间的等价性。 该结果应用于有界箭图代数，提出了箭头集合的可消除性概念，并证明了这种类型的连续箭头消除会导致代数的独特定义的箭头约化版本。 在相反的方向上，我们通过可消除乘法双模对广义箭头消除代数进行了表征，并引入了平凡单箭头扩张作为一种新的组合构造，用于添加箭头同时保持Finitistic（以及整体）维数的有限性。 所有这些新方法都通过各种具体的有界箭图代数进行了说明，根据我们目前的知识，确认 $\mathsf{(FDC)}$之前似乎难以处理。 显示英文摘要

摘要： We present new techniques for removing arrows of bound quiver algebras, reducing thus the Finitistic Dimension Conjecture $\mathsf{(FDC)}$ for a given algebra to a smaller one. Unlike the classic arrow removal operation of Green-Psaroudakis-Solberg, our methods allow for removing arrows even when they occur in every generating set for the defining admissible ideal of the algebra. Our first main result establishes an equivalence for the finiteness of the finitistic (and global) dimensions of a ring $\Lambda$ and its quotient $\Lambda/K$, under specific homological and structural conditions on the ideal $K$, in the broader context of left artinian rings. The application of this result to bound quiver algebras suggests the notion of removability for sets of arrows, and we prove that successive arrow removals of this sort lead to a uniquely defined arrow reduced version of the algebra. Towards the opposite direction, we characterize generalized arrow removal algebras via removable multiplicative bimodules, and introduce trivial one-arrow extensions as a novel combinatorial construction for adding arrows while preserving the finiteness of the finitistic (and global) dimensions. All these new techniques are illustrated through various concrete bound quiver algebras, for which confirming the $\mathsf{(FDC)}$ had previously seemed intractable to the best of our knowledge.