Abstract:In this paper we develop new reduction techniques for testing the finiteness of the finitistic dimension of a finite dimensional algebra over a field. Viewing the latter algebra as a quotient of a path algebra, we propose two operations on the quiver of the algebra, namely arrow removal and vertex removal. The former gives rise to cleft extensions and the latter to recollements. These two operations provide us new practical methods to detect algebras of finite finitistic dimension. We illustrate our methods with many examples.

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中文翻译：

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有限维的约简技术

摘要：在本文中，我们开发了新的约简技术，用于测试域上有限维代数的有限维数的有限性。将后一个代数视为路径代数的商，我们提出了对代数颤动的两个操作，即箭头去除和顶点去除。前者引起裂隙扩展，后者引起重新碰撞。这两个操作为我们提供了新的实用方法来检测有限维数的代数。我们用许多例子来说明我们的方法。

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