More than 30 years ago, M. Auslander proved, for Cohen–Macaulay orders over a complete regular local ring of dimension d, that existence of almost split sequences is equivalent to the presence of an isolated singularity. CM-finite Cohen–Macaulay orders form an important special class. They have been studied intensively for \(d\le 2\), with scattered results for \(d>2\). More recently, the question of CM-finiteness in its widest sense has become relevant for exact categories arising in commutative algebra, non-commutative singularity theory, Gorenstein homological algebra, and related topics. In the paper, two types of criteria for CM-finiteness are established which extend previously known results to arbitrary dimension. The first type of criteria deals with Krull–Schmidt categories with almost split sequences. It is shown that finite CM-type is closely related, but not equivalent to finiteness with respect to L-functors. The second type of criteria appeals to non-commutative crepant resolutions.

30 多年前，M. Auslander 证明，对于 Cohen-Macaulay 阶在维数为d的完整规则局部环上，几乎分裂的序列的存在等同于孤立奇点的存在。CM 有限 Cohen-Macaulay 阶构成一个重要的特殊类。他们已经对\(d\le 2\)进行了深入研究，\(d>2\) 的结果分散. 最近，最广泛意义上的 CM 有限性问题已与交换代数、非交换奇点理论、Gorenstein 同调代数和相关主题中出现的精确范畴相关。在本文中，建立了两种类型的 CM 有限性标准，将先前已知的结果扩展到任意维度。第一类标准处理具有几乎分裂序列的 Krull-Schmidt 类别。结果表明，有限 CM 类型与L函子密切相关，但不等同于有限性。第二种标准适用于非交换式蠕变分辨率。