Abstract For any exact category A with splitting idempotents, a maximal exact category T ( A ) containing A as a biresolving subcategory, is constructed. Important types of exact categories, including n-tilting torsion classes, categories of Cohen-Macaulay modules over a Cohen-Macaulay order, or categories of Gorenstein projectives, are shown to be of the form T ( A ) . The quotient category T ( A ) / A in the sense of Grothendieck always exists and carries a triangulated structure. More generally, it is proved that any biresolving subcategory A of an exact category M gives a triangulated localization M / A . For example, the unbounded derived category D ( A ) of an exact category A is obtained directly, with no passage through a homotopy category of complexes. As applications, some recent developments related to Gorenstein projectivity, non-commutative crepant resolutions, singularity categories, and Cohen-Macaulay representations are extended and improved in the new framework. For example, the concept of non-commutative resolution of a noetherian Frobenius category is extended to arbitrary exact categories, which leads to an overarching connection with representation dimension of exact categories and n-tilting.

中文翻译：

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精确范畴的无环闭包及其三角剖分

摘要 对于具有分裂幂等的任何精确类别 A，构造包含 A 作为双解子类别的最大精确类别 T ( A )。精确类别的重要类型，包括 n 倾斜扭转类别、Cohen-Macaulay 阶上的 Cohen-Macaulay 模块类别或 Gorenstein 射影类别，显示为 T ( A ) 形式。格洛腾迪克意义上的商范畴 T ( A ) / A 始终存在并带有三角结构。更一般地，证明了精确类别 M 的任何双解子类别 A 都给出了三角定位 M / A 。例如，直接得到精确范畴 A 的无界派生范畴 D ( A )，不经过配合物的同伦范畴。作为应用，与 Gorenstein 投影相关的一些最新进展，非交换蠕变分辨率、奇点类别和 Cohen-Macaulay 表示在新框架中得到扩展和改进。例如，将诺特 Frobenius 范畴的非交换分解的概念扩展到任意精确范畴，这导致与精确范畴的表示维和 n 倾斜的总体联系。