Abstract The paper exhibits three constitutive exact subcategories of Adelman's free abelian category Ab ( A ) over a Quillen exact category A , which provide an intrinsic description of Ab ( A ) and encapsulate the mechanism of (finite or infinite) tilting. In general, two of these subcategories are left and right abelian, respectively, and a tilting adjunction takes place between them if they are abelian. The third category Im ( A ) contains the acyclic closure T ( A ) of A as an exact subcategory of those objects which arise as images of the morphisms in an exact complex over A . If the exact structure of A is trivial, T ( A ) consists of the Gorenstein projectives over A , a category that does not depend on an embedding of A into some ambient abelian category. The construction of Ab ( A ) and its intrinsic description sheds some light upon old and new concepts and results concerning Gruson-Jensen duality, resolving subcategories, representations of Cohen-Macaulay orders, Gorenstein projectivity, non-commutative resolutions, and representation dimension. Applications to infinite tilting and existence of derived equivalences will be given in a forthcoming publication.

中文翻译：

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精确范畴的阿贝尔闭包

摘要 本文展示了 Adelman 自由阿贝尔范畴 Ab ( A ) 在 Quillen 精确范畴 A 上的三个本构精确子范畴，它们提供了 Ab ( A ) 的内在描述并封装了（有限或无限）倾斜的机制。通常，其中两个子类别分别是左阿贝尔和右阿贝尔，如果它们是阿贝尔，则它们之间会发生倾斜附加。第三类 Im ( A ) 包含 A 的非循环闭包 T ( A ) 作为那些对象的精确子类别，这些对象作为 A 上的精确复形中态射的图像出现。如果 A 的确切结构是微不足道的，则 T ( A ) 由 A 上的 Gorenstein 射影组成，该类别不依赖于将 A 嵌入到某个环境阿贝尔类别中。Ab ( A ) 的构造及其内在描述揭示了有关 Gruson-Jensen 对偶性、解析子类别、Cohen-Macaulay 阶数的表示、Gorenstein 投影性、非交换分辨率和表示维数的新旧概念和结果。无限倾斜的应用和派生等价的存在将在即将出版的出版物中给出。