The abelian closure of an exact category

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 closureT(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.