The acyclic closure of an exact category and its triangulation

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.