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We study singularity categories of exact categories with a focus on thoseassociated to a complete hereditary cotorsion pair. As an application weidentify a non-affine analogue of the singularity category of a Gorensteinlocal ring; with this Buchweitz's classic equivalence of three categories overGorenstein local rings has been generalized to schemes, a project started byMurfet and Salarian more than ten years ago. As another application we use theframework to characterize rings of finite finitistic dimension.

The singularity category of an exact category applied to characterize Gorenstein schemes