Cotorsion pairs and a K-theory localization theorem

We show that a complete hereditary cotorsion pair (C,C^⊥) in an exact category E, together with a subcategory Z⊆E containing C^⊥, determines a Waldhausen category structure on the exact category C, in which Z is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the K-theory of exact categories A⊆B to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require A to be a Serre subcategory, which produces new examples. Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.