## Abstract

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.

Original language | English (US) |
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Article number | 106399 |

Journal | Journal of Pure and Applied Algebra |

Volume | 224 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2020 Elsevier B.V.

## Keywords

- Algebraic K-theory
- Cotorsion pair
- Exact category
- Localization
- Waldhausen category