## Abstract

We define higher semiadditive algebraic K-theory, a variant of algebraic K-theory that takes into account higher semiadditive structure, as enjoyed for example by the - and -local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if is a ring spectrum of height, then its semiadditive K-theory is of height. Under further hypothesis on, which are satisfied for example by the Lubin-Tate spectrum, we show that its semiadditive algebraic K-theory is of height exactly. Finally, we connect semiadditive K-theory to -localized K-theory, showing that they coincide for any -invertible ring spectrum and for the completed Johnson-Wilson spectrum.

Original language | American English |
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Pages (from-to) | 237-287 |

Number of pages | 51 |

Journal | Compositio Mathematica |

Volume | 160 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2023 |

### Bibliographical note

Publisher Copyright:© 2023 The Author(s).

## Keywords

- algebraic K-theory
- ambidexterity
- chromatic homotopy theory
- higher semiadditivity
- redshift