Higher semiadditive algebraic K-theory and redshift

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.