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 K(n)and T(n)-local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if R is a ring spectrum of height ≤ n, then its semiadditive algebraic K-theory is of height ≤ n + 1. Under further hypothesis on R, which are satisfied for example by the Lubin-Tate spectrum En, we show that its semiadditive algebraic K-theory is of height exactly n + 1. Finally, we show that for any p-invertible ring spectrum, its semiadditive algebraic K-theory coincides with the T(1)-localization of its ordinary algebraic K-theory.

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