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 $\mathrm {K}(n)$ - and $\mathrm {T}(n)$ -local categories. We prove that it satisfies a form of the redshift conjecture. Namely, that if $R$ is a ring spectrum of height $\leq n$ , then its semiadditive K-theory is of height $\leq n+1$. Under further hypothesis on $R$ , which are satisfied for example by the Lubin–Tate spectrum $\mathrm {E}_n$ , we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $\mathrm {T}(n+1)$ -localized K-theory, showing that they coincide for any $p$ -invertible ring spectrum and for the completed Johnson–Wilson spectrum $\widehat {\mathrm {E}(n)}$. [ABSTRACT FROM AUTHOR]