Higher Semiadditive Algebraic K-Theory and Redshift
Shay Ben-Moshe, Tomer M. Schlank
TL;DR
The paper introduces higher semiadditive algebraic K-theory, building a framework that merges semiadditivity with algebraic K-theory for stable $ig$∞-categories and algebras. It proves a redshift phenomenon where semiadditive height can increase by at most one under a suitable categorification, and, under hypotheses such as the presence of $p$-th roots of unity, achieves exact height $n+1$ (notably for Lubin–Tate spectra $E_n$). It also connects this theory to chromatically localized K-theory, showing that for $p$-invertible spectra the higher semiadditive K-theory coincides with $T(1)$-local K-theory, and that for completed Johnson–Wilson spectra the results land in $T(n+1)$-local spheres. The work develops two equivalent constructions of $K^{[m]}$, proves lax symmetric monoidal structures, and uses the notions of atomic objects and Day convolution to enable a robust multiplicative framework. Overall, this advances the redshift program by giving a concrete, higher-categorical K-theory that tracks chromatic height and descent properties in a semiadditive setting, with concrete identifications in important cases like $E_n$ and $ ext{widehat{E}(n)}$.
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 $K(n)$- 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 $\leq n$, 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 $E_n$, we show that its semiadditive algebraic K-theory is of height exactly $n+1$. Finally, we connect semiadditive K-theory to $T(n+1)$-localized K-theory, showing that they coincide for any $p$-invertible ring spectrum and for the completed Johnson-Wilson spectrum $\widehat{E(n)}$.