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 $\mathrm {K}(n)$- 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 $\widehat {\mathrm {E}(n)}$.
Subject
Algebra and Number Theory
Reference40 articles.
1. Hyperdescent and étale K-theory;Clausen;Invent. Math,2021
2. HL13 Hopkins, M. and Lurie, J. , Ambidexterity in $K(n)$ -local stable homotopy theory, Preprint (2013), https://www.math.ias.edu/~lurie/papers/Ambidexterity.pdf.
3. Multiplicative structures on algebraic K-theory;Barwick;Doc. Math,2015
4. On the K-theory of algebraically closed fields;Suslin;Invent. Math,1983
5. Bousfield localization functors and Hopkins’ chromatic splitting conjecture
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