Higher Semiadditive Algebraic K-Theory and Redshift

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 $\le n$, then its semiadditive K-theory is of height $\le n+1$. 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 $\widehat{E(n)}$.

v2: Added Theorem D and E concerning the higher semiadditive K-theory of completed Johnson-Wilson at any height, and other small improvements. v1: 44 pages, comments are welcome!

References in corpus (1)

• Naturality of the $\infty$-Categorical Enriched Yoneda Embedding

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