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arXiv · 2111.10203

Higher Semiadditive Algebraic K-Theory and Redshift

Shay Ben-Moshe·Tomer M. Schlank
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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 $\widehat{E(n)}$.

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BibTeXRIS

Shay Ben-Moshe, Tomer M. Schlank. 2021-11-04. Higher Semiadditive Algebraic K-Theory and Redshift. https://doi.org/10.1112/s0010437x23007595

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