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Higher semiadditive algebraic K-theory and redshift
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 2 / February 2024
- Published online by Cambridge University Press:
- 15 December 2023, pp. 237-287
- Print publication:
- February 2024
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- Article
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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 
$\widehat {\mathrm {E}(n)}$.
