Higher Semiadditive Algebraic K-Theory and Redshift
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866913194520346624 |
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| author | Ben-Moshe, Shay Schlank, Tomer M. |
| author_facet | Ben-Moshe, Shay Schlank, Tomer M. |
| contents | 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 $\widehat{E(n)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10203 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Higher Semiadditive Algebraic K-Theory and Redshift Ben-Moshe, Shay Schlank, Tomer M. K-Theory and Homology Algebraic Topology 19D55, 55P42, 18N60 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 $\widehat{E(n)}$. |
| title | Higher Semiadditive Algebraic K-Theory and Redshift |
| topic | K-Theory and Homology Algebraic Topology 19D55, 55P42, 18N60 |
| url | https://arxiv.org/abs/2111.10203 |