A homological approach to chromatic complexity of algebraic K-theory
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866915318538960896 |
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| author | Angelini-Knoll, Gabriel Quigley, J. D. |
| author_facet | Angelini-Knoll, Gabriel Quigley, J. D. |
| contents | The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1908_09164 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A homological approach to chromatic complexity of algebraic K-theory Angelini-Knoll, Gabriel Quigley, J. D. Algebraic Topology K-Theory and Homology 55P43, 18F25, 55T99, 19D55 The family of Thom spectra $y(n)$ interpolates between the sphere spectrum and the mod two Eilenberg--MacLane spectrum. Computations of Mahowald, Ravenel, Shick, and the authors show that the associative ring spectrum $y(n)$ has type $n$. Using trace methods, we give evidence that algebraic K-theory preserves this chromatic complexity. Our approach sheds light on the chromatic complexity of topological negative cyclic homology and topological periodic cyclic homology, which approximate algebraic K-theory and are of independent interest. Our main contribution is a homological approach that can be applied in great generality, such as to associative ring spectra $R$ without additional structure whose coefficient rings are not completely understood. |
| title | A homological approach to chromatic complexity of algebraic K-theory |
| topic | Algebraic Topology K-Theory and Homology 55P43, 18F25, 55T99, 19D55 |
| url | https://arxiv.org/abs/1908.09164 |