Algebraic K-theory of real topological K-theory
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911709372874752 |
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| author | Angelini-Knoll, Gabriel Ausoni, Christian Rognes, John |
| author_facet | Angelini-Knoll, Gabriel Ausoni, Christian Rognes, John |
| contents | We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_11463 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Algebraic K-theory of real topological K-theory Angelini-Knoll, Gabriel Ausoni, Christian Rognes, John Algebraic Topology K-Theory and Homology 19D50, 19D55, 55Q51, 55P43, 14F30 (Primary) 19E20, 13D03, 55N15, 55Q10, 55T25 (Secondary) We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology. |
| title | Algebraic K-theory of real topological K-theory |
| topic | Algebraic Topology K-Theory and Homology 19D50, 19D55, 55Q51, 55P43, 14F30 (Primary) 19E20, 13D03, 55N15, 55Q10, 55T25 (Secondary) |
| url | https://arxiv.org/abs/2309.11463 |