Algebraic K-theory of elliptic cohomology
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913742953906176 |
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| author | Angelini-Knoll, Gabriel Ausoni, Christian Culver, Dominic Leon Höning, Eva Rognes, John |
| author_facet | Angelini-Knoll, Gabriel Ausoni, Christian Culver, Dominic Leon Höning, Eva Rognes, John |
| contents | We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_05890 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Algebraic K-theory of elliptic cohomology Angelini-Knoll, Gabriel Ausoni, Christian Culver, Dominic Leon Höning, Eva Rognes, John Algebraic Topology K-Theory and Homology Primary 19D50, 19D55, 55P43, 55Q51, Secondary 55N20, 55N34, 55N91, 55Q10, 55T25 We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner. |
| title | Algebraic K-theory of elliptic cohomology |
| topic | Algebraic Topology K-Theory and Homology Primary 19D50, 19D55, 55P43, 55Q51, Secondary 55N20, 55N34, 55N91, 55Q10, 55T25 |
| url | https://arxiv.org/abs/2204.05890 |