Stable comodule deformations and the synthetic Adams-Novikov spectral sequence
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918132747075584 |
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| author | Baer, J. Francis Johnson, Maxwell Marek, Peter |
| author_facet | Baer, J. Francis Johnson, Maxwell Marek, Peter |
| contents | We study the Adams-Novikov spectral sequence in $\mathbb{F}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{BP}$ and its cooperations to identify the synthetic Adams-Novikov $\mathrm{E}_2$-page, computed in a range with a synthetic algebraic Novikov spectral sequence. We then identify deformations associated to the Cartan-Eilenberg and algebraic Novikov spectral sequences in terms of stable comodule categories, categorifying an algebraic Novikov spectral sequence result of Gheorghe-Wang-Xu. We then apply Isaksen-Wang-Xu methods in $\mathbb{F}_2$-synthetic spectra to deduce differentials in the $p=2$ synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Stable comodule deformations and the synthetic Adams-Novikov spectral sequence Baer, J. Francis Johnson, Maxwell Marek, Peter Algebraic Topology 55Q10, 55T15, 55Q45, 55P42 We study the Adams-Novikov spectral sequence in $\mathbb{F}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{BP}$ and its cooperations to identify the synthetic Adams-Novikov $\mathrm{E}_2$-page, computed in a range with a synthetic algebraic Novikov spectral sequence. We then identify deformations associated to the Cartan-Eilenberg and algebraic Novikov spectral sequences in terms of stable comodule categories, categorifying an algebraic Novikov spectral sequence result of Gheorghe-Wang-Xu. We then apply Isaksen-Wang-Xu methods in $\mathbb{F}_2$-synthetic spectra to deduce differentials in the $p=2$ synthetic Adams-Novikov for the sphere, producing almost entirely algebraic computations through the 45-stem. |
| title | Stable comodule deformations and the synthetic Adams-Novikov spectral sequence |
| topic | Algebraic Topology 55Q10, 55T15, 55Q45, 55P42 |
| url | https://arxiv.org/abs/2402.14274 |