Filtered formal groups, Cartier duality, and derived algebraic geometry
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913162745348096 |
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| author | Moulinos, Tasos |
| author_facet | Moulinos, Tasos |
| contents | We develop a notion of formal groups in the filtered setting and describe a duality relating these to a specified class of filtered Hopf algebras. We then study a deformation to the normal cone construction in the setting of derived algebraic geometry. Applied to the unit section of a formal group $\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degeneration of $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity result on complete filtrations, which, in particular, identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this in a special case, together with the aforementioned notion of Cartier duality, to recover the filtration on the filtered circle of [MRT19]. Finally, we investigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_10262 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Filtered formal groups, Cartier duality, and derived algebraic geometry Moulinos, Tasos Algebraic Geometry Algebraic Topology K-Theory and Homology We develop a notion of formal groups in the filtered setting and describe a duality relating these to a specified class of filtered Hopf algebras. We then study a deformation to the normal cone construction in the setting of derived algebraic geometry. Applied to the unit section of a formal group $\widehat{\mathbb{G}}$, this provides a $\mathbb{G}_m$-equivariant degeneration of $\widehat{\mathbb{G}}$ to its tangent Lie algebra. We prove a unicity result on complete filtrations, which, in particular, identifies the resulting filtration on the coordinate algebra of this deformation with the adic filtration on the coordinate algebra of $\widehat{\mathbb{G}}$. We use this in a special case, together with the aforementioned notion of Cartier duality, to recover the filtration on the filtered circle of [MRT19]. Finally, we investigate some properties of $\widehat{\mathbb{G}}$-Hochschild homology set out in loc. cit., and describe "lifts" of these invariants to the setting of spectral algebraic geometry. |
| title | Filtered formal groups, Cartier duality, and derived algebraic geometry |
| topic | Algebraic Geometry Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2101.10262 |