The Synthetic Hilbert Additive Group Scheme
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909657127190528 |
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| author | Hedenlund, Alice Moulinos, Tasos |
| author_facet | Hedenlund, Alice Moulinos, Tasos |
| contents | We construct a lift of the degree filtration on the integer valued polynomials to (even MU-based) synthetic spectra. Namely, we construct a bialgebra in modules over the evenly filtered sphere spectrum which base-changes to the degree filtration on the integer valued polynomials. As a consequence, we may lift the Hilbert additive group scheme to a spectral group scheme over $\mathbb{A}^1/\mathbb{G}_m$. We study the cohomology of its deloopings, and show that one obtains a lift of the filtered circle, studied in [MRT22]. At the level of quasi-coherent sheaves, one obtains lifts synthetic lifts of the $\mathbb{Z}$-linear $\infty$-categories of $S^1_{\mathrm{fil}}$-representations. Our constructions crucially rely on the use of the even filtration of Hahn--Raksit--Wilson; it is linearity with respect to the even filtered sphere that powers the results of this work. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Synthetic Hilbert Additive Group Scheme Hedenlund, Alice Moulinos, Tasos Algebraic Geometry Algebraic Topology 14A30 We construct a lift of the degree filtration on the integer valued polynomials to (even MU-based) synthetic spectra. Namely, we construct a bialgebra in modules over the evenly filtered sphere spectrum which base-changes to the degree filtration on the integer valued polynomials. As a consequence, we may lift the Hilbert additive group scheme to a spectral group scheme over $\mathbb{A}^1/\mathbb{G}_m$. We study the cohomology of its deloopings, and show that one obtains a lift of the filtered circle, studied in [MRT22]. At the level of quasi-coherent sheaves, one obtains lifts synthetic lifts of the $\mathbb{Z}$-linear $\infty$-categories of $S^1_{\mathrm{fil}}$-representations. Our constructions crucially rely on the use of the even filtration of Hahn--Raksit--Wilson; it is linearity with respect to the even filtered sphere that powers the results of this work. |
| title | The Synthetic Hilbert Additive Group Scheme |
| topic | Algebraic Geometry Algebraic Topology 14A30 |
| url | https://arxiv.org/abs/2411.17441 |