The Synthetic Hilbert Additive Group Scheme

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, w...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Authors: Hedenlund, Alice, Moulinos, Tasos
Format: Article
Language:English
Subjects:
Filtration
Homology
Integers
Lifts
Linearity
Polynomials
Sheaves
Online Access:Get full text
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Summary: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.
ISSN:2331-8422