The Synthetic Hilbert Additive Group Scheme

• Alice Hedenlund ^[1] ; Tasos Moulinos ^[2]
  1. [1] Norwegian University of Science and Technology

     Norwegian University of Science and Technology

     Noruega

     
  2. [2] University of Paris-Saclay

     University of Paris-Saclay

     Arrondissement de Palaiseau, Francia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01085-y
• Enlaces
  • Texto completo
• Resumen

  • 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 A1/Gm. We study the cohomology of its deloopings, and show that one obtains a lift of the filtered circle, studied in [25]. At the level of quasi-coherent sheaves, one obtains synthetic lifts of the Z-linear ∞- categories of S1 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.

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