The Synthetic Hilbert Additive Group Scheme

TL;DR

This paper constructs a spectral lift of the Hilbert additive group scheme and the degree filtration on integer valued polynomials, utilizing the even filtration in synthetic spectra to advance the understanding of spectral group schemes and their cohomology.

Contribution

It introduces a novel construction of a spectral lift of the Hilbert additive group scheme and the degree filtration, leveraging the even filtration in synthetic spectra.

Findings

Lift of the Hilbert additive group scheme to a spectral group scheme over $\

Construction of a lift of the filtered circle in spectral terms

Synthetic lifts of $\

Abstract

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.