Abstract:
The Hilbert additive group arises in algebraic geometry as the affine group scheme coming from the free binomial ring on one generator. This can be endowed with the structure of a "filtered stack" by noting that the free binomial ring on one generator can be identified with integer-valued polynomials, which of course carries a canonical filtration by degree. This filtered stack is intimately related to the HKR filtration on Hochschild homology, as shown in work of Moulinos-Robalo-Toën.
With a view towards geometrizing the motivic filtrations on topological Hochschild homology, we extend the filtered Hilbert additive group to the setting of spectral algebraic geometry to an object that we call the synthetic Hilbert additive group. This crucially relies on the even filtration introduced by Hahn-Raksit-Wilson and the view of even synthetic spectra as modules over the even sphere. This is joint work with Tasos Moulinos.