Extending torsors on the punctured Spec(A_inf)

TL;DR

This paper proves that torsors under certain group schemes over punctured Fontaine's ring extend to the entire spectrum, with applications to affine Grassmannians and affine flag varieties.

Contribution

It extends torsors under parahoric group schemes on punctured spectra of Fontaine's ring to the whole spectrum, generalizing previous results and applying descent methods.

Findings

Torsors extend from punctured to full spectrum of Fontaine's ring.

Extension results apply to Kisin-Pappas ring $rak{S}$ and equal characteristic cases.

Constructs a canonical map from $B^+_{dR}$-affine Grassmannian to Witt vector affine flag variety.

Abstract

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine's ring $A_{\mathrm{inf}}$, extend to the whole spectrum. Using descent we can extend a similar result for the ring $\mathfrak{S}$ of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the $B^+_{\mathrm{dR}}$-affine Grassmannian to the Witt vector affine flag variety.