Truncations and the Motive of the Stack of Local $G$-Shtukas

TL;DR

This paper computes the rational motive of the stack of local G-shtukas for split reductive groups, linking it to the motive of G-zips, and utilizes recent advances in the theory of truncated local G-shtukas.

Contribution

It explicitly relates the motive of local G-shtukas to G-zips using truncated shtukas and advanced categorical tools, providing a new explicit computation in the field.

Findings

Rational motive of local G-shtukas expressed in terms of G-zips

Utilizes truncated local G-shtukas by Viehmann-Wedhorn

Employs the $*$- and $!$-adjunction for pro-systems

Abstract

We compute the rational motive of the stack of local $G$-shtukas, for a split reductive group $G$, representing compactly supported cohomology in terms of the motive of the stack of $G$-zips. This result makes explicit use of the truncated version of local $G$-shtukas established by Viehmann-Wedhorn and the theoretical background on the $*$- and $!$-adjunction for pro-systems of algebraic objects established by the author.