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

Can Yaylali

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.

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

Abstract

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

Paper Structure

This paper contains 6 sections, 14 theorems, 37 equations.

Table of Contents

  1. The stack of local $G$-shtukas
  2. Motives on pro- and ind-Artin stacks
  3. Recollection on the theory of motives
  4. Complements on motives on ind-pro-Artin stacks
  5. Motives on local $G$-shtukas
  6. Application to the stack of local $G$-shtukas

Key Result

Theorem 1

Let $G$ be a split reductive group and $T$ a split maximal torus contained in $G$. Let $\mu\in X_{*}(T)_{+}$ be a dominant cocharacter and let $\mathrm{Sht}_{G}^{\leq\mu}$ denote the closureSee Section sec-GSht for the notation. of the stack of local $G$-shtukas of type $\mu$. Then in $\mathrm{DM}(\mathbb{F}_{p})$. In particular, it is a Tate motive.

Theorems & Definitions (37)

  • Theorem 1: \ref{['thm.main']}
  • Proposition 2: cf. Section \ref{['sec-complement']}
  • Definition 1.1: HV
  • Definition 1.2: VW
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 1.7
  • proof
  • ...and 27 more