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Additivity and Double Coset formulae for the Motivic and Étale Becker-Gottlieb transfer

Roy Joshua, Pablo Pelaez

Abstract

In this paper, which is a continuation of earlier work by the first author and Gunnar Carlsson, one of the first results we establish is the additivity of the motivic Becker-Gottlieb transfer, as well as their étale realizations. This extends the additivity results the authors already established for the corresponding traces. We then apply this to derive several important consequences: for example, in addition to obtaining the analogues of various double coset formulae known in the classical setting of algebraic topology, we also obtain applications to Brauer groups of homogeneous spaces associated to reductive groups over separably closed fields. We also consider the relationship between the transfer on schemes provided with a compatible action by a $1$-parameter subgroup and the transfer associated to the fixed point scheme of the $1$-parameter subgroup.

Additivity and Double Coset formulae for the Motivic and Étale Becker-Gottlieb transfer

Abstract

In this paper, which is a continuation of earlier work by the first author and Gunnar Carlsson, one of the first results we establish is the additivity of the motivic Becker-Gottlieb transfer, as well as their étale realizations. This extends the additivity results the authors already established for the corresponding traces. We then apply this to derive several important consequences: for example, in addition to obtaining the analogues of various double coset formulae known in the classical setting of algebraic topology, we also obtain applications to Brauer groups of homogeneous spaces associated to reductive groups over separably closed fields. We also consider the relationship between the transfer on schemes provided with a compatible action by a -parameter subgroup and the transfer associated to the fixed point scheme of the -parameter subgroup.

Paper Structure

This paper contains 10 sections, 31 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The $\rm G$-equivariant pre-transfer
  3. The additivity of the pre-transfer
  4. Mayer-Vietoris and additivity of the transfer
  5. The rigidity property, Motivic Tubular Neighborhoods and Henselization along smooth closed subschemes
  6. Motivic tubular Neighborhoods and Henselization along smooth closed subschemes
  7. More on Nisnevich neighborhoods
  8. Applications of the Additivity (and Multiplicativity) of the transfer
  9. The generic torus slice theorem and applications
  10. Double Coset formulae

Key Result

Theorem \oldthetheorem

Let $\rm U_+ {\overset {\rm j_+} \rightarrow} \rm X {\overset {\rm k_+} \rightarrow} \rm X/\rm U = Cone(\rm j) \rightarrow \rm S^1 \wedge \rm U_+$ denote a cofiber sequence as in additivity.0 where all the above simplicial presheaves are provided with actions by a linear algebraic group $\rm G$ so Moreover if $\it {tr}(f)$, $\it {tr}(g)$ and $\it {tr}(h)$ denote the induced transfer maps in any

Theorems & Definitions (77)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • ...and 67 more