[Paper Review] Additivity and Double Coset formulae for the Motivic and Étale Becker-Gottlieb transfer

Roy Joshua⋆, Pablo Pelaez|arXiv (Cornell University)|Jul 5, 2020

Algebraic structures and combinatorial models38 references4 citations

TL;DR

This paper establishes the additivity of the motivic and étale Becker-Gottlieb transfer and its associated trace, proving foundational properties that enable new applications in motivic homotopy theory. It resolves a long-standing conjecture on the Euler characteristic of G/N(T) in the Grothendieck-Witt ring and derives analogues of classical double coset formulae, with applications to Brauer groups of homogeneous spaces over separably closed fields.

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.

Motivation & Objective

• To establish the additivity of the motivic and étale Becker-Gottlieb transfer and its associated trace in the context of algebraic geometry over perfect fields.
• To resolve the conjecture that the Euler characteristic of G/N(T) for a split reductive group G and its maximal torus normalizer N(T) is 1 in the Grothendieck-Witt ring.
• To derive analogues of classical double coset formulae in the motivic and étale settings for reductive group actions.
• To explore the relationship between transfers on schemes with 1-parameter subgroup actions and those on fixed point schemes.
• To apply these results to compute Brauer groups of homogeneous spaces associated to reductive groups over separably closed fields.

Proposed method

• Utilizes G-equivariant spectra and non-equivariant spectra to define the motivic and étale Becker-Gottlieb transfer via pre-transfer maps and Borel construction modifications.
• Applies the framework of [CJ19] to lift motivic ring spectra such as Σ_T and its localizations/completions to equivariant spectra.
• Employs the trace map construction to relate the transfer to generalized equivariant motivic cohomology theories.
• Derives double coset formulae by analyzing compositions of transfer maps under group inclusions and using filtration arguments on classifying spaces.
• Applies the additivity of the trace and transfer to compute the Euler characteristic of G/N(T) in the Grothendieck-Witt ring via spectral sequence and colimit arguments.
• Uses the projection maps f_{n-j,j} and their compositions with classifying space inclusions to show weak-equivalence of the total map to the wedge of truncated classifying spaces.

Experimental results

Research questions

• RQ1 Is the motivic Becker-Gottlieb transfer additive in the motivic and étale settings over perfect fields of arbitrary characteristic?
• RQ2 Does the Euler characteristic of G/N(T) for a split reductive group G and its maximal torus normalizer N(T) equal 1 in the Grothendieck-Witt ring?
• RQ3 Can double coset formulae for the transfer be generalized from classical algebraic topology to the motivic and étale contexts?
• RQ4 How do transfers on schemes with 1-parameter subgroup actions relate to transfers on the fixed point schemes?
• RQ5 What are the implications of the additivity of the transfer and trace for Brauer groups of homogeneous spaces over separably closed fields?

Key findings

• The motivic and étale Becker-Gottlieb transfer is additive, and so is its associated trace, in the stable motivic homotopy category over a perfect field.
• The Euler characteristic of G/N(T) for a split reductive group G and the normalizer of a split maximal torus N(T) is proven to be 1 in the Grothendieck-Witt ring.
• Analogues of classical double coset formulae are established in the motivic and étale settings, extending results from classical algebraic topology.
• The composition of the map f_{n-j,j} with the inclusion of BGL_j into BGL_n induces the projection onto the j-th component in the wedge sum of truncated classifying spaces.
• The map Π_{0≤j≤n} f_{n-j,j} : Σ_T BGL_{n,+} → ⋁_{0≤j≤n} Σ_T BGL_{j,+} is a weak equivalence, confirming the filtration structure of the motivic spectrum.
• The results extend to the infinite case via homotopy colimits, and the étale version follows by similar reasoning, completing the proof in both settings.