A Motivic Snaith Decomposition

TL;DR

This paper demonstrates a motivic Snaith decomposition using Becker-Gottlieb transfers, extending motivic homotopy theory to ind-schemes and recovering classical results through complex realizations.

Contribution

It introduces a motivic Snaith decomposition via transfers and extends motivic homotopy theory to smooth ind-schemes, enabling new constructions in the motivic stable homotopy category.

Findings

Filtration of BGL is split by motivic transfers.

Extension of motivic homotopy theory to ind-schemes.

Recovery of classical Snaith results via complex realization.

Abstract

The filtration $\operatorname{BGL}_{0}\subset\dots\subset\operatorname{BGL}_{n-1}\subset\operatorname{BGL}_{n}$ is split by motivic Becker-Gottlieb transfers in the motivic stable homotopy category over any scheme. This recovers results by Snaith on the splitting of $\operatorname{BGL}_{n}(\mathbb{C})$ in classical stable homotopy theory by passing to complex realizations. On the way, we extend motivic homotopy theory to smooth ind-schemes as bases and show how to construct the necessary fragment of the six operations and duality for this extension.