Weil restriction, normal bundles and motivic Thom spaces

TL;DR

This paper provides a geometric approach to motivic homotopy theory by studying Weil restriction of schemes, demonstrating its compatibility with vector bundles, normal bundles, and Thom spaces without relying on advanced categorical machinery.

Contribution

It introduces an elementary geometric method for Weil restriction in motivic homotopy theory, extending results on normal bundles and establishing compatibility with Thom spaces and Thom classes.

Findings

Weil restriction preserves vector bundles.

Extension of normal bundle results via Weil restriction.

Compatibility of Weil restriction with Thom spaces and Thom classes.

Abstract

Recent developments in motivic homotopy theory, especially the construction of norm functors by Bachmann and Hoyois, rely heavily on the machinery of infinite categories. In this paper, we take a purely geometric and elementary approach via the Weil restriction of schemes -- the fundamental geometric operation underlying these norm functors -- without invoking highly abstract categorical methods. We show that the Weil restriction preserves vector bundles and extend an existing result on normal bundles. We then construct the Weil restriction functor on the unstable motivic homotopy category and prove its compatibility with Thom spaces. Finally, in the setting of effective motives and the associated cohomology theories, we show that the Weil restriction sends Thom classes to Thom classes.