Strongly A^1-invariant sheaves (after F. Morel)

TL;DR

This paper discusses strongly A^1-invariant sheaves in motivic homotopy theory, proving a key theorem by Fabien Morel that relates strongly and strictly A^1-invariant sheaves over perfect fields, and explores their applications.

Contribution

It provides a complete proof of Morel's theorem and outlines applications of strongly A^1-invariant sheaves in motivic homotopy theory.

Findings

Proof of Morel's theorem confirming the invariance property

Clarification of the relationship between strongly and strictly A^1-invariant sheaves

Outline of applications in motivic homotopy theory

Abstract

Strongly (respectively strictly) A1-invariant sheaves are foundational for motivic homotopy theory over fields. They are sheaves of (abelian) groups on the Nisnevich site of smooth varieties over a field k, with the property that their zeroth and first Nisnevich cohomology sets (respectively all Nisnevich cohomology groups) are invariant under replacing a variety X by the affine line over X. A celebrated theorem of Fabien Morel states that if the base field k is perfect, then any strongly A1-invariant sheaf of abelian groups is automatically strictly A1-invariant. The aim of these lecture notes is twofold: (1) provide a complete proof if this result, and (2) outline some of its applications.