Notes on A^1-contractibility and A^1-excision

TL;DR

This paper establishes conditions under which smooth schemes over perfect fields are A^1-contractible and proves an excision theorem for A^1-homotopy sheaves, advancing the understanding of A^1-homotopy theory.

Contribution

It proves that A^1-n-connected smooth schemes over perfect fields are A^1-contractible and establishes an excision result for A^1-homotopy sheaves.

Findings

Smooth A^1-n-connected schemes are A^1-contractible.

An excision theorem for A^1-homotopy sheaves is proven.

Results deepen the understanding of A^1-homotopy theory.

Abstract

We prove that a smooth scheme of dimension $n$ over a perfect field is A^1-weakly equivalent to a point if it is A^1-n-connected. We also prove an excision result for A^1-homotopy sheaves over a perfect field.