Zeroth $\mathbb{A}^1$-homology of smooth proper varieties

TL;DR

This paper provides an explicit formula for the zeroth $ ext{A}^1$-homology sheaf of smooth proper varieties and offers a simplified proof of a key theorem related to birational localization.

Contribution

It introduces a direct formula for the zeroth $ ext{A}^1$-homology sheaf and simplifies the proof of a significant theorem in birational geometry.

Findings

Explicit formula for zeroth $ ext{A}^1$-homology sheaf

Simplified proof of Kahn-Sujatha theorem

Clarifies hom sets in birational localization

Abstract

We give an explicit formula for the zeroth $\mathbb{A}^1$-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of smooth varieties.