We give an explicit formula for the zeroth A¹-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.

I am grateful to Shuji Saito for supporting me to learn A¹-homotopy theory. I would like to thank Bruno Kahn for pointing out the necessity of a restriction on DVRs. I would like to thank Aravind Asok for helpful comments on the A¹-locality of classifying spaces. I would also like to thank Anand Sawant and Amit Hogadi for encouraging me to write up the paper. Finally, I would like to thank the referee for many helpful comments.