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

Abstract

We give an explicit formula for the zeroth 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.

Unramified logarithmic Hodge–Witt cohomology and -invariance

“…We close this introduction with a brief discussion of related works. Shimizu [27] and Koizumi [21] have obtained some results resembling our moving lemma (Theorem 3.5) in A 1 -homotopy theory. Ayoub [4] has considered the notion of P 1 -localisation, which is much more sophisticated than our P 𝑛 -Suslin complex introduced in Section 4.…”