On the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve

TL;DR

This paper offers a cohomological perspective on the zeroth stable $ extbf{A}^1$-homotopy group of smooth curves, linking it to Milnor--Witt K-theory and Suslin homology, and provides explicit computational tools.

Contribution

It introduces a cohomological interpretation of the zeroth stable $ extbf{A}^1$-homotopy group for smooth curves, connecting it to Milnor--Witt K-theory and Suslin homology, with explicit Gersten-type complexes.

Findings

The zeroth stable $ extbf{A}^1$-homotopy group is isomorphic to a first cohomology group of a Milnor--Witt related sheaf.

For algebraically closed fields, it coincides with the zeroth Suslin homology group.

Reobtains a version of Suslin's rigidity theorem.

Abstract

We provide a cohomological interpretation of the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve over an infinite perfect field. We show that this group is isomorphic to the first Nisnevich (or Zariski) cohomology group of a certain sheaf closely related to the first Milnor--Witt $\mathrm{K}$-theory sheaf. This cohomology group can be computed using an explicit Gersten-type complex. We show that if the base field is algebraically closed then the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curve coincides with the zeroth Suslin homology group that was identified by Suslin and Voevodsky with a relative Picard group. As a consequence we reobtain a version of Suslin's rigidity theorem.