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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 anexplicit Gersten-type complex. We show that if the base field is algebraicallyclosed then the zeroth stable $\mathbb{A}^1$-homotopy group of a smooth curvecoincides with the zeroth Suslin homology group that was identified by Suslinand Voevodsky with a relative Picard group. As a consequence we reobtain aversion of Suslin's rigidity theorem.

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