Abstract

For an ${\mathbb{𝔸}}^1$-connected pointed simplicial sheaf $\mathcal{𝒳}$ over a perfect field $k$, we prove that the Hurewicz map ${\pi }_1^{{\mathbb{𝔸}}^1}(\mathcal{𝒳})\to H_1^{{\mathbb{𝔸}}^1}(\mathcal{𝒳})$ is surjective. We also observe that the Hurewicz map for ${\mathbb{P}}_k^1$ is the abelianization map. In the course of proving this result, we also show that for any morphism $\varphi$ of strongly ${\mathbb{𝔸}}^1$-invariant sheaves of groups, the image and kernel of $\varphi$ are also strongly ${\mathbb{𝔸}}^1$-invariant.

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