Corrigendum to “$\mathbb {A}^1$-connectedness in reductive algebraic groups”

Strong A1${\mathbb {A}}^1$‐invariance of A1${\mathbb {A}}^1$‐connected components of reductive algebraic groups

“…Combining Theorem 3.2 with the results of Asok–Hoyois–Wendt and Gille, we get the following generalizations of [10, Theorem 3.6] and [10, Theorem 1], which we record below. Corollary Let $$k$$ be an arbitrary field and let $$G$$ be a semisimple, simply connected algebraic group over $$k$$.…”

“…In the case $$G$$ is anisotropic, the group $$G(k)^+$$ is trivial. However, it was shown in [9] (see also [10, Theorem 3.6]) that over an infinite perfect field $$k$$, there is a bijection $$$$\begin{equation*} \pi _0^{{\mathbb {A}}^1}(G)(k) \xrightarrow {\simeq } G(k)/R. \end{equation*}$$$$We first observe that the proof of this result given in [9] can be appropriately modified to make it work over an arbitrary base field.…”

“…Proof As in [10, Proof of Theorem 3.6], we are reduced to showing that $$\pi _0^{{\mathbb {A}}^1}(G)(k) \rightarrow G(k)/R$$ is a bijection, where $$G$$ is semisimple, simply connected and absolutely almost simple over $$k$$. If $$G$$ is isotropic, then the desired result follows from (3.1).…”

“…Since $$\pi _0^{{\mathbb {A}}^1}(G)$$ is a birational sheaf, there exists $$\beta \in G(X_L)$$ such that $$\beta |_{U_L}$$ and $$\widetilde{\alpha }$$ map to the same element in $$\pi _0^{{\mathbb {A}}^1}(G)(U_L)$$. We will identify sections of $$\pi _0^{{\mathbb {A}}^1}(G)$$ over fields with $$R$$‐equivalence classes, using [9, Main Theorem] and [10, Theorem 3.6]. Hence, $$\beta |_{U_L}$$ and $$\widetilde{\alpha }$$ are $$R$$‐equivalent.…”