I have been reading [Asok and Doran - $\mathbb A^1$-homotopy groups, excision, and solvable quotients](https://reader.elsevier.com/reader/sd/pii/S0001870809000450). In the Example 2.17, page 1155, there is a claim that for $m>1$ and $n\geq 1$, $\pi\_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu\_n)\simeq\mathbb{G}\_m/\mathbb{G}\_m^n$.


Let me explain the notation quickly:


$\mu\_n$ is the group scheme of $n^\text{th}$ roots of unity acting on $\mathbb{A}^m-0$ by diagonal action. So $\mathbb{A}^m-0/\mu\_n$ is a smooth scheme. $\mathbb{G}\_m/\mathbb{G}\_m^n$ is the cokernel of the $n^\text{th}$-power map $\mathbb{G}\_m\xrightarrow{(-)^n} \mathbb{G}\_m$ in the category of Nisnevich sheaves of abelian groups.


My question is how to see the isomorphism $\pi\_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu\_n)\simeq\mathbb{G}\_m/\mathbb{G}\_m^n$?


Comments are most welcome!



$\pi_0^{\mathbb{A}^1}(\mathbb{A}^m-0/\mu_n)\simeq\mathbb{G}_m/\mathbb{G}_m^n$
Over a field of characteristic coprime to $n$, the classifying space ${\rm B}\_{\rm et}\mu\_n$ is $\mathbb{A}^1$-local. In this situation, any $\mu\_n$-covering $\widetilde{X}\to X$ is an $\mathbb{A}^1$-covering space and provides a classifying morphism $X\to {\rm B}\_{\rm et}\mu\_n$. From the Kummer sequence $\mu\_n\to \mathbb{G}\_{\rm m}\to \mathbb{G}\_{\rm m}$ we get a fiber sequence ${\rm B}\_{\rm et}\mu\_n\to {\rm B}\mathbb{G}\_{\rm m}\to {\rm B}\mathbb{G}\_{\rm m}$. This implies that the sheaf of $\mathbb{A}^1$-connected components is $\mathbb{G}\_{\rm m}/\mathbb{G}\_{\rm m}^n$ (the Nisnevich sheaf quotient) which we can alternatively write as the Nisnevich sheafification of ${\rm H}^1\_{\rm et}(-,\mu\_n)$. (This is in the Morel-Voevodsky paper.)


Now turn to $\mathbb{A}^m\setminus\{0\}$. The $\mu\_n$-covering $\mathbb{A}^m\setminus\{0\}\to\mathbb{A}^m\setminus\{0\}/\mu\_n$ is classified by a map $\mathbb{A}^m\setminus\{0\}/\mu\_n\to {\rm B}\_{\rm et}\mu\_n$ and the more precise claim is that the induced map on $\pi\_0^{\mathbb{A}^1}$ is an isomorphism of Nisnevich sheaves.


For surjectivity, we consider an embedding $\mathbb{G}\_{\rm m}\to\mathbb{A}^m\setminus\{0\}$ as $i$-th component. This is a $\mu\_n$-equivariant map if we take the componentwise action on $\mathbb{A}^m\setminus\{0\}$. Therefore, we get an induced map on quotient schemes $\mathbb{G}\_{\rm m}\to \mathbb{A}^m\setminus\{0\}/\mu\_n$. On sheaves of $\mathbb{A}^1$-connected components we get induced maps
$$
\mathbb{G}\_{\rm m}\to \pi\_0^{\mathbb{A}^1}\mathbb{A}^m\setminus\{0\}/\mu\_n\to \mathbb{G}\_{\rm m}/\mathbb{G}\_{\rm m}^n
$$
such that the composition is the natural quotient map. This proves the surjectivity.


We check injectivity locally in the Nisnevich topology. For that, let $R$ be a henselian local ring. A morphism ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}/\mu\_n$ is represented by a tuple $(x\_1,\dots,x\_m)$ with $x\_i\in R$ such that at least one of the $x\_i$ is not in the maximal ideal of $R$. Two such tuples $(x\_1,\dots,x\_m)$ and $(y\_1,\dots,y\_m)$ give the same morphism if the tuples $(x\_1^n,\dots,x\_m^n)$ and $(y\_1^n,\dots,y\_m^n)$ agree. Now we want to show that any map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu\_n$ (corresponding to a tuple $(x\_1,\dots,x\_m)$) factors through a coordinate embedding $\mathbb{G}\_{\rm m}\to\mathbb{A}^m\setminus\{0\}/\mu\_n$ used before. One entry $x\_i$ is not contained in the maximal ideal, assume it's the first. Then $(x\_1,t\cdot x\_2,\dots,t\cdot x\_m)$ provides a naive $\mathbb{A}^1$-homotopy from the given tuple to $(x\_1,0,\dots,0)$. So up to $\mathbb{A}^1$-homotopy, any map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu\_n$ factors through a coordinate embedding $\mathbb{G}\_{\rm m}\to \mathbb{A}^m\setminus\{0\}$. If such a homotopy class maps to the base point of $\mathbb{G}\_{\rm m}/\mathbb{G}\_{\rm m}^n$ then by the identifications above, it must lift through the $\mu\_n$-covering $\mathbb{G}\_{\rm m}\xrightarrow{n} \mathbb{G}\_{\rm m}$. But since we have a morphism from the covering $\mathbb{G}\_{\rm m}\to \mathbb{G}\_{\rm m}$ to the covering $\mathbb{A}^m\setminus\{0\}\to \mathbb{A}^m\setminus\{0\}/\mu\_n$, the map ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}/\mu\_n$ lifts (up to $\mathbb{A}^1$-homotopy) through a map ${\rm Spec}R\to \mathbb{A}^m\setminus\{0\}$. But the target there is $\mathbb{A}^1$-connected, meaning that the original map ${\rm Spec} R\to \mathbb{A}^m\setminus\{0\}/\mu\_n$ must be null-homotopic, showing injectivity.


