\mathbb{G}_m^n$

8 events

when toggle format	what		by	license	comment
May 24, 2021 at 5:17	vote	accept	Evans Gambit
May 17, 2021 at 11:26	answer	added	Matthias Wendt		timeline score: 3
May 11, 2021 at 3:46	comment	added	Evans Gambit		@MatthiasWendt: That will be great, thanks!
May 10, 2021 at 17:45	comment	added	Matthias Wendt		You're absolutely right, sorry about the $m=2$ mistake. Maybe I should try to give more detailed explanations in an answer, not a series of comments. Might take me some time though...
May 10, 2021 at 9:30	comment	added	Evans Gambit		@MatthiasWendt: Thanks a lot for your comments. For $m=2$, $\mathbb{A}^2-0$ is not $\mathbb{A}^1$-simply connected as $\pi_1^{\mathbb{A}^1}(\mathbb{A}^2-0)\simeq K_2^{MW}$. For $m>2$, could you please indicate how to see that the map $\mathbb{A}^m-0/\mu_n\to B_{et}\mu_n$, induces isomorphism on $\pi_0$. Also, I suppose for $m=2$ there should be a separate argument, then?
May 9, 2021 at 18:24	comment	added	Matthias Wendt		Here's a short sketch of the relevant steps: $\mathbb{A}^m-0/\mu_n$ is an approximation of the classifying space of $\mu_n$. For $m>1$ the universal covering $\mathbb{A}^m-0$ is $\mathbb{A}^1$-simply-connected, then the map $\mathbb{A}^m-0/\mu_n\to B_{et}\mu_n$ (which classifies the $\mu_n$-covering) induces an isomorphism on $\pi_0$. The sheaf of connected components of $B_{et}\mu_n$ is $H^1_{et}(-,\mu_n)\cong \mathbb{G}_m/\mathbb{G}_m^n$.
May 9, 2021 at 12:17	history	edited	LSpice	CC BY-SA 4.0	Name of this paper
May 9, 2021 at 10:58	history	asked	Evans Gambit	CC BY-SA 4.0