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This is a repost of my question here. Do we know the structure of the étale fundamental group $\pi^\text{et}_1(\mathbb{G}_{m,K^\text{sep}})$ of the multiplicative group, for a given field $K$? For ...

Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...

Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,...,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. ...

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map: $$ \pi_1^{ab}(...