[過去ﾛｸﾞ] Interｰuniversal geometryとABC予想(応用スレ)51 (1002ﾚｽ)
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376(1): 2021/02/11(木)21:33 ID:xRkvTpwx(20/21) AAS
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つづき

Examples and theorems
The most basic example of a fundamental group is π1(Spec k), the fundamental group of a field k. Essentially by definition, the fundamental group of k can be shown to be isomorphic to the absolute Galois group Gal (ksep / k). More precisely, the choice of a geometric point of Spec (k) is equivalent to giving a separably closed extension field K, and the fundamental group with respect to that base point identifies with the Galois group Gal (K / k). This interpretation of the Galois group is known as Grothendieck's Galois theory.

More generally, for any geometrically connected variety X over a field k (i.e., X is such that Xsep := X ×k ksep is connected) there is an exact sequence of profinite groups

1 → π1(Xsep, x) → π1(X, x) → Gal(ksep / k) → 1.

The pro-etale fundamental group
Bhatt & Scholze (2015, §7) have introduced a variant of the etale fundamental group called the pro-etale fundamental group. It is constructed by considering, instead of finite etale covers, maps which are both etale and satisfy the valuative criterion of properness. For geometrically unibranch schemes (e.g., normal schemes), the two approaches agree, but in general the pro-etale fundamental group is a finer invariant: its profinite completion is the etale fundamental group.
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