[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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425(1): 2021/02/13(土)19:26 ID:wXktx3pj(13/18) AAS
>>424
つづき

Affine schemes over a field of characteristic p （標数ｐで”Affine schemes”）
It turns out that every affine scheme {\displaystyle X\subset \mathbf {A} _{k}^{n}}{\displaystyle X\subset \mathbf {A} _{k}^{n}} is a {\displaystyle K(\pi ,1)}K(\pi ,1)-space, in the sense that the etale homotopy type of {\displaystyle X}X is entirely determined by its etale homotopy group.[5] Note {\displaystyle \pi =\pi _{1}^{et}(X,{\overline {x}})}{\displaystyle \pi =\pi _{1}^{et}(X,{\overline {x}})} where {\displaystyle {\overline {x}}}{\overline {x}} is a geometric point.

Further topics（”From a category-theoretic point of view”）
From a category-theoretic point of view, the fundamental group is a functor
{Pointed algebraic varieties} → {Profinite groups}.
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426(2): 2021/02/13(土)21:41 ID:wXktx3pj(14/18) AAS
>>424-425

機械翻訳

外部ﾘﾝｸ:en.wikipedia.org
Etale fundamental group

Examples and theorems

Schemes over a field of characteristic zero
For a scheme X that is of finite type over C, the complex numbers, there is a close relation between the etale fundamental group of X and the usual, topological, fundamental group of X(C), the complex analytic space attached to X. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π1(X). This is a consequence of the Riemann existence theorem, which says that all finite etale coverings of X(C) stem from ones of X. In particular, as the fundamental group of smooth curves over C (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups.
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