[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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424(2): 2021/02/13(土)19:25 ID:wXktx3pj(12/18) AAS
>>423
補足
”Etale fundamental group”で、下記の場合分け大事だね
外部ﾘﾝｸ:en.wikipedia.org
Etale fundamental group
Examples and theorems
Schemes over a field of characteristic zero（標数0）
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.
Schemes over a field of positive characteristic and the tame fundamental group（標数正で”tame”）
For an algebraically closed field k of positive characteristic, the results are different, since Artin?Schreier coverings exist in this situation. For example, the fundamental group of the affine line {\displaystyle \mathbf {A} _{k}^{1}}{\mathbf A}_{k}^{1} is not topologically finitely generated. The tame fundamental group of some scheme U is a quotient of the usual fundamental group of U which takes into account only covers that are tamely ramified along D, where X is some compactification and D is the complement of U in X.[3][4] For example, the tame fundamental group of the affine line is zero.
省1
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}.
The inverse Galois problem asks what groups can arise as fundamental groups (or Galois groups of field extensions). Anabelian geometry, for example Grothendieck's section conjecture, seeks to identify classes of varieties which are determined by their fundamental groups.[6]
Friedlander (1982) studies higher etale homotopy groups by means of the etale homotopy type of a scheme.
(引用終り)
省1
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.
省8
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