Interｰuniversal geometryとABC予想(応用スレ)51

[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: １３２人目の素数さん [] 2021/02/13(土) 19:25:51 ID:wXktx3pj

>>423
補足
”Etale fundamental group”で、下記の場合分け大事だね

https://en.wikipedia.org/wiki/%C3%89tale_fundamental_group
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.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1610452199/424

425: １３２人目の素数さん [] 2021/02/13(土) 19:26:19 ID:wXktx3pj

>>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)
以上

http://rio2016.5ch.net/test/read.cgi/math/1610452199/425

426: １３２人目の素数さん [] 2021/02/13(土) 21:41:40 ID:wXktx3pj

>>424-425

機械翻訳

https://en.wikipedia.org/wiki/%C3%89tale_fundamental_group
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.

標数ゼロの場の上のスキーム
（Google訳（以下同じ））
複素数であるC上で有限型のスキームXの場合、エタール基本群Xと、Xに付加された複素解析空間であるX（C）の通常の位相幾何学的基本群との間には密接な関係があります。 この場合に一般的に呼ばれる代数的基本群は、π1（X）の有限型の完了です。 これは、リーマン存在定理の結果であり、X（C）のすべての有限エタール射影はXのものに由来するというものです。特に、C上の滑らかな曲線の基本群（つまり、開いたリーマン面）はよく理解されています。 ; これは代数的基本群を決定します。 より一般的には、標数的閉体の拡張が同型の基本群を誘発するため、標数的閉体の任意の代数的閉体に対する適切なスキームの基本群が知られています。

Schemes over a field of positive characteristic and the tame fundamental group
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  {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.

正の特性と飼いならされた基本群の分野にわたるスキーム
正の標数の代数的閉体kの場合、Artin-Schreier被覆がこの状況に存在するため、結果は異なります。 たとえば、アフィン線{A} _ {k} ^ {1}の基本群はトポロジー的に有限生成されません。 あるスキームUの飼いならされた基本群は、Dに沿って飼いならされた分岐のみを考慮したUの通常の基本群の商です。ここで、Xはコンパクト化であり、DはXのUの補数です。[3] [ 4] たとえば、アフィン線の飼いならされた基本群はゼロです。

つづく

http://rio2016.5ch.net/test/read.cgi/math/1610452199/426

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