Inter-universal geometry と ABC予想 (応援スレ) 44

Inter-universal geometry と ABC予想 (応援スレ) 44at MATH

Inter-universal geometry と ABC予想 (応援スレ) 44 - 暇つぶし2ch602:めん）記号のコピペミスを直すのは面倒なので正確にはpdfを見てね >Note that perfectoid fields can be of characteristic 0 or p. ということらしい Perfectoid Spaces and their Applications https://www.math.uni-bonn.de/people/scholze/ICM.pdf >Definition 3.1. A perfectoid field is a complete topological field K, whose topology comes from a nonarchimedean norm | · | : K → R≥0 with dense image, such >that |p| < 1 and, letting OK = {x ∈ K | |x| ≤ 1} be the ring of integers, the >Frobenius map Φ : OK/p → OK/p is surjective. >Examples include the completions of Qp(p1/p∞), Qp(µp∞), Qp and Fp((t))(t1/p∞), >Fp((t)). Note that perfectoid fields can be of characteristic 0 or p. In the first case, >they contain Qp naturally, as |p| < 1. Note that Qp is not a perfectoid field (although Zp/p = Fp has a surjective Frobenius map), because | · | : Qp → R≥0 has >discrete image 0 ∪ pZ ⊂ R≥0. In characteristic p, perfectoid fields are the same >thing as perfect complete nonarchimedean fields. >By a construction of Fontaine, one can take any perfectoid field K, and produce >a perfectoid field K[ of characteristic p, called the tilt of K. First, one defines >OK[ = lim←－ΦOK/p, and then defines K[ as the fraction field of OK[ . It comes with >a natural norm, with respect to which OK[ ⊂ K[is the ring of integers. In fact, >one has the following alternative description of K[.

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