現代数学の系譜工学物理雑談古典ガロア理論も読む63

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む63 (1002ﾚｽ)
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449: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/04/12(金) 20:57:59.67 ID:aUo1NtT0

>>448
>生成点（英語版）(generic point)

https://en.wikipedia.org/wiki/Generic_point
Generic point
(抜粋)
In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.

In scheme theory, the spectrum of an integral domain has a unique generic point, which is the minimal prime ideal.

Contents
1 Definition and motivation
2 Examples
3 History

History

In the foundational approach of Andre Weil, developed in his Foundations of Algebraic Geometry, generic points played an important role, but were handled in a different manner.
For an algebraic variety V over a field K, generic points of V were a whole class of points of V taking values in a universal domain Ω, an algebraically closed field containing K but also an infinite supply of fresh indeterminates.
This approach worked, without any need to deal directly with the topology of V (K-Zariski topology, that is), because the specializations could all be discussed at the field level (as in the valuation theory approach to algebraic geometry, popular in the 1930s).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1553946643/449

450: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [sage] 2019/04/12(金) 20:58:57.51 ID:aUo1NtT0

>>449

つづき

This was at a cost of there being a huge collection of equally generic points. Oscar Zariski, a colleague of Weil's at Sao Paulo just after World War II, always insisted that generic points should be unique. (This can be put back into topologists' terms: Weil's idea fails to give a Kolmogorov space and Zariski thinks in terms of the Kolmogorov quotient.)

In the rapid foundational changes of the 1950s Weil's approach became obsolete.
In scheme theory, though, from 1957, generic points returned: this time a la Zariski. For example for R a discrete valuation ring, Spec(R) consists of two points, a generic point (coming from the prime ideal {0}) and a closed point or special point coming from the unique maximal ideal.
For morphisms to Spec(R), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point.
Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists.
Other local rings have unique generic and special points, but a more complicated spectrum, since they represent general dimensions. The discrete valuation case is much like the complex unit disk, for these purposes.)
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1553946643/450

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