現代数学の系譜工学物理雑談古典ガロア理論も読む79

[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79 (1002ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

ﾘﾛｰﾄﾞ規制です｡10分ほどで解除するので､他のﾌﾞﾗｳｻﾞへ避難してください｡

405: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/14(土) 10:28:37.87 ID:s6Tab8iq

>>404
つづき

Introduction
Many writers have mused about algebraic geometry over deeper bases than the
ring Z of integers. Although there are several, possibly unrelated reasons for this,
here I will mention just two. The first is that the combinatorial nature of enumeration formulas in linear algebra over finite fields Fq as q tends to 1 suggests that,
just as one can work over all finite fields simultaneously by using algebraic geometry over Z, perhaps one could bring in the combinatorics of finite sets by working
over an even deeper base, one which somehow allows q = 1. It is common, following Tits [60], to call this mythical base F1, the field with one element. (See also
Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis.
With the analogy between integers and polynomials in mind, we might hope that
Spec Z would be a kind of curve over Spec F1, that Spec Z ?F1 Z would not only
make sense but be a surface bearing some kind of intersection theory, and that we
could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function
fields.1 Of course, since Z is the initial object in the category of rings, any theory
of algebraic geometry over a deeper base would have to leave the usual world of
rings and schemes.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1573769803/405

406: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/12/14(土) 10:29:03.86 ID:s6Tab8iq

>>405
つづき

The most obvious way of doing this is to consider weaker algebraic structures
than rings (commutative, as always), such as commutative monoids, and to try using them as the affine building blocks for a more rigid theory of algebraic geometry.
This has been pursued in a number of papers, which I will cite below. Another natural approach is motived by the following question, first articulated by Soul´e [57]:
Which rings over Z can be defined over F1? Less set-theoretically, on a ring over
Z, what should descent data to F1 be?
The main goal of this paper is to show that a reasonable answer to this question
is a Λ-ring structure, in the sense of Grothendieck’s Riemann?Roch theory [31].
More precisely, we show that a Λ-ring structure on a ring can be thought of as
descent data to a deeper base in the precise sense that it gives rise to a map from
the big ´etale topos of Spec Z to a Λ-equivariant version of the big ´etale topos of
Spec Z, and that this deeper base has many properties expected of the field with
one element. Not only does the resulting algebraic geometry fit into the supple
formalism of topos theory, it is also arithmetically rich?unlike the category of sets,
say, which is the deepest topos of all. For instance, it is closely related to global
class field theory, complex multiplication, and crystalline cohomology.
(引用終り)
以上

http://rio2016.5ch.net/test/read.cgi/math/1573769803/406

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.031s