現代数学の系譜工学物理雑談古典ガロア理論も読む79

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182: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/27(水) 07:49:43.49 ID:qnEhNItW

>>181

つづき

Contents
1 General classification of p-adic representations
2 Period rings and comparison isomorphisms in arithmetic geometry

General classification of p-adic representations
Let K be a local field with residue field k of characteristic p. In this article, a p-adic representation of K (or of GK, the absolute Galois group of K) will be a continuous representation ρ : GK→ GL(V), where V is a finite-dimensional vector space over Qp.
The collection of all p-adic representations of K form an abelian category denoted  \mathrm {Rep} _{\mathbf {Q} _{p}}(K)}{\mathrm  {Rep}}_{{{\mathbf  {Q}}_{p}}}(K) in this article.
p-adic Hodge theory provides subcollections of p-adic representations based on how nice they are, and also provides faithful functors to categories of linear algebraic objects that are easier to study. The basic classification is as follows:[2]

{Rep} _{\mathrm {cris} }(K)\subsetneq  {Rep} _{st}(K)\subsetneq  {Rep} _{dR}(K)\subsetneq  {Rep} _{HT}(K)\subsetneq  {Rep} _{\mathbf {Q} _{p}}(K)}
where each collection is a full subcategory properly contained in the next. In order, these are the categories of crystalline representations, semistable representations, de Rham representations, Hodge?Tate representations, and all p-adic representations.
In addition, two other categories of representations can be introduced, the potentially crystalline representations Reppcris(K) and the potentially semistable representations Reppst(K).
The latter strictly contains the former which in turn generally strictly contains Repcris(K); additionally, Reppst(K) generally strictly contains Repst(K), and is contained in RepdR(K) (with equality when the residue field of K is finite, a statement called the p-adic monodromy theorem).

つづく

http://rio2016.5ch.net/test/read.cgi/math/1573769803/182

183: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/27(水) 07:50:27.01 ID:qnEhNItW

>>182

つづき

Period rings and comparison isomorphisms in arithmetic geometry
The general strategy of p-adic Hodge theory, introduced by Fontaine, is to construct certain so-called period rings[3] such as BdR, Bst, Bcris, and BHT which have both an action by GK and some linear algebraic structure and to consider so-called Dieudonne modules

D_{B}(V)=(B\otimes _{\mathbf {Q} _{p}}V)^{G_{K}}}
(where B is a period ring, and V is a p-adic representation) which no longer have a GK-action, but are endowed with linear algebraic structures inherited from the ring B.
In particular, they are vector spaces over the fixed field  E:=B^{G_{K}}}E:=B^{{G_{K}}}.[4] This construction fits into the formalism of B-admissible representations introduced by Fontaine.
For a period ring like the aforementioned ones B? (for ? = HT, dR, st, cris), the category of p-adic representations Rep?(K) mentioned above is the category of B?-admissible ones, i.e. those p-adic representations V for which

\dim _{E}D_{B_{\ast }}(V)=\dim _{\mathbf {Q} _{p}}V}
or, equivalently, the comparison morphism

\alpha _{V}:B_{\ast }\otimes _{E}D_{B_{\ast }}(V)\longrightarrow B_{\ast }\otimes _{\mathbf {Q} _{p}}V}
is an isomorphism.

This formalism (and the name period ring) grew out of a few results and conjectures regarding comparison isomorphisms in arithmetic and complex geometry:

If X is a proper smooth scheme over C, there is a classical comparison isomorphism between the algebraic de Rham cohomology of X over C and the singular cohomology of X(C)
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1573769803/183

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