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{{DISPLAYTITLE:''B''-admissible representation}}
In [[mathematics]], the formalism of '''''B''-admissible representations''' provides constructions of [[full subcategory|full]] [[Tannakian category|Tannakian]] [[subcategory|subcategories]] of the category of [[group representation|representations]] of a [[group (mathematics)|group]] ''G'' on [[finite-dimensional]] [[vector space]]s over a given [[field (mathematics)|field]] ''E''. In this theory, ''B'' is chosen to be a so-called '''(''E'', ''G'')-regular ring''', i.e. an [[Algebra over a ring|''E''-algebra]] with an [[linear representation|''E''-linear action]] of ''G'' satisfying certain conditions given below. This theory is most prominently used in [[p-adic Hodge theory|''p''-adic Hodge theory]] to define important subcategories of [[Galois representation|''p''-adic Galois representations]] of the [[absolute Galois group]] of [[local field|local]] and [[global field]]s.

==(''E'', ''G'')-rings and the functor ''D''==
Let ''G'' be a group and ''E'' a field. Let Rep(''G'') denote a non-trivial [[strictly full subcategory]] of the Tannakian category of ''E''-linear representations of ''G'' on finite-dimensional vector spaces over ''E'' stable under [[subobject]]s, [[quotient object]]s, [[direct sum of representations|direct sums]], [[tensor product]]s, and [[dual representation|duals]].<ref>Of course, the entire category of representations can be taken, but this generality allows, for example if ''G'' and ''E'' have [[topological space|topologies]], to only consider [[continuous (topology)|continuous]] representations.</ref>

An '''(''E'', ''G'')-ring''' is a [[commutative ring]] ''B'' that is an ''E''-algebra with an ''E''-linear action of ''G''. Let ''F'' = ''BG'' be the [[G-invariant|''G''-invariants]] of ''B''. The [[covariant functor]] ''DB'' : Rep(''G'') → Mod''F'' defined by
:<math>D_B(V):=(B\otimes_EV)^G</math>
is ''E''-linear (Mod''F'' denotes the category of [[module (mathematics)|''F''-modules]]). The inclusion of ''DB''(V) in ''B'' ⊗''EV'' induces a homomorphism
:<math>\alpha_{B,V}:B\otimes_FD_B(V)\longrightarrow B\otimes_EV</math>
called the '''comparison morphism'''.<ref>A [[contravariant functor|contravariant]] formalism can also be defined. In this case, the functor used is <math>D_B^\ast(V):=\mathrm{Hom}_G(V,B)</math>, the ''G''-invariant linear homomorphisms from ''V'' to ''B''.</ref>

==Regular (''E'', ''G'')-rings and ''B''-admissible representations==
An (''E'', ''G'')-ring ''B'' is called '''regular''' if
#''B'' is [[reduced ring|reduced]];
#for every ''V'' in Rep(''G''), α''B,V'' is [[injective function|injective]];
#every ''b'' ∈ ''B'' for which the line ''bE'' is ''G''-stable is [[invertible element|invertible]] in ''B''.
The third condition implies ''F'' is a field. If ''B'' is a field, it is automatically regular.

When ''B'' is regular,
:<math>\dim_FD_B(V)\leq\dim_EV</math>
with equality if, and only if, α''B,V'' is an [[isomorphism]].

A representation ''V'' ∈ Rep(''G'') is called '''''B''-admissible''' if α''B,V'' is an isomorphism. The full subcategory of ''B''-admissible representations, denoted Rep''B''(''G''), is Tannakian.

If ''B'' has extra structure, such as a [[filtration (mathematics)|filtration]] or an ''E''-linear [[endomorphism]], then ''DB''(''V'') inherits this structure and the functor ''DB'' can be viewed as taking values in the corresponding category.

==Examples==
*Let ''K'' be a field of [[characteristic (algebra)|characteristic]] ''p'' (a prime), and ''Ks'' a [[separable closure]] of ''K''. If ''E'' = '''F'''''p'' (the [[finite field]] with ''p'' elements) and ''G'' = Gal(''Ks''/''K'') (the absolute Galois group of ''K''), then ''B'' = ''Ks'' is a regular (''E'', ''G'')-ring. On ''Ks'' there is an injective [[Frobenius endomorphism]] σ : ''Ks'' → ''Ks'' sending ''x'' to ''xp''. Given a representation ''G'' → GL(''V'') for some finite-dimensional '''F'''''p''-vector space ''V'', <math>D=D_{K_s}(V)</math> is a finite-dimensional vector space over ''F''=(''K''''s'')''G'' = ''K'' which inherits from ''B'' = ''Ks'' an injective function φ''D'' : ''D'' → ''D'' which is σ-semilinear (i.e. φ(''ad'') = σ(''a'')φ(''d'') for all a ∈ ''K'' and all d ∈ ''D''). The ''Ks''-admissible representations are the continuous ones (where ''G'' has the [[Krull topology]] and ''V'' has the [[discrete topology]]). In fact, <math>D_{K_s}</math> is an [[equivalence of categories]] between the ''Ks''-admissible representations (i.e. continuous ones) and the finite-dimensional vector spaces over ''K'' equipped with an injective σ-semilinear φ.

==Potentially ''B''-admissible representations==
A '''potentially ''B''-admissible representation''' captures the idea of a representation that becomes ''B''-admissible when [[restriction (mathematics)|restricted]] to some [[subgroup]] of ''G''.

==Notes==
{{reflist}}

==References==
*{{Citation
| last=Fontaine
| first=Jean-Marc
| author-link=Jean-Marc Fontaine
| contribution=Représentations ''p''-adiques semi-stables
| editor-last=Fontaine
| editor-first=Jean-Marc
| editor-link=Jean-Marc Fontaine
| title=Périodes p-adiques
| publisher=Société Mathématique de France
| location=Paris
| year=1994
| mr=1293969
| series=Astérisque
| volume=223
| pages=113–184
}}

[[Category:Representation theory of groups]]

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