# B-admissible representation

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

## (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 subobjects, quotient objects, direct sums, tensor products, and duals.
An (E, G)-ring is a commutative ring B that is an E-algebra with an E-linear action of G. Let F = B$^G$ be the G-invariants of B. The covariant functor D$_B$ : Rep(G) → Mod$_F$ defined by
    $D_{B}(V):=(B\otimes _{E}V)^{G}$
is E-linear (Mod$_F$ denotes the category of F-modules). The inclusion of D$_B$(V) in B ⊗$_E$V induces a homomorphism
    $\alpha _{B,V}:B\otimes _{F}D_{B}(V)\longrightarrow B\otimes _{E}V$
called the comparison morphism.

## Regular (E, G)-rings and B-admissible representations

An (E, G)-ring B is called regular if
* B is reduced;
* for every V in Rep(G), α$_{B,V}$ is injective;
* every b ∈ B for which the line bE is G-stable is invertible in B.
The third condition implies F is a field. If B is a field, it is automatically regular.
When B is regular,
    $\dim _{F}D_{B}(V)\leq \dim _{E}V$
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 or an E-linear endomorphism, then D$_B$(V) inherits this structure and the functor D$_B$ can be viewed as taking values in the corresponding category.

## Examples

* Let K be a field of characteristic p (a prime), and K$_s$ a separable closure of K. If E = F$_p$ (the finite field with p elements) and G = Gal(K$_s$/K) (the absolute Galois group of K), then B = K$_s$ is a regular (E, G)-ring. On K$_s$ there is an injective Frobenius endomorphism σ : K$_s$ → K$_s$ sending x to x$^p$. Given a representation G → GL(V) for some finite-dimensional F$_p$-vector space V, $D=D_{K_{s}}(V)$ is a finite-dimensional vector space over F=(K$_s$)$^G$ = K which inherits from B = K$_s$ 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 K$_s$-admissible representations are the continuous ones (where G has the Krull topology and V has the discrete topology). In fact, $D_{K_{s}}$ is an equivalence of categories between the K$_s$-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 restricted to some subgroup of G.