B-admissible representation explained

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.^[1]

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 ⊗} ^{G} ^{EV)} is E-linear (Mod_F denotes the category of F-modules). The inclusion of D_B(V) in B ⊗_EV induces a homomorphism\alpha _{B,V} :B ⊗ _{FD} _{B(V)\longrightarrow} B ⊗ _{EV} called the comparison morphism.^[2]

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

An (E, G)-ring B is called regular if

1. B is reduced;
2. for every V in Rep(G), α_B,V is injective;
3. 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 _{FD} _{B(V)\leq\dim} _{EV} 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

Notes and References

1. Of course, the entire category of representations can be taken, but this generality allows, for example if G and E have topologies, to only consider continuous representations.
2. A contravariant formalism can also be defined. In this case, the functor used is D ^{\ast(V): = Hom} _{G(V,B)} , the G-invariant linear homomorphisms from V to B.