B-admissible representation

Last updated March 12, 2019

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.

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(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]

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In mathematics, the tensor product $V \otimes W$ of two vector spaces $V$ and $W$ is itself a vector space, endowed with the operation of bilinear composition, denoted by $\otimes$ , from ordered pairs in the Cartesian product $V \times W$ onto $V \otimes W$ in a way that generalizes the outer product. The tensor product of $V$ and $W$ is the vector space generated by the symbols $v \otimes w$ , with $v \in V$ and $w \in W$ , in which the relations of bilinearity are imposed for the product operation $\otimes$ , and no other relations are assumed to hold. The tensor product space is thus the "freest" such vector space, in the sense of having the fewest constraints.

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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

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

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 ⊗_EV induces a homomorphism

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring and a multiplication is defined between elements of the ring and elements of the module.

\alpha _{B,V}:B\otimes _{F}D_{B}(V)\longrightarrow B\otimes _{E}V

called the comparison morphism.^[2]

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.

In mathematics, an isomorphism is a homomorphism or morphism that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

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.

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In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is automorphism. For example, an endomorphism of a vector space $V$ is a linear map $f : V \to V$ , and an endomorphism of a group $G$ is a group homomorphism $f : G \to G$ . In general, we can talk about endomorphisms in any category. In the category of sets, endomorphisms are functions from a set S to itself.

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.

Notes

↑ 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.
↑ A contravariant formalism can also be defined. In this case, the functor used is $D_{B}^{\ast }(V):=\mathrm {Hom} _{G}(V,B)$ , the G-invariant linear homomorphisms from V to B.

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References

Fontaine, Jean-Marc (1994), "Représentations p-adiques semi-stables", in Fontaine, Jean-Marc, Périodes p-adiques, Astérisque, 223, Paris: Société Mathématique de France, pp. 113–184, MR 1293969

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[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 $D_{B}^{\ast }(V):=\mathrm {Hom} _{G}(V,B)$ , the G-invariant linear homomorphisms from V to B.