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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<sup>G</sup> be the G-invariants of B. The covariant functor D<sub>B</sub> : Rep(G) → Mod<sub>F</sub> defined by

is E-linear (Mod<sub>F</sub> denotes the category of F-modules). The inclusion of D<sub>B</sub>(V) in B ⊗<sub>E</sub>V induces a homomorphism

called the comparison morphism.

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<sub>B,V</sub> 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,

with equality if, and only if, α<sub>B,V</sub> is an isomorphism.

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

If B has extra structure, such as a filtration or an E-linear endomorphism, then D<sub>B</sub>(V) inherits this structure and the functor D<sub>B</sub> can be viewed as taking values in the corresponding category.

Examples

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

<span id="potentiallyadmissible"></span>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

References