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 -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 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 -ring is a commutative ring B that is an E-algebra with an E-linear action of G. Let F = BG be the G-invariants of B. The covariant functor DB : Rep → ModF defined by
is E-linear. The inclusion of DB in B ⊗EV induces a homomorphism
called the comparison morphism.

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

An -ring B is called regular if
  1. B is reduced;
  2. for every V in Rep, αB,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,
with equality if, and only if, αB,V is an isomorphism.
A representation V ∈ Rep is called B-admissible if αB,V is an isomorphism. The full subcategory of B-admissible representations, denoted RepB, is Tannakian.
If B has extra structure, such as a filtration or an E-linear endomorphism, then DB inherits this structure and the functor DB can be viewed as taking values in the corresponding category.

Examples

A potentially B-admissible representation captures the idea of a representation that becomes B-admissible when restricted to some subgroup of G.