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" = "BG" be the "G"-invariants of "B". The covariant functor "DB" : Rep("G") → Mod"F" defined by
is "E"-linear (Mod"F" denotes the category of "F"-modules). The inclusion of "DB"(V) in "B" ⊗"EV" induces a homomorphism
called the comparison morphism.
Regular ("E", "G")-rings and "B"-admissible representations.
An ("E", "G")-ring "B" is called regular if
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"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 "DB"("V") inherits this structure and the functor "DB" can be viewed as taking values in the corresponding category.
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".