Filtrations and Growth of $\mathbb G$-modules

Author name not available (Why is that?)

Abstract: We investigate rational representations (

m a t h b b G

-modules) of affine group schemes

m a t h b b G

over a field of positive characteristic

p

. For any subspace

X s u b s e t m a t h c a l O (m a t h b b G)

, we consider the abelian subcategory

M o d (m a t h b b G, X) s u b s e t M o d (m a t h b b G)

of ``

X

-comodules" and the left exact functor

(-)_{X} : M o d (m a t h b b G) o M o d (m a t h b b G, X)

which is right adjoint to the inclusion functor. We employ ``ascending converging sequences"

X_{i}

of subspaces of

m a t h c a l O (m a t h b b G)

to provide functorial filtrations

M_{X_{i}}

of each

m a t h b b G

-module

M

. For any subspace

X s u b s e t m a t h c a l O (m a t h b b G)

,

M o d (m a t h b b G, X) s u b s e t M o d (m a t h b b G)

is naturally identified with the category

C o M o d (m a t h c a l O (m a t h b b G)_{X})

of comodules for the coalgebra

m a t h c a l O (m a t h b b G)_{X} s u b s e t X

. This construction gives a corrected version of the filtration of exponential degree for a linear algebraic group of exponential type. We provide two types of ascending, converging sequences of finite dimensional sub-coalgebras for an arbitrary affine group scheme, one is a generalization to all affine group schemes of Jantzen's truncated subcategories for reductive algebraic groups and the other is a generalization to all affine group schemes of the grading of polynomial functions for classical groups. Any ascending converging sequence

X_{i}

of subspaces of

m a t h c a l O (m a t h b b G)

provides a test for injectivity for

m a t h b b G

-modules. When each

X_{i}

is a sub-coalgebra of

m a t h c a l O (m a t h b b G)

,

X_{i}

determine a filtration of Hochschild cohomology. The subcategories

C o H t (m a t h b b G) s u b s e t M o d (m a t h b b G), C o F i n (m a t h b b G) s u b s e t M o d (m a t h b b G)

of

m a t h b b G

-modules of finite coheight and of cofinite

m a t h b b G

-modules do not depend upon the choice of

X_{i}

provided that each

X_{i}

finite dimensional.

Mathematics Subject Classification ID

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