Vol. 61, No. 2, 1975

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On linear representations of affine groups. I

Manfred Wischnewsky

Vol. 61 (1975), No. 2, 551–572

The category of linear representations of an affine group is isomorphic to the category of comodules over a k-Hopf-algebra where k denotes a commutative ring. The category of C-comodules Comod-C over an arbitrary k-coalgebra C is comonadic over the category k-Mod of k-modules. It is complete, cocomplete and has a cogenerator. The C-comodules whose cardinality max (cardk, 0) generate the category Comod-C. Comod-C is in general not abelian but can nicely be embedded into an AB-4 category. Comod-C is a tensored and cotensored k-Mod-category (enriched over k-Mod) with a canonical (E,M)-factorization which is the factorization in k-mod if and only if C is flat. Comod-C has free C-comodules if and only if C is finitely generated and projective. Furthermore I give numerous examples and counterexamples as well as the explicit description of all constructions, in particular of the limits in Comod-C which was not known even for coalgebras over fields.

Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 18E05
Received: 16 May 1975
Published: 1 December 1975
Manfred Wischnewsky