Module and Comodule Categories - a Survey
نویسنده
چکیده
The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule structure of algebras and comodule categories are sketched. 1. Module theory: Homological classification, the category σ[M ], Morita equivalence, the functor ring, Morita dualities, decompositions, torsion theories, trace functor. 2. Bimodule structure of an algebra: Multiplication algebra, Azumaya rings, biregular algebras, central closure of semiprime algebras. 3. Coalgebras and comodules: C-comodules and C∗-modules, σ-decomposition, rational functor, right semiperfect coalgebras, duality for comodules. 4. Bialgebras and bimodules: The category MB , coinvariants, B as projective generator inMB , fundamental theorem for Hopf algebras, semiperfect Hopf algebras. 5. Comodule algebras: (A-H)-bimodules, smash product A#H∗, coinvariants, A as progenerator in MA . 6. Group actions and module algebras: Group actions on algebras, A∗GA as a progenerator in σ[A∗GA], module algebras, smash product A#H, A#HA as a progenerator in σ[A#HA].
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