ABSTRACT
For a long period the theory of modules over rings on the one hand and comodules and Hopf modules for coalgebras and bialgebras on the other side developed quite independently. In this talk we want to outline how ideas from module theory can be applied to enrich the theory of comodules and vice versa. For this we consider A-corings C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter19_309_1.tif"/> with grouplike elements over a ring A, in particular Galois corings. If A is right self-injective it turns out that C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter19_309_2.tif"/> is a Galois coring if and only if for any injective comodule N the canonical map Hom C ( A , N ) ⊗ B A → N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter19_309_3.tif"/> is an isomorphism, where B = End C ( A ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter19_309_4.tif"/> , the ring of coinvariants of A. Together with flatness of bA this characterises A as generator in the category of right C https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187629/ac445a68-1635-4dfb-97a7-7653d5efb83a/content/inq_chapter19_309_5.tif"/> -comodules. This is a special case of the fact that over any ring A, an A-module M is a generator in the category σ[M] (objects are A-modules subgenerated by M) if and only if M is flat as module over its endomorphism ring S and the evaluation map M ⊗s Hom(M, N) → N is an isomorphism for injective modules N in σ[M].
