On the category of comodules over corings Robert Wisbauer Abstract It is well known that the category M C of right comodules over an A-coring C , A an associative ring, is a subcategory of the category of left modules * C M over the dual ring ∗ C . The main purpose of this note is to show that M C is a full subcatgeory in * C M if and only if C is locally projective as a left A-module. 1 Introduction For any coassociative coalgebra C over a commutative ring R, the convolu- tion product turns the dual module C ∗ = Hom R (C,R) into an associative R-algebra. The category M C of right comodules is an additive subcate- gory of the category C * M of left C ∗ -modules. M C is an abelian (in fact a Grothendieck) category if and only if C is flat as an R-module. Moreover, M C coincides with C * M if and only if C is finitely generated and projective as an R-module (e.g. [11, Corollary 33]). In case C is projective as an R-module, M C is a full subcategory of C * M and coincides with σ[ C * C ], the category of submodules of C -generated C ∗ -modules (e.g. [9, 3.15, 4.3]). It was well understood from examples that projectivity of C as an R-module was not necessary to achieve M C = σ[ C * C ] and that the equality holds provided C satisfies the α-condition, i.e., the canonical maps N ⊗ R C → Hom ZZ (C ∗ ,N ) are injective for all R-modules N (e.g. [1, Satz 2.2.13], [2, Section 2], [10, 3.2]). It will follow from our results that this condition is in fact equivalent to M C = σ[ C * C ] and also to C being locally projective as an R-module. We do investigate the questions and results mentioned above in the more general case of comodules over any A-coring, A an associative ring, and it will turn out that the above observations remain valid almost literally in this extended setting. 1