Localising embeddings of comodule categories with applications to tame and Euler coalgebras

Given a K-coalgebra C and an injective left C-comodule E, we construct a coalgebra CE and fully faithful left exact embedding of comodule categories such that the image of □E is the subcategory C-ComodE consisting of the comodules M with an injective presentation 0→M→E0→E1, where E0 and E1 are direct sums of direct summands of the comodule E. The functor □E preserves the indecomposability, the injectivity, and is right adjoint to the restriction functor . Applications to the study of tame coalgebras, Betti numbers, and cosyzygy comodules of simple comodules over a left Euler coalgebra C are given. A localising reduction to countably dimensional Euler coalgebras is presented.