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

Abstract

Given a K-coalgebra C and an injective left C-comodule E, we construct a coalgebra C E and fully faithful left exact embedding □ E : C E -Comod → C - Comod of comodule categories such that the image of □ E is the subcategory C - Comod E consisting of the comodules M with an injective presentation 0 → M → E 0 → E 1 , where E 0 and E 1 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 res E : C - Comod → C E - Comod . 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.