Localization in Coalgebras. Stable Localizations and Path Coalgebras

Abstract

We study localizing and colocalizing subcategories of a comodule category of a coalgebra C over a field, using the correspondence between localizing subcategories and equivalence classes of idempotent elements in the dual algebra C*. In this framework, we give a useful description of the localization functor by means of the Morita–Takeuchi context defined by the quasi-finite injective cogenerator of the localizing subcategory. Applying this description; first we characterize that a localizing subcategory 𝒯, with associated idempotent element e ∈ C*, is colocalizing if and only if eC is a quasi-finite eCe-comodule and, in addition, 𝒯 is perfect whenever eC is injective. And second, we prove that a localizing subcategory 𝒯 is stable if and only if e is a semicentral idempotent element of C*. We apply the theory to path coalgebras and obtain, in particular, that the “localized” coalgebra of a path coalgebra is again a path coalgebra.