Localization in Coalgebras. Stable Localizations and Path Coalgebras
نویسندگان
چکیده
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 , with associated idempotent element e ∈ C∗, is colocalizing if and only if eC is a quasi-finite eCe-comodule and, in addition, is perfect whenever eC is injective. And second, we prove that a localizing subcategory 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.
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