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 , 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.

Key Words:

• Coalgebras
• Localization

Mathematics Subject Classification:

• Primary 16
• Secondary 16, 13

ACKNOWLEDGMENTS

Research partially supported by BMF2001-2823, MTM2004-08125, and FQM-266 (Junta de Andalucía Research Group).

Notes

Communicated by R. Wisbauer.