Localization in Coalgebras. Stable Localizations and Path Coalgebras

Localization in right (*)-serial coalgebras

“…Meanwhile, two idempotent elements e, f ∈ C * are equivalent, e ∼ f, if the injective right C-comodules Ce and Cf are equivalent. Thus, there is a bijective correspondence between the localizing subcategories of M C and the equivalent classes of idempotent elements in C * (see also [8,10,21]). Throughout, we abbreviate T e to stand for the localizing subcategory T Ce of M C .…”

Serial coalgebras and their valued Gabriel quivers

“…[24]) is given by (1) e ⊗ ex (2) e and eCe (exe) = C (x) (2) using the sigma-notation of [31]. Throughout we denote by T e the localizing subcategory associated to the idempotent e. For completeness, we recall from [4] (see also [15]) the following description of the localizing functors. We recall that, given an idempotent e ∈ C * , for each right C-comodule M, the vector space eM is endowed with a structure of right eCe-comodule given by (1) ⊗ x (0) using the sigma-notation of [31].…”

“…In [4,15] and [33], localizing subcategories are described by means of idempotents in the dual algebra C * . In particular, it is proved that the quotient category M C /T is the category of right comodules over the coalgebra eCe, where e ∈ C * is an idempotent associated to the localizing subcategory T (that is, E = Ce, where E is the injective right C-comodule associated to the localizing subcategory T ).…”

On path coalgebras of quivers with relations

“…For this purpose, in Section 2, we establish a general framework using the weak * topology on the dual algebra to treat the problem in an elementary context. In Section 3, a result of [7] allows us to obtain a more manageable basis of a relation coalgebra which we use in Section 4 to give a criterion for deciding whether or not a relation subcoalgebra is a path coalgebra of a quiver with relations.…”

“…The aim of this section is to obtain a more manageable basis for a relation subcoalgebra of a path coalgebra. For more information and technical properties of subcoalgebras see [7].…”