Localization in Coalgebras: Applications to Finiteness Conditions

Abstract

We introduce the notion of right strictly quasi-finite coalgebras, as coalgebras with the property that the class of quasi-finite right comodules is closed under factor comodules, and study its properties. A major tool in this study is the local techniques, in the sense of abstract localization.

Triangular matrix coalgebras and applications

“…It is also shown there that the second inclusion is strict ([GNT, Example 1.5]), and it is also remarked that the inclusion between Artinian comodules and quasi-finite comodules is also strict ( [GNT,Remark 1.7]). The three classes coincide over an almost connected coalgebra (i.e.…”

“…as left comodule). It also shows that a coalgebra can be left co-Noetherian but not right co-Noetherian, and left strictly quasi-finite but not right strictly quasi-finite (another example of right but not left strictly quasi-finite coalgebra is contained in [GNT,Example 2.7]). …”

When is a Coalgebra a Generator?

“…By [7] C is left coFrobenius so in particular is a fqcF-coalgebra. By Proposition 2.6 C is a generator in C M. Now, we prove that C is not a generator in M C .…”

“…Theorem B [7] Let C be a coalgebra. With notation as before the following statements are equivalent A it follows that T i (C) has finite length and by [7,Corollary 2.11] it follows that C is left semiperfect.…”

“…Section 3 is devoted to present a positive answer to our problem for left strictly quasi-finite coalgebras by using localizations in the category of comodules as developed in [7].…”