Abstract
Localisation is an important technique in ring theory and yields the construction of various rings of quotients. Colocalisation in comodule categories has been investigated by some authors (see Jara et al., Commun. Algebra, 34(8):2843–2856, 2006 and Nastasescu and Torrecillas, J. Algebra, 185:203–220, 1994). Here we look at possible coalgebra covers π : D → C that could play the rôle of a coalgebra colocalisation. Codense covers will dualise dense (or rational) extensions; a maximal codense cover construction for coalgebras with projective covers is proposed. We also look at a dual non-singularity concept for modules which turns out to be the comodule-theoretic property that turns the dual algebra of a coalgebra into a non-singular ring. As a corollary we deduce that hereditary coalgebras and hence path coalgebras are non-singular in the above sense. We also look at coprime coalgebras and Hopf algebras which are non-singular as coalgebras.
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Lomp, C., Rodrigues, V. Covering Coalgebras and Dual Non-singularity. Appl Categor Struct 16, 195–211 (2008). https://doi.org/10.1007/s10485-006-9059-y
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DOI: https://doi.org/10.1007/s10485-006-9059-y
Key words
- localisation of coalgebras
- non-singular coalgebras
- hereditary coalgebras
- path coalgebras
- copolyform modules
- maximal ring of quotients