by Ardizzoni, A.

The construction of the cotensor coalgebra for an “abelian monoidal” category $\mathcal{M}$ which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in $\mathcal{M}$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras $\widetilde{D}$, where D is a coseparable coalgebra in $\mathcal{M}$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $\widetilde{D}$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T^c_{D}(D\wedge_E D/D)$ if and only if it is a cotensor coalgebra $T^c_{D}(N)$, where N is a certain D-bicomodule in $\mathcal{M}$. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.

DOI: 10.1007/s10468-008-9089-2
Online Date: 5/30/2008
Print publication date: 10/1/2008
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