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Keywords:
coalgebra objects; Grothendieck categories
coalgebra objects; Grothendieck categories
Summary:
We develop a theory of coalgebra objects and comodules that is internal to any $k$-linear Grothendieck category, where $k$ is a commutative noetherian ring. We begin with a counterpart in $k$-linear Grothendieck categories for the finite dual construction of a $k$-algebra and the comodules over it. In the second part of the paper, we construct "coalgebra objects" inside a Grothendieck category. These are not coalgebras in an explicit sense, but enjoy several categorical properties arising in the classical theory of coalgebras, such as those of semiperfect or quasi-co-Frobenius coalgebras. In particular, this construction works in any Grothendieck category and there is no need for a monoidal structure in order to define these coalgebra objects.
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