Tertiary decomposition in Grothendieck categories
The category of groupoid graded modules
We introduce the abelian category R-gr of groupoid graded modules and give an answer to the following general question: If U: R-gr → R-mod denotes the functor which associates to any graded left R-module M the underlying ungraded structure U(M), when does either of the following two implications hold: (I) M has property X ⇒ U(M) has property X; (II) U(M) has property X ⇒ M has property X? We treat the cases when X is one of the properties: direct summand, free, finitely generated, finitely presented,...
The Hermitian Morita theorems.
Topological linear compactness for Grothendieck categories. Theorem of Tychonoff. Applications to coalgebras.
We show the Tychonoff's theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra.