243 research outputs found
Calabi-Yau coalgebras
We provide a construction of minimal injective resolutions of simple
comodules of path coalgebras of quivers with relations. Dual to Calabi-Yau
condition of algebras, we introduce the Calabi-Yau condition to coalgebras.
Then we give some descriptions of Calabi-Yau coalgebras with lower global
dimensions. An appendix is included for listing some properties of cohom
functors
Dualities of artinian coalgebras with applications to noetherian complete algebras
A duality theorem of the bounded derived category of quasi-finite comodules
over an artinian coalgebra is established. Let be a noetherian complete
basic semiperfect algebra over an algebraically closed field, and be its
dual coalgebra. If is Artin-Schelter regular, then the local cohomology of
is isomorphic to a shift of twisted bimodule with
a coalgebra automorphism. This yields that the balanced dualinzing
complex of is a shift of the twisted bimodule . If
is an inner automorphism, then is Calabi-Yau
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
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