Morita–Takeuchi Contexts Acting on Graded Coalgebras

Pseudo-dualizing complexes of bicomodules and pairs of t-structures

“…We recall that for any right C-comodule N and k-vector space V the vector space Hom k (N, V ) has a natural left C-contramodule structure [ Finiteness and quasi-finiteness conditions for coalgebras and comodules were studied in [29,30,10,21] and many other papers. The next two lemmas are well-known and included here for the reader's convenience.…”

“…(Such comodules were called "quasi-finite" in [29,10].) Recall that a left C-comodule is said to be finitely cogenerated [29,30,21] if it is a subcomodule of a cofree left C-comodule C ⊗ k V with a finite-dimensional vector space of cogenerators V . Any finitely cogenerated C-comodule is quasi-finitely cogenerated, while the cofree left C-comodule C ⊗ k V cogenerated by an infinitedimensional k-vector space V is not quasi-finitely cogenerated when C = 0.…”

“…Over a left quasi-co-Noetherian coalgebra C, quasifinitely cogenerated left C-comodules form an abelian category. Recall that a coalgebra C is said to be left co-Noetherian [30,10,21] if any quotient comodule of a finitely cogenerated left C-comodule is finitely cogenerated. It is clear from Proposition 2.3 that any left co-Noetherian coalgebra is left quasi-co-Noetherian.…”

Localization in Coalgebras: Applications to Finiteness Conditions

“…Co-noetherian comodules were investigated in [15] and [14]. In [14, Definition 3.2], strictly quasi-finite comodules are also called co-notetherian.…”

Dedualizing Complexes of Bicomodules and MGM Duality Over Coalgebras

“…The finiteness conditions on coalgebras were, of course, traditionally discussed in the language of comodules [37,6]. Some of the dual-analogous contramodule conditions lead to equivalent conditions on the coalgebra.…”