Localization on (necessarily) topological coalgebras and cohomology
The development of cohomology theories for coalgebras has been hampered by the lack of a good concept of localization. The authors study localization of topological coalgebras. The topologies studied need not be linear, and the completions used depend on the topologies used. For a topological coalgebra \(C\), the localizing set \(S\) is a multiplicatively closed subset of the center \(Z(C')\) of the topological dual algebra. The localization \(C_{[S]}\) need not be a flat \(C\)-comodule, but conditions are given when it will be. The authors are interested in the cohomology \(\text{Hoch}^*\), the analogue of Hochschild homology of algebras, which commutes with their localization. After some examples, they study the category of complete topological left \(C\)-comodules and then study localization, proving some local-global results. The main theorem says that under some technical conditions, if \(C_{[S]}\) is \(C\)-coflat, and \(M\) a \(C_{[S]}\)-bicomodule (hence a \(C\)-bicomodule), then there is a natural isomorphism between \(\text{Hoch}^*(_SM_S,C_{[S]})\) and \(_S\text{Hoch}^*(M,C)_S\).
scientific article; zbMATH DE number 1230352