Localization on (necessarily) topological coalgebras and cohomology
type: Item
instance of: scholarly article
title: Localization on (necessarily) topological coalgebras and cohomology
zbMATH Open document ID: 0936.16032
published in: Journal of Algebra
publication date: 1999-11-16T00:00:00Z
review text: 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 \(_S\text{Hoch}^*(M,C)_S\).
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1230352
zbMATH Keywords: complete topological left comodules
zbMATH Keywords: localizations of topological coalgebras
zbMATH Keywords: completions
zbMATH Keywords: cohomology
zbMATH Keywords: Hochschild homology
zbMATH Keywords: local-global results
author: Marco A. Farinati
author: Andrea Solotar
MaRDI profile type: Publication
full work available at URL: jabr.1998.7490
OpenAlex ID: W2165982931
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DOI: 10.1006/JABR.1998.7490
description: scientific article; zbMATH DE number 1230352