Topological Hochschild homology and cohomology of \(A_{\infty }\) ring spectra
type: Item
instance of: scholarly article
title: Topological Hochschild homology and cohomology of \(A_{\infty }\) ring spectra
zbMATH Open document ID: 1149.55006
author: Vigleik Angeltveit
published in: Geometry \& Topology
publication date: 2008-05-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0612164
review text: The topological Hochschild (co)homology \(\text{THH}(A)\) of an \(A_\infty\) ring spectrum \(A\) can be defined after replacing \(A\) with a weakly equivalent \(S\)-algebra \(\widetilde A\). But, it is hard to see how the \(A_\infty\) structure on \(A\) affects the \(S\)-algebra structure on \(\widetilde A\). The author first gives an alternative definition of \(\text{THH}(A)\) in terms of associahedra and cyclohedra. Then the author gives an obstruction theory for endowing a spectrum with an \(A_\infty\) structure, and show that a large class of spectra can be given an \(A_\infty\) structure. Next the author defines two notions of commutativity for an \(A_n\) structure on a spectrum. It turns out that these notions are important to study \(\text{THH}(A)\). Finally, the author gives some sample calculations. The author also compares THH with Morava \(K\)-theory over different ground rings.
reviewed by: Yutaka Hemmi
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 18D50
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 55S35
zbMATH DE Number: 5275138
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: \(A_\infty\) structure
zbMATH Keywords: commutativity
zbMATH Keywords: associahedra
zbMATH Keywords: cyclohedra
zbMATH Keywords: Morava \(K\)-theory
MaRDI profile type: Publication
OpenAlex ID: W1981440585
arXiv ID: math/0612164
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DOI: 10.2140/GT.2008.12.987
description: scientific article