The smash product for derived categories in stable homotopy theory
type: Item
instance of: scholarly article
title: The smash product for derived categories in stable homotopy theory
zbMATH Open document ID: 1246.55010
author: Michael A. Mandell
published in: Advances in Mathematics
publication date: 2012-07-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.0006
reviewed by: Friedrich Wilhelm Bauer
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D50
zbMATH DE Number: 6060946
zbMATH Keywords: formal group law
zbMATH Keywords: ring spectrum
MaRDI profile type: Publication
OpenAlex ID: W2001957061
arXiv ID: 1004.0006
cites work: Iterated monoidal categories
cites work: The multiplication on BP
cites work: Homotopy invariant algebraic structures on topological spaces
cites work: Tensor product of operads and iterated loop spaces
cites work: Q4716283
cites work: Koszul duality for operads
cites work: Symmetric spectra
cites work: Q4867806
cites work: Q3159161
cites work: Equivariant stable homotopy theory. With contributions by J. E. McClure
cites work: Modules in monoidal model categories
cites work: Model categories of diagram spectra
cites work: The geometry of iterated loop spaces
cites work: Pairings of categories and spectra
cites work: Q3147710
DOI: 10.1016/J.AIM.2012.04.012
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review text: Many decades ago, when we all were much younger, one got for several reasons interested in what was called ``higher categories''. In the approach of the present paper it started with the fact that certain multiplications where not strictly associative but only associative up to homotopy. In the course of time one came down with \(E_n\)-ring spectra \(R\), where \(E_n,\;n \in \mathbb N \cup \{ \infty\}\), denotes a hierarchy of structures. In a previous paper [The multiplication on \(BP\), \url{arXiv:1101.0023}] the present author together with \textit{M. Basterra} investigated the Brown-Peterson spectrum \(BP\), reaching the conclusion that \(BP\) is an \(E_4\) spectrum. Whether however \(BP\) is an \(E_\infty\) spectrum is not known.NEWLINENEWLINE Let \(R\) be a ring spectrum, then the homotopy category of the category of \(R\)-modules (the latter carrying the structure of a homotopy) is called the derived category \(\mathcal D_R\). There is a balanced product between the derived category of right and left \(R\)-modules \(\wedge_R\). An \(E_1\) ring spectrum has a derived category of left modules. An \(E_\infty\) ring spectrum has a derived category having a symmetric monoidal product.NEWLINENEWLINE The main issue of the present paper is the investigation of properties of \(E_n\) for low \(n\):NEWLINENEWLINE (1) Let \(R\) be an \(E_2\) ring spectrum.NEWLINENEWLINEThe derived category of left modules \(\mathcal D_R\) is equivalent to the derived category of right modules \(\mathcal D_{R^{op}}\) having a closed monoidal product NEWLINE\[NEWLINE\wedge_R: \mathcal D_R \times \mathcal D_R \longrightarrow \mathcal D_RNEWLINE\]NEWLINE extending the balanced product.NEWLINENEWLINE (2) If \(R\) is a \(E_3\) ring spectrum, \(\wedge_R\) has a braiding.NEWLINENEWLINE (3) If \(R\) is a \(E_4\) spectrum \(\mathcal D_R\) is a closed symmetric monoidal category.NEWLINENEWLINE The paper contains a wealth of technical details; the reviewer would wish for some more words about the mo…
description: scientific article; zbMATH DE number 6060946