Support theory via actions of tensor triangulated categories
type: Item
instance of: scholarly article
title: Support theory via actions of tensor triangulated categories
zbMATH Open document ID: 1280.18010
author: Greg Stevenson
publication date: 2013-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1105.4692
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 18E35
zbMATH DE Number: 6211855
zbMATH Keywords: tensor triangulated category
zbMATH Keywords: support
zbMATH Keywords: localization
zbMATH Keywords: telescope conjecture
zbMATH Keywords: local-to-global-principle
MaRDI profile type: Publication
OpenAlex ID: W2963065974
arXiv ID: 1105.4692
DOI: 10.1515/CRELLE-2012-0025
published in: Journal für die Reine und Angewandte Mathematik
review text: Let \(\mathcal K\) be a triangulated category and \(\mathcal T\) be a tensor triangulated category. In the first part of the paper the author gives a definition of what is meant by the action of \(\mathcal T\) on \(\mathcal K\). In the case when \(\mathcal T\) is rigidly-compactly generated and \(\mathcal K\) is compactly generated, it is shown that this action gives rise to a notion of supports which gives the possibility to extend recent work of D. J.~Benson, S. K.~Iyengar and H.~Krause and of P.~Balmer and G.~Favi. The author states a suitable version of the local-to-global principle (originally due to Benson, Iyengar and Krause) for actions of triangulated categories which is shown to hold under very general assumptions. Next, he formulates a relative version of the telescope conjecture (roughly saying that certain smashing subcategories of \(\mathcal K\) are generated by compact objects of \(\mathcal K\)), and gives sufficient conditions for it to hold.
reviewed by: Andrei Marcus
description: scientific article; zbMATH DE number 6211855