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Proceedings of the American Mathematical Society Series B

ISSN 2330-1511



Homotopy theory of modules over diagrams of rings

Authors: J. P. C. Greenlees and B. Shipley
Journal: Proc. Amer. Math. Soc. Ser. B 1 (2014), 89-104
MSC (2010): Primary 55U35, 55P60, 55P42, 55P91
Published electronically: September 3, 2014
MathSciNet review: 3254575
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Abstract: Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories $ \mathcal {M}(s)$ (as $ s$ runs through the diagram), we consider the category of diagrams where the object $ X(s)$ at $ s$ comes from $ \mathcal {M}(s)$. We develop model structures on such categories of diagrams and Quillen adjunctions that relate categories based on different diagram shapes.

Under certain conditions, cellularizations (or right Bousfield localizations) of these adjunctions induce Quillen equivalences. As an application we show that a cellularization of a category of modules over a diagram of ring spectra (or differential graded rings) is Quillen equivalent to modules over the associated inverse limit of the rings. Another application of the general machinery here is given in work by the authors on algebraic models of rational equivariant spectra. Some of this material originally appeared in the preprint ``An algebraic model for rational torus-equivariant stable homotopy theory'', arXiv:1101.2511, but has been generalized here.

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Additional Information

J. P. C. Greenlees
Affiliation: Department of Pure Mathematics, The Hicks Building, University of Sheffield, Sheffield, S3 7RH, United Kingdom

B. Shipley
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 508 SEO m/c 249, 851 S. Morgan Street, Chicago, Illinois 60607-7045

Received by editor(s): September 26, 2013
Received by editor(s) in revised form: November 10, 2013, and March 20, 2014
Published electronically: September 3, 2014
Additional Notes: The first author is grateful for support under EPSRC grant No. EP/H040692/1
This material is based upon work by the second author supported by the National Science Foundation under grant No. DMS-1104396
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2014 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)

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