The algebraic chromatic splitting conjecture for Noetherian ring spectra

The algebraic chromatic splitting conjecture for Noetherian ring spectra Math. Z. (2018) 290:1359–1375 https://doi.org/10.1007/s00209-018-2066-5 Mathematische Zeitschrift The algebraic chromatic splitting conjecture for Noetherian ring spectra 1 2 3 Tobias Barthel · Drew Heard · Gabriel Valenzuela Received: 22 August 2016 / Accepted: 6 March 2018 / Published online: 5 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract We formulate a version of Hopkins’ chromatic splitting conjecture for an arbitrary structured ring spectrum R, and prove it whenever π R is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups. 1 Introduction In his seminal talk [20], Hopkins presents the global structure of the stable homotopy category in parallel to the structure of the derived category D of a Noetherian commutative ring R. In both cases, the thick subcategories of compact objects are classified in terms of a support theory, which in turn is based on a spectrum of certain prime objects. In the algebraic case, Neeman [29] shows that this spectrum can be taken to be the Zariski spectrum Spec(R) of prime ideals in R, while the corresponding result in homotopy theory has been worked out previously by Devinatz, Hopkins, and Smith [15,21]. Based http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

The algebraic chromatic splitting conjecture for Noetherian ring spectra

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2018 by Springer-Verlag GmbH Germany, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-018-2066-5
Publisher site
See Article on Publisher Site

Abstract

Math. Z. (2018) 290:1359–1375 https://doi.org/10.1007/s00209-018-2066-5 Mathematische Zeitschrift The algebraic chromatic splitting conjecture for Noetherian ring spectra 1 2 3 Tobias Barthel · Drew Heard · Gabriel Valenzuela Received: 22 August 2016 / Accepted: 6 March 2018 / Published online: 5 June 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract We formulate a version of Hopkins’ chromatic splitting conjecture for an arbitrary structured ring spectrum R, and prove it whenever π R is Noetherian. As an application, these results provide a new local-to-global principle in the modular representation theory of finite groups. 1 Introduction In his seminal talk [20], Hopkins presents the global structure of the stable homotopy category in parallel to the structure of the derived category D of a Noetherian commutative ring R. In both cases, the thick subcategories of compact objects are classified in terms of a support theory, which in turn is based on a spectrum of certain prime objects. In the algebraic case, Neeman [29] shows that this spectrum can be taken to be the Zariski spectrum Spec(R) of prime ideals in R, while the corresponding result in homotopy theory has been worked out previously by Devinatz, Hopkins, and Smith [15,21]. Based

Journal

Mathematische ZeitschriftSpringer Journals

Published: Jun 5, 2018

References

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