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 , 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  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
Mathematische Zeitschrift – Springer Journals
Published: Jun 5, 2018
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