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K-theoretic obstructions to bounded t-structures

  • Benjamin Antieau
  • David Gepner
  • Jeremiah Heller
Article
  • 61 Downloads

Abstract

Schlichting conjectured that the negative K-groups of small abelian categories vanish and proved this for noetherian abelian categories and for all abelian categories in degree \(-1\). The main results of this paper are that \(\mathrm {K}_{-1}(E)\) vanishes when E is a small stable \(\infty \)-category with a bounded t-structure and that \(\mathrm {K}_{-n}(E)\) vanishes for all \(n\geqslant 1\) when additionally the heart of E is noetherian. It follows that Barwick’s theorem of the heart holds for nonconnective K-theory spectra when the heart is noetherian. We give several applications, to non-existence results for bounded t-structures and stability conditions, to possible K-theoretic obstructions to the existence of the motivic t-structure, and to vanishing results for the negative K-groups of a large class of dg algebras and ring spectra.

Mathematics Subject Classification

Primary: 16E45 18E30 19D35 Secondary: 16P40 18E10 55P43 

Notes

Acknowledgements

BA and JH thank the Hausdorff Institute for Mathematics in Bonn and DG thanks the Max Planck Institute for Mathematics: these were our hosts during the summer of 2015, when this project was conceived. BA thanks Akhil Mathew for several conversations that summer at HIM, especially about bounded t-structures for compact modules over cochain algebras. BA thanks John Calabrese, Denis-Charles Cisinski, Michael Gröchenig, Jacob Lurie, Matthew Morrow, Marco Schlichting, Jesse Wolfson, and Matthew Woolf for conversations and emails about material related to this paper. DG thanks Andrew Blumberg and Markus Spitzweck for conversations about material related to this paper. Both BA and DG would especially like to thank Benjamin Hennion for explaining Tate objects and the subtleties behind excisive squares. We all are very grateful for detailed, helpful comments from an anonymous referee. We also thank the UIC Visitors’ Fund, Purdue University, UIUC, and Lars Hesselholt for supporting collaborative visits in 2016.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Benjamin Antieau
    • 1
  • David Gepner
    • 2
  • Jeremiah Heller
    • 3
  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA
  2. 2.School of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  3. 3.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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