The \(\ell \)-adic trace formula for dg-categories and Bloch’s conductor conjecture

  • Bertrand Toën
  • Gabriele VezzosiEmail author


Building on the recent paper (Blanc et al. preprint, arXiv:1607.03012), we present an \(\ell \)-adic trace formula for smooth and proper dg-categories over a base \(\mathbb {E}_\infty \)-algebra B. We also give a variant when B is just an \(\mathbb {E}_2\)-algebra. As an application of this trace formula, we propose a strategy of proof of Bloch’s conductor conjecture. This is a research announcement and detailed proofs will appear elsewhere.



We are grateful to our co-authors Anthony Blanc and Marco Robalo, for stimulating discussions that led to the joint work [2], and subsequently to the present paper. We also wish to thank Takeshi Saito for a very useful email exchange, and the Max-Planck-Institut für Mathematik in Bonn for providing a perfect scientific environment while the mathematics related to this paper was conceived.


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© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.CNRSUniversité Paul SabatiérToulouseFrance
  2. 2.DIMAIUniversità di FirenzeFlorenceItaly

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