A note on Grothendieck’s standard conjectures of type $C^+$ and $D$
- 12 January 2018
- journal article
- research article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 146 (4), 1389-1399
- https://doi.org/10.1090/proc/13955
Abstract
Grothendieck conjectured in the sixties that the even Kunneth projector (with respect to a Weil cohomology theory) is algebraic and that the homological equivalence relation on algebraic cycles coincides with the numerical equivalence relation. In this note we extend these celebrated conjectures from smooth projective schemes to the broad setting of smooth proper dg categories. As an application, we prove that Grothendieck's conjectures are invariant under homological projective duality. This leads to a proof of Grothendieck's original conjectures in the case of intersections of quadrics and linear sections of determinantal varieties. Along the way, we also prove the case of quadric fibrations and intersections of bilinear divisors.Keywords
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This publication has 6 references indexed in Scilit:
- Homological projective duality for determinantal varietiesAdvances in Mathematics, 2016
- Noncommutative numerical motives, Tannakian structures, and motivic Galois groupsJournal of the European Mathematical Society, 2016
- Noncommutative motives of Azumaya algebrasJournal of the Institute of Mathematics of Jussieu, 2014
- Notes on Motives in Finite CharacteristicPublished by Springer Science and Business Media LLC ,2009
- Derived categories of quadric fibrations and intersections of quadricsAdvances in Mathematics, 2008
- Homological projective dualityPublications mathématiques de l'IHÉS, 2007