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A note on Grothendieck's standard conjectures of type $ \mathrm{C}^+$ and $ \mathrm{D}$ in positive characteristic


Author: Gonçalo Tabuada
Journal: Proc. Amer. Math. Soc. 147 (2019), 5039-5054
MSC (2010): Primary 14A22
DOI: https://doi.org/10.1090/proc/14768
Published electronically: September 20, 2019
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Abstract: Making use of topological periodic cyclic homology, we extend Grothendieck's standard conjectures of type $ \mathrm {C}^+$ and $ \mathrm {D}$ (with respect to crystalline cohomology theory) from smooth projective schemes to smooth proper dg categories in the sense of Kontsevich. As a first application, we prove Grothendieck's original conjectures in the new cases of linear sections of determinantal varieties. As a second application, we prove Grothendieck's (generalized) conjectures in the new cases of ``low-dimensional'' orbifolds. Finally, as a third application, we establish a far-reaching noncommutative generalization of Berthelot's cohomological interpretation of the classical zeta function and of Grothendieck's conditional approach to ``half'' of the Riemann hypothesis. Along the way, following Scholze, we prove that the topological periodic cyclic homology of a smooth proper scheme $ X$ agrees with the crystalline cohomology theory of $ X$ (after inverting the characteristic of the base field).


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Additional Information

Gonçalo Tabuada
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139; and Departamento de Matemática, FCT, UNL, Portugal; and Centro de Matemática e Aplicações (CMA), FCT, UNL, Portugal
Email: tabuada@math.mit.edu

DOI: https://doi.org/10.1090/proc/14768
Received by editor(s): May 29, 2018
Published electronically: September 20, 2019
Additional Notes: The author was partially supported by an NSF CAREER Award #1350472 and by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2019 (Centro de Matemática e Aplicações)
Communicated by: Jerzy Weyman
Article copyright: © Copyright 2019 American Mathematical Society