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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
MathSciNet review: 4021067
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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
MR Author ID: 751291
Email: tabuada@math.mit.edu

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