A note on Grothendieck’s standard conjectures of type 𝐶⁺ and 𝐷 in positive characteristic

Author:

Tabuada Gonçalo

Abstract

Making use of topological periodic cyclic homology, we extend Grothendieck’s standard conjectures of type C + \mathrm {C}^+ and D \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 X agrees with the crystalline cohomology theory of X X (after inverting the characteristic of the base field).

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference29 articles.

1. Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems;Auel, Asher;J. Math. Pures Appl. (9),2014

2. Lecture Notes in Mathematics, Vol. 407;Berthelot, Pierre,1974

3. Topological Hochschild homology and integral 𝑝-adic Hodge theory;Bhatt, Bhargav;Publ. Math. Inst. Hautes \'{E}tudes Sci.,2019

4. A. Blumberg and M. Mandell, The strong Künneth theorem for topological periodic cyclic homology, available at arXiv:1706.06846.

5. A remark on hyperplane sections of rational normal scrolls;Conca, Aldo;Bull. Math. Soc. Sci. Math. Roumanie (N.S.),2017

Cited by 3 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. HPD-invariance of the Tate conjecture(s);Journal of Noncommutative Geometry;2023-02-07

2. Noncommutative Weil conjecture;Advances in Mathematics;2022-08

3. Noncommutative counterparts of celebrated conjectures;-theory in Algebra, Analysis and Topology;2020

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3