Noncommutative Riemann hypothesis
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Abstract:
In this note, making use of noncommutative $l$-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first application, we prove that the generalized Riemann hypothesis is invariant under derived equivalences and homological projective duality. As a second application, we prove the noncommutative generalized Riemann hypothesis in some new cases.References
- A. I. Bondal and M. M. Kapranov, Framed triangulated categories, Mat. Sb. 181 (1990), no. 5, 669–683 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 1, 93–107. MR 1055981, DOI 10.1070/SM1991v070n01ABEH001253
- A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, arXiv:9506012, 1995.
- Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
- Pierre Deligne, La conjecture de Weil I, Publ. Math. Inst. Hautes Études Sci. 43 (1974), 273–307.
- Alexander Grothendieck, Formule de Lefschetz et rationalité des fonctions $L$, Séminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1995, pp. Exp. No. 279, 41–55 (French). MR 1608788
- D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006. MR 2244106, DOI 10.1093/acprof:oso/9780199296866.001.0001
- Bernhard Keller, On differential graded categories, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 151–190. MR 2275593
- Maxim Kontsevich, XI Solomon Lefschetz Memorial Lecture series: Hodge structures in non-commutative geometry, Non-commutative geometry in mathematics and physics, Contemp. Math., vol. 462, Amer. Math. Soc., Providence, RI, 2008, pp. 1–21. Notes by Ernesto Lupercio. MR 2444365, DOI 10.1090/conm/462/09058
- Maxim Kontsevich, Noncommutative motives, Talk at the IAS on the occasion of the $61^{\mathrm {st}}$ birthday of Pierre Deligne, 2005, Video available at the webpage http://video.ias.edu/Geometry-and-Arithmetic.
- Maxim Kontsevich, Triangulated categories and geometry, Course at École Normale Supérieure, Paris, 1998, Notes available at the webpage www.math.uchicago.edu/mitya/langlands.html.
- Alexander Kuznetsov, Calabi-Yau and fractional Calabi-Yau categories, J. Reine Angew. Math. 753 (2019), 239–267. MR 3987870, DOI 10.1515/crelle-2017-0004
- Alexander Kuznetsov, Semiorthogonal decompositions in algebraic geometry, Proceedings of the International Congress of Mathematicians, Vol. II, Seoul, 2014, pp. 635–660.
- Alexander Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci. (2007), no. 105, 157–220.
- Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque 228 (1995), 4, 121–163. Columbia University Number Theory Seminar (New York, 1992). MR 1330931
- Matilde Marcolli and Gonçalo Tabuada, Noncommutative Artin motives, Selecta Math. (N.S.) 20 (2014), no. 1, 315–358. MR 3147418, DOI 10.1007/s00029-013-0131-9
- Dmitri Orlov, Finite-dimensional differential graded algebras and their geometric realizations, Adv. Math. 366 (2020), 107096, 33. MR 4072798, DOI 10.1016/j.aim.2020.107096
- Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2016), 59–105. MR 3545926, DOI 10.1016/j.aim.2016.07.014
- B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie, 1859.
- Jean-Pierre Serre, Facteurs locaux des fonctions zêta des varietés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou. 11e année: 1969/70. Théorie des nombres. Fasc. 1: Exposés 1 à 15; Fasc. 2: Exposés 16 à 24, Secrétariat Math., Paris, 1970, pp. 15 (French). MR 3618526
- Jean-Pierre Serre, Zeta and $L$ functions, Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963) Harper & Row, New York, 1965, pp. 82–92. MR 0194396
- Gonçalo Tabuada, Noncommutative motives, University Lecture Series, vol. 63, American Mathematical Society, Providence, RI, 2015. With a preface by Yuri I. Manin. MR 3379910, DOI 10.1090/ulect/063
- G. Tabuada and M. Van den Bergh, Motivic Atiyah-Segal completion theorem, arXiv:2009.08448, 2021.
- Gonçalo Tabuada and Michel Van den Bergh, Noncommutative motives of Azumaya algebras, J. Inst. Math. Jussieu 14 (2015), no. 2, 379–403. MR 3315059, DOI 10.1017/S147474801400005X
- Richard P. Thomas, Notes on homological projective duality, Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97, Amer. Math. Soc., Providence, RI, 2018, pp. 585–609. MR 3821163
- R. W. Thomason, Algebraic $K$-theory and étale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552. MR 826102, DOI 10.24033/asens.1495
- C. Soulé, Operations on étale $K$-theory. Applications, Algebraic $K$-theory, Part I (Oberwolfach, 1980) Lecture Notes in Math., vol. 966, Springer, Berlin, 1982, pp. 271–303. MR 689380, DOI 10.1007/BFb0062180
- A. Weil, Théorie élémentaire des correspondences sur une courbe, Sur les courbes algébriques et les variétés qui s’en déduisent, Hermann, 1948.
Additional Information
- Gonçalo Tabuada
- Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
- Email: goncalo.tabuada@warwick.ac.uk
- Received by editor(s): May 25, 2021
- Received by editor(s) in revised form: October 8, 2021
- Published electronically: March 16, 2022
- Additional Notes: The author was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement number 892994
- Communicated by: Julie Bergner
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 2385-2404
- MSC (2020): Primary 11S40, 14A22, 18D20
- DOI: https://doi.org/10.1090/proc/15874
- MathSciNet review: 4399257