Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T16:43:24.724Z Has data issue: false hasContentIssue false
  • English
  • Français

tt-geometry of Tate motives over algebraically closed fields

Part of: Abelian categories Categories with structure Homological algebra (Co)homology theory Cycles and subschemes $K$-theory in geometry

Published online by Cambridge University Press:  06 September 2019

Martin Gallauer
Martin Gallauer*
Affiliation:
Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK URL: http://people.maths.ox.ac.uk/gallauer email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study Tate motives with integral coefficients through the lens of tensor triangular geometry. For some base fields, including $\overline{\mathbb{Q}}$ and $\overline{\mathbb{F}_{p}}$, we arrive at a complete description of the tensor triangular spectrum and a classification of the thick tensor ideals.

Keywords

tt-geometrymotiveTate motiveclassificationcohomology

MSC classification

Primary: 14C15: (Equivariant) Chow groups and rings; motives 14F42: Motivic cohomology; motivic homotopy theory 19E15: Algebraic cycles and motivic cohomology 18E30: Derived categories, triangulated categories 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories 18G99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 10 , October 2019 , pp. 1888 - 1923
Copyright
© The Author 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ayoub, J., A guide to (étale) motivic sheaves , in Proceedings of the International Congress of Mathematicians, Seoul 2014, Vol. II, eds Jang, S. Y., Kim, Y. R., Lee, D.-W. and Yie, I. (Kyung Moon Sa, Seoul, 2014).Google Scholar
Ayoub, J., La réalisation étale et les opérations de Grothendieck , Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), 1–145.Google Scholar
Balmer, P., The spectrum of prime ideals in tensor triangulated categories , J. Reine Angew. Math. 588 (2005), 149–168.Google Scholar
Balmer, P., Spectra, spectra, spectra – tensor triangular spectra versus Zariski spectra of endomorphism rings , Algebr. Geom. Topol. 10 (2010), 1521–1563.Google Scholar
Balmer, P., Tensor triangular geometry , in Proceedings of the International Congress of Mathematicians, Vol. II (Hindustan Book Agency, New Delhi, 2010), 85–112.Google Scholar
Balmer, P., On the surjectivity of the map of spectra associated to a tensor-triangulated functor , Bull. Lond. Math. Soc. 50 (2018), 487–495.Google Scholar
Bloch, S., Algebraic cycles and the Lie algebra of mixed Tate motives , J. Amer. Math. Soc. 4 (1991), 771–791.Google Scholar
Bloch, S., The moving lemma for higher Chow groups , J. Algebraic Geom. 3 (1994), 537–568.Google Scholar
Bloch, S. and Kříž, I., Mixed Tate motives , Ann. of Math. (2) 140 (1994), 557–605.Google Scholar
Brown, F., Mixed Tate motives over ℤ , Ann. of Math. (2) 175 (2012), 949–976.Google Scholar
Cisinski, D.-C. and Déglise, F., Triangulated categories of mixed motives, Springer Monographs in Mathematics, to appear. Preprint (2012), arXiv:0912.2110v3.Google Scholar
Cisinski, D.-C. and Déglise, F., Integral mixed motives in equal characteristic , Doc. Math. (2015), 145–194; Extra Volume: Alexander S. Merkurjev’s sixtieth birthday.Google Scholar
Cisinski, D.-C. and Déglise, F., Étale motives , Compos. Math. 152 (2016), 556–666.Google Scholar
Deligne, P. and Goncharov, A. B., Groupes fondamentaux motiviques de Tate mixte , Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 1–56.Google Scholar
Dyer, M., Exact subcategories of triangulated categories, (2005), available at https://www3.nd.edu/∼dyer/papers/index.html.Google Scholar
Gallauer, M., Tensor triangular geometry of filtered modules , Algebra Number Theory 12 (2018), 1975–2003.Google Scholar
Geisser, T. and Levine, M., The Bloch–Kato conjecture and a theorem of Suslin–Voevodsky , J. Reine Angew. Math. 530 (2001), 55–103.Google Scholar
Gillam, W., Localization of ringed spaces , Adv. Pure Math. 1 (2011), 250–263.Google Scholar
Harder, G., Die Kohomologie S-arithmetischer Gruppen über Funktionenkörpern , Invent. Math. 42 (1977), 135–175.Google Scholar
Hasemeyer, C. and Hornbostel, J., Motives and etale motives with finite coefficients , J. K-Theory 34 (2005), 195–207.Google Scholar
Ivorra, F., Réalisation l-adique des motifs triangulés géométriques. I , Doc. Math. 12 (2007), 607–671.Google Scholar
Iwanari, I., Bar construction and tannakization , Publ. Res. Inst. Math. Sci. 50 (2014), 515–568.Google Scholar
Keller, B., Derived categories and their uses, Handbook of Algebra (Elsevier/North Holland, Amsterdam, 1996), 671–701.Google Scholar
Kelly, S., Some observations about motivic tensor triangulated geometry over a finite field, Preprint (2016), arXiv:1608.02913.Google Scholar
Kelly, S., Voevodsky motives and ldh-descent , Astérisque 391 (2017).Google Scholar
Kříž, I. and May, J. P., Operads, algebras, modules and motives , Astérisque 233 (1995).Google Scholar
Levine, M., Tate motives and the vanishing conjectures for algebraic K-theory , in Algebraic K-theory and algebraic topology, NATO ASI Series, vol. 407, ed. Goerss, J. F. J. P. G. (Kluwer Academic, Dordrecht, 1992), 113–146.Google Scholar
Levine, M., Bloch’s higher Chow groups revisited , in K-theory (Strasbourg, 1992), Astérisque, (1994), 235–320.Google Scholar
Mazza, C., Voevodsky, V. and Weibel, C., Lecture Notes on Motivic Cohomology, Clay Mathematics Monographs, vol. 2 (American Mathematical Society, Clay Mathematics Institute, Cambridge, MA, 2006).Google Scholar
Milnor, J., Algebraic K-theory and quadratic forms , Invent. Math. 9 (1969/70), 318–344.Google Scholar
Nesterenko, Y. P. and Suslin, A. A., Homology of the general linear group over a local ring, and Milnor’s K-theory , Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), 121–146.Google Scholar
Peter, T. J., Prime ideals of mixed Artin–Tate motives , J. K-Theory 11 (2013), 331–349.Google Scholar
Porta, M., Universal property of triangulated derivators via Keller’s towers, Preprint (2015),arXiv:1512.02691.Google Scholar
Positselski, L., Mixed Artin–Tate motives with finite coefficients , Mosc. Math. J. 11 (2011), 317–402; 407–408.Google Scholar
Suslin, A. and Voevodsky, V., Bloch–Kato conjecture and motivic cohomology with finite coefficients , in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548 (Kluwer Academic, Dordrecht, 2000), 117–189.Google Scholar
Terasoma, T., Mixed Tate motives and multiple zeta values , Invent. Math. 149 (2002), 339–369.Google Scholar
Thornton, R., The homogeneous spectrum of Milnor–Witt K-theory , J. Algebra 459 (2016), 376–388.Google Scholar
Totaro, B., Milnor K-theory is the simplest part of algebraic K-theory , J. K-Theory 6 (1992), 177–189.Google Scholar
Voevodsky, V., Triangulated categories of motives over a field , inCycles, transfers, and motivic homology theories, Annals of Mathematics Studies (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Voevodsky, V., Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic , Int. Math. Res. Not. IMRN 2002(7) (2002), 351–355.Google Scholar
Wildeshaus, J., f-catégories, tours et motifs de Tate , C. R. Math. Acad. Sci. Paris 347 (2009), 1337–1342.Google Scholar