The spectrum of Artin motives

Balmer, Paul; Gallauer, Martin

doi:10.1090/tran/9306

The spectrum of Artin motives
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by Paul Balmer and Martin Gallauer;

Trans. Amer. Math. Soc. 378 (2025), 1733-1754

DOI: https://doi.org/10.1090/tran/9306

Published electronically: October 31, 2024

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Abstract:

We analyze the tt-geometry of derived Artin motives, via modular representation theory of profinite groups. To illustrate our methods, we discuss Artin motives over a finite field, in which case we also prove stratification.

References

Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math. 588 (2005), 149–168. MR 2196732, DOI 10.1515/crll.2005.2005.588.149
Paul Balmer, Tensor triangular geometry, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 85–112. MR 2827786
Paul Balmer, On the surjectivity of the map of spectra associated to a tensor-triangulated functor, Bull. Lond. Math. Soc. 50 (2018), no. 3, 487–495. MR 3829735, DOI 10.1112/blms.12158
S. Balchin, D. Barnes, and T. Barthel, Profinite equivariant spectra and their tensor-triangular geometry, Preprint, arXiv:2401.01878, 2024.
Tobias Barthel, Natàlia Castellana, Drew Heard, Niko Naumann, Luca Pol, and Beren Sanders, Descent in tensor triangular geometry, Triangulated categories in representation theory and beyond—the Abel Symposium 2022, Abel Symp., vol. 17, Springer, Cham, [2024] ©2024, pp. 1–56. MR 4786501, DOI 10.1007/978-3-031-57789-5_{1}
T. Barthel, N. Castellana, D. Heard, and B. Sanders, Cosupport in tensor triangular geometry, Preprint, arXiv:2303.13480, 2023.
Paul Balmer and Giordano Favi, Generalized tensor idempotents and the telescope conjecture, Proc. Lond. Math. Soc. (3) 102 (2011), no. 6, 1161–1185. MR 2806103, DOI 10.1112/plms/pdq050
Paul Balmer and Martin Gallauer, Permutation modules and cohomological singularity, Comment. Math. Helv. 97 (2022), no. 3, 413–430. MR 4468990, DOI 10.4171/cmh/534
Paul Balmer and Martin Gallauer, Three real Artin-Tate motives, Adv. Math. 406 (2022), Paper No. 108535, 68. MR 4445121, DOI 10.1016/j.aim.2022.108535
Paul Balmer and Martin Gallauer, Finite permutation resolutions, Duke Math. J. 172 (2023), no. 2, 201–229. MR 4541331, DOI 10.1215/00127094-2022-0041
Paul Balmer and Martin Gallauer, Permutation modules, Mackey functors, and Artin motives, Representations of algebras and related structures, EMS Ser. Congr. Rep., EMS Press, Berlin, [2023] ©2023, pp. 37–75. MR 4693637
Paul Balmer and Martin Gallauer, The geometry of permutation modules, Preprint available on authors’ webpages, 2023.
Tobias Barthel, Drew Heard, and Beren Sanders, Stratification in tensor triangular geometry with applications to spectral Mackey functors, Camb. J. Math. 11 (2023), no. 4, 829–915. MR 4650265, DOI 10.4310/cjm.2023.v11.n4.a2
David J. Benson, Srikanth B. Iyengar, and Henning Krause, Stratifying modular representations of finite groups, Ann. of Math. (2) 174 (2011), no. 3, 1643–1684. MR 2846489, DOI 10.4007/annals.2011.174.3.6
Paul Balmer, Henning Krause, and Greg Stevenson, Tensor-triangular fields: ruminations, Selecta Math. (N.S.) 25 (2019), no. 1, Paper No. 13, 36. MR 3911737, DOI 10.1007/s00029-019-0454-2
Andreas W. M. Dress, Notes on the theory of representations of finite groups. Part I: The Burnside ring of a finite group and some AGN-applications, Universität Bielefeld, Fakultät für Mathematik, Bielefeld, 1971. With the aid of lecture notes, taken by Manfred Küchler. MR 360771
Max Dickmann, Niels Schwartz, and Marcus Tressl, Spectral spaces, New Mathematical Monographs, vol. 35, Cambridge University Press, Cambridge, 2019. MR 3929704, DOI 10.1017/9781316543870
Peng Du and Alexander Vishik, On the Balmer spectrum of Morel-Voevodsky category, Preprint, arXiv:2309.09077, 2023.
Martin Gallauer, Tensor triangular geometry of filtered modules, Algebra Number Theory 12 (2018), no. 8, 1975–2003. MR 3892970, DOI 10.2140/ant.2018.12.1975
Martin Gallauer, tt-geometry of Tate motives over algebraically closed fields, Compos. Math. 155 (2019), no. 10, 1888–1923. MR 4010428, DOI 10.1112/s0010437x19007528
Martin Gallauer, A note on Tannakian categories and mixed motives, Bull. Lond. Math. Soc. 53 (2021), no. 1, 119–129. MR 4224517, DOI 10.1112/blms.12405
Wulf-Dieter Geyer, Unendliche algebraische Zahlkörper, über denen jede Gleichung auflösbar von beschränkter Stufe ist, J. Number Theory 1 (1969), 346–374 (German). MR 282946, DOI 10.1016/0022-314X(69)90050-X
Jeremiah Heller and Kyle M. Ormsby, Primes and fields in stable motivic homotopy theory, Geom. Topol. 22 (2018), no. 4, 2187–2218. MR 3784519, DOI 10.2140/gt.2018.22.2187
Henning Krause, Decomposing thick subcategories of the stable module category, Math. Ann. 313 (1999), no. 1, 95–108. MR 1666825, DOI 10.1007/s002080050252
Henning Krause, The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), no. 5, 1128–1162. MR 2157133, DOI 10.1112/S0010437X05001375
Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224, DOI 10.1007/BF02698831
Tobias J. Peter, Prime ideals of mixed Artin-Tate motives, J. K-Theory 11 (2013), no. 2, 331–349. MR 3061000, DOI 10.1017/is013001031jkt215
Luis Ribes and Pavel Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, Springer-Verlag, Berlin, 2010. MR 2599132, DOI 10.1007/978-3-642-01642-4
Alexander Vishik, Isotropic and numerical equivalence for Chow groups and Morava K-theories, Invent. Math. 237 (2024), no. 2, 779–808. MR 4768634, DOI 10.1007/s00222-024-01267-z
Vladimir Voevodsky, Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. MR 1764202
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136