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Motivic homological stability for configuration spaces of the line. (English) Zbl 1375.14085
Summary: We lift the classical theorem of V. I. Arnol’d [Tr. Mosk. Mat. O.-va 21, 27–46 (1970; Zbl 0208.24003)] on homological stability for configuration spaces of the plane to the motivic world. More precisely, we prove that the schemes of unordered configurations of points in the affine line satisfy stability with respect to the motivic \(t\)-structure on mixed Tate motives.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
18E30 Derived categories, triangulated categories (MSC2010)
55R80 Discriminantal varieties and configuration spaces in algebraic topology
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[1] Arnol’d, Certain topological invariants of algebraic functions, Tr. Mosk. Mat. Obs. 21 pp 27– (1970)
[2] DOI: 10.1093/qmath/hat039 · Zbl 1306.55010 · doi:10.1093/qmath/hat039
[3] Cantero, On homological stability for configuration spaces on closed background manifolds, Doc. Math. 20 pp 753– (2015) · Zbl 1352.55008
[4] Cazanave, Algebraic homotopy classes of rational functions, Ann. Sci. Éc. Norm. Supér. (4) 45 pp 511– (2012) · Zbl 1419.14025 · doi:10.24033/asens.2172
[5] D.-C. Cisinski and F. Déglise , ’Triangulated categories of mixed motives’, Preprint, 2009, arXiv:0912.2110.
[6] Déglise, Around the Gysin triangle. II, Doc. Math. 13 pp 613– (2008) · Zbl 1152.14019
[7] B. Farb and J. Wolfson , ’Topology and arithmetic of resultants, I: spaces of rational maps’, Preprint, 2015, arXiv:1506.02713. · Zbl 1379.55016
[8] B. Fresse , ’Homotopy of operads and Grothendieck–Teichmüller groups’, Research Monograph, http://math.univ-lille1.fr/fresse/OperadHomotopyBook/ . · Zbl 1375.55007
[9] DOI: 10.1007/s002220050014 · Zbl 0957.19003 · doi:10.1007/s002220050014
[10] S. Gelfand and Y. Manin , Methods of homological algebra, 2nd edn, Springer Monographs in Mathematics (Springer, Berlin, 2003). · Zbl 1006.18001 · doi:10.1007/978-3-662-12492-5
[11] G. Horel , ’Profinite completion of operads and the Grothendieck–Teichmüller group’, Preprint, 2015, arXiv:1504.01605.
[12] B. Kahn , ’Weight filtration and mixed tate motives’, Proceedings of the Algebraic Geometry Symposium of Kinosaki, 2000, https://webusers.imj-prg.fr/ bruno.kahn/preprints/prep.html .
[13] B. Kahn , ’Algebraic \(K\) -theory, algebraic cycles and arithmetic geometry’, Handbook of K-theory (Springer, Berlin, 2005) 351–428. · Zbl 1115.19003
[14] S. Lang , Algebra, 3rd edn, Graduate Texts in Mathematics 211 (Springer, New York, 2002).
[15] DOI: 10.1007/s00229-012-0587-5 · Zbl 1284.14014 · doi:10.1007/s00229-012-0587-5
[16] M. Levine , ’Tate motives and the vanishing conjectures for algebraic K-theory’, Algebraic K-theory and algebraic topology (Springer, Berlin, 1993) 167–188. · Zbl 0885.19001 · doi:10.1007/978-94-017-0695-7_7
[17] M. Levine , ’K-theory and motivic cohomology of schemes, I’, Preprint, 2004, http://www.esaga.uni-due.de/marc.levine/pub/ .
[18] C. Mazza , V. Voevodsky and C. Weibel , Lecture notes on motivic cohomology, Clay Mathematics Monographs 2 (American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006). · Zbl 1115.14010
[19] DOI: 10.1016/0040-9383(75)90038-5 · Zbl 0296.57001 · doi:10.1016/0040-9383(75)90038-5
[20] DOI: 10.1016/j.aim.2008.05.013 · Zbl 1180.14015 · doi:10.1016/j.aim.2008.05.013
[21] DOI: 10.1007/BF01390197 · Zbl 0267.55020 · doi:10.1007/BF01390197
[22] DOI: 10.1007/BF02392088 · Zbl 0427.55006 · doi:10.1007/BF02392088
[23] I. Soudères , ’A motivic Grothendieck–Teichmüller group’, Preprint, 2015, arXiv:1502.05640. · Zbl 1391.14040
[24] M. Spitzweck , ’A commutative \(\textbf {P}^1\) -spectrum representing motivic cohomology over Dedekind domains’, Preprint, 2012, arXiv:1207.4078.
[25] DOI: 10.4007/annals.2011.174.1.11 · Zbl 1236.14026 · doi:10.4007/annals.2011.174.1.11
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