Bachmann, Tom

Cancellation theorem for motivic spaces with finite flat transfers

Doc. Math. 26, 1121-1144 (2021)
DOI: 10.25537/dm.2021v26.1121-1144
Communicated by Thomas Geisser

Summary

We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of Voevodsky's cancellation theorem.

Mathematics Subject Classification

14F42, 19E15

Keywords/Phrases

transfers, cancellation theorem, motives

References

  • [AGP16]. Alexey Ananyevskiy, Grigory Garkusha, and Ivan Panin. Cancellation theorem for framed motives of algebraic varieties. Adv. Math. 383, Article ID 107681, 38 p., 2021, DOI 10.1090/jams/958, zbl 07304882, MR4232551, arxiv 1601.06642.
  • [AHW17]. Aravind Asok, Marc Hoyois, and Matthias Wendt. Affine representability results in \(\mathbb{a}^1 \) -homotopy theory, I: Vector bundles. Duke Math. J. 166(10):1923-1953, 2017, DOI 10.1215/00127094-0000014X, zbl 1401.14118, MR3679884, arxiv 1506.07093.
  • [Bac17]. Tom Bachmann. The generalized slices of hermitian K-theory. J. Topol. 10(4):1124-1144, 2017, DOI 10.1112/topo.12032, zbl 1453.14065, MR3743071, arxiv 1610.01346.
  • [Bac19]. Tom Bachmann. Affine grassmannians in \(\mathbb{A}^1\)-homotopy theory. Selecta Math. (N.S.) 25(2):paper 25, 2019, DOI 10.1007/s00029-019-0471-1, zbl 1445.14037, MR3925100, arxiv 1801.08471.
  • [Bar17]. Clark Barwick. Spectral Mackey functors and equivariant algebraic \(K\)-theory, I. Adv. Math. 304(2):646-727, 2017, DOI 10.1016/j.aim.2016.08.043, zbl 1348.18020, MR3558219, arxiv 1404.0108.
  • [BE19]. Tom Bachmann and Elden Elmanto. Notes on motivic infinite loop space theory, 2019, arxiv 1912.06530.
  • [BF17]. Tom Bachmann and Jean Fasel. On the effectivity of spectra representing motivic cohomology theories, 2017. arxiv 1710.00594.
  • [BH20]. Tom Bachmann and Marc Hoyois. Norms in motivic homotopy theory. Accepted for publication in Asterisque, 2020, arxiv 1711.03061.
  • [EHK{\etalchar{+}}17]. Elden Elmanto, Marc Hoyois, Adeel A. Khan, Vladimir Sosnilo, and Maria Yakerson. Motivic infinite loop spaces, 2017, arxiv 1711.05248.
  • [FØ17]. J. Fasel and P. A. Østvær. A cancellation theorem for Milnor-Witt correspondences, 2017, arxiv 1708.06098.
  • [GGN16]. David Gepner, Moritz Groth, and Thomas Nikolaus. Universality of multiplicative infinite loop space machines. Algebr. Geom. Topol. 15(6):3107-3153, 2016, DOI 10.2140/agt.2015.15.3107, zbl 1336.55006, MR3450758, arxiv 1305.4550.
  • [GP15]. Grigory Garkusha and Ivan Panin. Homotopy invariant presheaves with framed transfers. Camb. J. Math. 8(1):1-94, 2020, DOI 10.4310/CJM.2020.v8.n1.a1, zbl 1453.14066, MR4085432, arxiv 1504.00884.
  • [{Gro}67]. A. Grothendieck. Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas. Rédigé avec la colloboration de Jean Dieudonné. Publ. Math., Inst. Hautes Étud. Sci. 32:1-361, 1967, zbl 0153.22301.
  • [HJN{\etalchar{+}}20]. Marc Hoyois, Joachim Jelisiejew, Denis Nardin, Burt Totaro, and Maria Yakerson. The Hilbert scheme of infinite affine space and algebraic K-theory, 2020, arxiv 2002.11439.
  • [Hoy15]. Marc Hoyois. From algebraic cobordism to motivic cohomology. J. Reine Angew. Math. 702:173-226, 2015, DOI 10.1515/crelle-2013-0038, zbl 1382.14006, MR3341470, arxiv 1210.7182.
  • [Hoy16]. Marc Hoyois. Cdh descent in equivariant homotopy \(K\)-theory. Doc. Math. 25:457-482, 2020, https://www.elibm.org/article/10012036, DOI 10.25537/dm.2020v25.457-482, zbl 1453.14068, MR4124487, arxiv 1604.06410.
  • [Hoy18]. Marc Hoyois. The localization theorem for framed motivic spaces. Compos. Math. 157(1):1-11, 2021, DOI 10.1112/S0010437X20007575, zbl 1455.14042, MR4215649, arxiv 1807.04253.
  • [Lev08]. Marc Levine. The homotopy coniveau tower. J. Topol. 1(1):217-267, 2008, DOI 10.1112/jtopol/jtm004, zbl 1154.14005, MR2365658, arxiv math/0510334.
  • [Lur09]. Jacob Lurie. Higher topos theory. Annals of Mathematics Studies 170. Princeton University Press, 2009, DOI 10.1515/9781400830558, zbl 1175.18001, MR2522659, arxiv math/0608040.
  • [Lur16]. Jacob Lurie. Higher algebra, May 2016.
  • [LYZR19]. Marc Levine, Yaping Yang, Gufang Zhao, and Joël Riou. Algebraic elliptic cohomology theory and flops, I. Math. Ann. 375(3):1823-1855, 2019, DOI 10.1007/s00208-019-01880-x, zbl 1433.14015, MR4023393, arxiv 1311.2159.
  • [Mor05]. Fabien Morel. The stable \(\mathbb{A}^1\)-connectivity theorems. K-Theory 35(1):1-68, 2005, DOI 10.1007/s10977-005-1562-7, zbl 1117.14023, MR2240215.
  • [Mor12]. Fabien Morel. \( \mathbb{A}^1\)-Algebraic Topology over a Field. Lecture Notes in Mathematics 2052. Springer Berlin Heidelberg, 2012, DOI 10.1007/978-3-642-29514-0, zbl 1263.14003, MR2934577.
  • [Rob15]. Marco Robalo. \(K\)-theory and the bridge from motives to noncommutative motives. Adv. Math. 269:399-550, 2015, DOI 10.1016/j.aim.2014.10.011, zbl 1315.14030, MR3281141, arxiv 1306.3795.
  • [SØ12]. Markus Spitzweck and Paul Arne Østvær. Motivic twisted K-theory. Algebr. Geom. Topol. 12(1):565-599, 2012, DOI 10.2140/agt.2012.12.565, zbl 1282.14040, MR2916287, arxiv 1008.4915.
  • [{Sta}18]. The Stacks Project Authors. Stacks Project, 2018. http://stacks.math.columbia.edu.
  • [Sus03]. Andrei Suslin. On the Grayson spectral sequence. Proc. Steklov Inst. Math. 241:202-237, 2003, zbl 1084.14025, MR2024054.
  • [Voe10]. Vladimir Voevodsky. Cancellation theorem. Doc. Math., Extra Volume: A. Suslin's Sixtieth Birthday:671-685, 2010, https://www.elibm.org/article/10011497, zbl 1202.14022, arxiv math/0202012.

Affiliation

Bachmann, Tom
LMU Munich, Mathematisches Institut, Theresienstr. 39, 80333 München, Germany

Downloads