Project description

One of the projects of this application proposes the construction of a homotopy coniveau tower over Dedekind schemes, aiming at a filtration of the algebraic K-theory spectrum over for example number rings with layers the motivic EM-spectrum. Motivic cohomology with finite coefficients which are invertible on the base is intimately related to étale cohomology. A precise statement is given by the Beilinson-Lichtenbaum conjecture, now proven for smooth schemes over fields and Dedekind domains. Thus insights into étale cohomology yield such for motivic cohomology. Of particular interest are purity statements. In the absolute case purity is known for étale cohomology thanks to the work of Gabber. A Gersten resolution for étale cohomology is missing so far over regular schemes. A positive solution to this question would yield purity statements for motivic cohomology, resulting in comparison statements for motivic cohomology defined via the motivic Eilenberg-MacLane spectrum on the one side and cycle complexes on the other side for regular schemes. We will also investigate extending the results of Hopkins-Morel-Hoyois for MGL to more general bases, without inverting residue characteristics. Other projects involve understanding relations between the slice spectra sequence for the sphere spectrum and various versions of Adams-Novikov spectral sequences.

Related publications

Published articles

Ananyevskiy, Alexey, Levine, Marc, Panin, Ivan. Witt sheaves and the $\eta$-inverted sphere spectrum. J. Topol. 10 (2017), no. 2, 370--385.

Fangzhou Jin, Enlin Yang, Künneth formulas for motives and additivity of traces. Adv. Math. 376 (2021), Paper No. 107446, 83 pp.

Frédéric Déglise, Fangzhou Jin, Adeel A. Khan, Fundamental classes in motivic homotopy theory. J. Eur. Math. Soc. (JEMS) 23 (2021), no. 12, 3935–3993.

Marc Levine, An overview of motivic homotopy theory. Acta Math. Vietnam. 41 (2016), no. 3, 379–407.

Oliver Röndigs, Markus Spitzweck, Paul Arne Østvær, Cellularity of hermitian K-theory and Witt theory ``K-Theory—Proceedings of the International Colloquium, Mumbai, 2016'', 35–40, Hindustan Book Agency, New Delhi, 2018.

Oliver Röndigs, Markus Spitzweck, Paul Arne Østvær, The motivic Hopf map solves the homotopy limit problem for K-theory Doc. Math. 23 (2018), 1405–1424.

Oliver Röndigs, Markus Spitzweck, Paul Arne Østvær, The first stable homotopy groups of motivic spheres, Ann. of Math. (2) 189 (2019), no. 1, 1–74.

J.D. Christensen, M. Frankland, Higher Toda brackets and the Adams spectral sequence in trianglated categories, Algebr. Geom. Topol. 17 (2017), no. 5, 2687–2735.

H.J. Baues, M. Frankland, Eilenberg-MacLane mapping algebras and higher distributivity up to homotopy, New York J. Math. 23 (2017), 1539–1580.

Preprints

Frédéric Déglise, Jean Fasel, Fangzhou Jin, Adeel A. Khan, On the rational motivic homotopy category, arXiv:2005.10147. Preprint 2020. To appear, Journal de l'École polytechnique

Fangzhou Jin, Heng Xie A Gersten complex on real schemes Preprint 2020. arXiv:2007.04625.

Lorenzo Mantovani, Localizations and completions in motivic homotopy theory. arXiv:1810.04134. Preprint October 2018

M. Frankland, A user's guide: Completed power operations for Morava E-theory, expository article at mathusersguides.org, 2016.

M. Frankland, M. Spitzweck, A possible approach to the Hopkins-Morel isomorphism over general base schemes, Work in progress.