Modules over algebraic cobordism

Elden Elmanto ; Marc Hoyois ; Adeel A. Khan ; Vladimir Sosnilo ; Maria Yakerson

Forum of Mathematics, Pi, Tome 8 (2020)

Résumé

We prove that the $\infty $-category of $\mathrm{MGL} $-modules over any scheme is equivalent to the $\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$-loop spaces, we deduce that very effective $\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$, $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$.

Détail

DOI : 10.1017/fmp.2020.13

Bibliographie

[AGP18] Ananyevskiy, A., Garkusha, G. and Panin, I., ‘Cancellation theorem for framed motives of algebraic varieties’, Preprint, 2018, .Google Scholar | arXiv

[Bar16] Barwick, C., ‘On the algebraic -theory of higher categories’, J. Topol. 9(1) (2016), 245–347.Google Scholar | DOI

[BEH+20] Bachmann, T., Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., ‘On the infinite loop spaces of algebraic cobordism and the motivic sphere’, Preprint, 2020, .Google Scholar | arXiv

[BF18] Bachmann, T. and Fasel, J., ‘On the effectivity of spectra representing motivic cohomology theories’, Preprint, 2018, .Google Scholar | arXiv

[BH20] Bachmann, T. and Hoyois, M., ‘Norms in motivic homotopy theory’, to appear in Astérisque (2020). Preprint at .Google Scholar | arXiv

[BHL17] Bhatt, B. and Halpern-Leistner, D., ‘Tannaka duality revisited’, Adv. Math. 316 (2017), 576–612.Google Scholar | DOI

[CD15] Cisinski, D.-C. and Déglise, F., ‘Integral mixed motives in equal characteristic’, Doc. Math., extra volume: Alexander S. Merkurjev’s Sixtieth Birthday (2015), 145–194.Google Scholar

[CMNN20] Clausen, D., Mathew, A., Naumann, N. and Noel, J., ‘Descent in algebraic -theory and a conjecture of Ausoni–Rognes’, J. Eur. Math. Soc. (JEMS) 22(4) (2020), 1149–1200.Google Scholar | DOI

[Dég18] Déglise, F., ‘Orientation theory in arithmetic geometry’, in -theory, Tata Institute of Fundamental Research Publications, 19 (Hindustan Book Agency, New Delhi, 2018), 241–350. Available at https://protect-eu.mimecast.com/s/8MGZCz6z6tnOn35uKRwK1?domain=bookstore.ams.org.Google Scholar

[DJK20] Déglise, F., Jin, F. and Khan, A. A., ‘Fundamental classes in motivic homotopy theory’, to appear in J. Eur. Math. Soc. (JEMS) (2020). Preprint at .Google Scholar | arXiv

[Dru20] Druzhinin, A., ‘Framed motives of smooth affine pairs’, Preprint, 2020, .Google Scholar | arXiv

[EHK+19] Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., ‘Motivic infinite loop spaces’, Preprint, 2019, .Google Scholar | arXiv

[EHK+20] Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., ‘Framed transfers and motivic fundamental classes’, J. Topol. 13(2) (2020), 460–500.Google Scholar | DOI

[EK20] Elmanto, E. and Kolderup, H., ‘On modules over motivic ring spectra’, Ann. K-Theory 5(2) (2020), 327–355.Google Scholar | DOI

[GMTW09] Galatius, S., Madsen, I., Tillmann, U. and Weiss, M., ‘The homotopy type of the cobordism category’, Acta Math. 202(2) (2009), 195–239.Google Scholar | DOI

[GN18] Garkusha, G. and Neshitov, A., ‘Fibrant resolutions for motivic Thom spectra’, Preprint, 2018, .Google Scholar | arXiv

[GNP18] Garkusha, G., Neshitov, A. and Panin, I., ‘Framed motives of relative motivic spheres’, Preprint, 2018, .Google Scholar | arXiv

[GP20a] Garkusha, G. and Panin, I., ‘Framed motives of algebraic varieties (after V. Voevodsky)’, to appear in J. Amer. Math. Soc. (2020). Preprint at .Google Scholar | arXiv

[GP20b] Garkusha, G. and Panin, I., ‘Homotopy invariant presheaves with framed transfers’, Camb. J. Math. 8(1) (2020), 1–94.Google Scholar | DOI

[Gro67] Grothendieck, A., Éléments de Géométrie Algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Publ. Math. I.H.É.S. 32 (1967). Available at https://protect-eu.mimecast.com/s/W7FNCA6l6tVGVgkiQpapc?domain=numdam.org.Google Scholar

[Gru72] Gruson, L., Une propriété des couples henséliens, Colloque d’algèbre commutative, exp. no 10, Publications des séminaires de mathématiques et informatique de Rennes, 1972. Available at https://protect-eu.mimecast.com/s/FgLkCBrmrSAzA1YU1oJRb?domain=numdam.org.Google Scholar

[Hau18] Haugseng, R., ‘Iterated spans and classical topological field theories’, Math. Z. 289(3) (2018), 1427–1488.Google Scholar | DOI

[Hoy20] Hoyois, M., ‘The localization theorem for framed motivic spaces’, to appear in Compos. Math. (2020). Preprint at .Google Scholar | arXiv

[Kha16] Khan, A. A., ‘Motivic homotopy theory in derived algebraic geometry’, Ph.D. thesis, Universität Duisburg-Essen, 2016. Available at https://www.preschema.com/thesis/thesis.pdf.Google Scholar

[KR19] Khan, A. A. and Rydh, D., ‘Virtual Cartier divisors and blow-ups’, Preprint, 2019, .Google Scholar | arXiv

[LM07] Levine, M. and Morel, F., Algebraic Cobordism (Springer, Location, 2007).Google Scholar

[LS16] Lowrey, P. and Schürg, T., ‘Derived algebraic cobordism’, J. Inst. Math. Jussieu 15 (2016), 407–443.Google Scholar | DOI

[Lur04] Lurie, J., ‘Derived Algebraic Geometry’, Ph.D. thesis, Massachusetts Institute of Technology, 2004. Available at https://www.math.ias.edu/~lurie/papers/DAG.pdf.Google Scholar

[Lur09] Lurie, J., ‘ -categories and the Goodwillie calculus I’, unpublished paper (2009). URL: https://www.math.ias.edu/~lurie/papers/GoodwillieI.pdf.Google Scholar

[Lur17a] Lurie, J., ‘Higher algebra’, unpublished paper (2017). URL: https://www.math.ias.edu/~lurie/papers/HA.pdf.Google Scholar

[Lur17b] Lurie, J., ‘Higher topos theory’, unpublished paper (2017). URL: https://www.math.ias.edu/~lurie/papers/HTT.pdf.Google Scholar

[Lur18] Lurie, J., ‘Spectral algebraic geometry’, unpublished paper (2018). URL: https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf.Google Scholar

[Nav16] Navarro, A., ‘Riemann–Roch for homotopy invariant -theory and Gysin morphisms’, Preprint, 2016, .Google Scholar | arXiv

[Nik17] Nikolaus, T., ‘The group completion theorem via localizations of ring spectra’, unpublished paper (2017). URL: https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/papers.html.Google Scholar

[Pan09] Panin, I., ‘Oriented cohomology theories of algebraic varieties II’, Homology Homotopy Appl. 11(1) (2009), 349–405.Google Scholar | DOI

[Qui71] Quillen, D., ‘Elementary proofs of some results of cobordism theory using Steenrod operations’, Adv. Math. 7(1) (1971), 29–56.Google Scholar | DOI

[Rap19] Raptis, G., ‘Some characterizations of acyclic maps’, J. Homotopy Relat. Struct. 14 (2019), 773–785.Google Scholar | DOI

[Rö08] Röndigs, O. and Østvær, P. A., ‘Modules over motivic cohomology’, Adv. Math. 219(2) (2008), 689–727.Google Scholar | DOI

[RW13] Randal-Williams, O., ‘“Group-completion”, local coefficient systems and perfection’, Q. J. Math. 64(3) (2013), 795–803.Google Scholar | DOI

[Ryd15] Rydh, D., ‘Noetherian approximation of algebraic spaces and stacks’, J. Algebra 422 (2015), 105–147.Google Scholar | DOI

[Spi18] Spitzweck, M., ‘A commutative -spectrum representing motivic cohomology over Dedekind domains’, Mém. Soc. Math. Fr. 157 (2018).Google Scholar

[Stacks] The Stacks Project Authors, ‘The Stacks Project’, URL: http://stacks.math.columbia.edu.Google Scholar

[TV08] Toën, B. and Vezzosi, G., ‘Homotopical Algebraic Geometry. II: Geometric stacks and applications’, Mem. Amer. Math. Soc. 193(902) (2008).Google Scholar

[Voe01] Voevodsky, V., ‘Notes on framed correspondences’, unpublished paper (2001). URL: http://www.math.ias.edu/vladimir/files/framed.pdf.Google Scholar

[Yak19] Yakerson, M., ‘Motivic stable homotopy groups via framed correspondences’, Ph.D. thesis, University of Duisburg-Essen, 2019. Available at https://duepublico2.uni-due.de/receive/duepublico_mods_00070044?q=iakerson.Google Scholar