Modules over algebraic cobordism. (English) Zbl 1458.14027

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

Forum Math. Pi 8, Paper No. e14, 44 p. (2020).

The article under review is a continuation of [E. Elmanto et. al. “Motivic infinite loop spaces”, Preprint, arXiv:1711.05248]. The main result is a concrete description of the \(\infty\)-category of modules over Voevodsky’s algebraic cobordism spectrum \(\operatorname{MGL}\) as motivic spectra with finite syntomic transfers \[ \mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}(S)) \simeq \mathbf{SH}^{\operatorname{fsyn}}(S) \] for an arbitrary base scheme \(S\) (Theorem 4.1.3). The proof of this result strongly relies on the flexible theory of framed correspondences developed in [loc. cit.]. One important result about these is the reconstruction theorem (Theorem 18 in [M. Hoyois, Compos. Math. 157, No. 1, 1–11 (2021; Zbl 1455.14042)]), which says that there is an equivalence of \(\infty\)-categories between motivic spectra and framed motivic spectra \[ \mathbf{SH}(S) \simeq \mathbf{SH}^\mathrm{fr}(S) \] over any scheme \(S\). This may be viewed as the “sphere spectrum version” of the above equivalence. The rough idea is that certain motivic spectra of interest may have a more geometric description as motivic spectra with framed correspondences. The second main result of this paper is an example of such a description. Namely for a so called smooth stable tangential structure (Definitions 3.3.1 and 3.3.8) \(\beta\) there is an equivalence of motivic spectra \[ M\beta \simeq \Sigma^\infty_{\mathbf{T},\operatorname{fr}} \mathcal{FQ}\operatorname{Sm}_S^\beta \] where the left hand side is the motivic Thom spectrum associated to \(\beta\) and the right hand side is the framed suspension spectrum of the moduli stack of finite quasi-smooth derived \(S\)-schemes with \(\beta\)-structure (Theorem 3.3.10). This general result has many concrete incarnations, a special case for example is the equivalence of motivic spectra \[ \operatorname{MGL}_S \simeq \Sigma^\infty_{T,\operatorname{fr}}\mathcal{FS}\mathrm{yn}_S, \] where \(\mathcal{FS}\mathrm{yn}_S\) is the moduli stack of finite syntomic \(S\)-schemes (Theorem 3.4.1). From this equivalence the authors then deduce the above description of the \(\infty\)-category of \(\operatorname{MGL}_S\)-modules. Furthermore the motivic recognition principle (Theorem 3.5.14 in [E. Elmanto et. al., “Motivic infinite loop spaces”, Preprint, arXiv:1711.05248]) ensures that all of the results mentioned above have a more concrete form when \(S\) is a perfect field \(k\). For example one gets an equivalence of \(\infty\)-categories \[ \mathbf{Mod}_{\operatorname{MGL}}(\mathbf{SH}^\mathrm{veff}(k)) \simeq \mathbf{H}^\mathrm{fsyn}(k)^\mathrm{gp} \] between very effective \(\operatorname{MGL}\)-modules and grouplike motivic spaces with finite syntomic transfers (Theorem 4.1.4). It is also noteworthy that for the proof of Theorem 4.1.3 the language of derived schemes, which is used throughout the paper, is essential even though the statement itself does not involve derived schemes.

Reviewer: Sebastian Wolf (Regensburg)

MSC:

14F42	Motivic cohomology; motivic homotopy theory
14D23	Stacks and moduli problems

Keywords:

algebraic cobordism; framed correspondences

Citations:

Zbl 1455.14042

References:

[1]	Ananyevskiy, A., Garkusha, G. and Panin, I., ‘Cancellation theorem for framed motives of algebraic varieties’, Preprint, 2018, arXiv:1601.06642v2.
[2]	Barwick, C., ‘On the algebraic \(K\)-theory of higher categories’, J. Topol.9(1) (2016), 245-347. · Zbl 1364.19001
[3]	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, arXiv:1911.02262v3. · Zbl 1444.14050
[4]	Bachmann, T. and Fasel, J., ‘On the effectivity of spectra representing motivic cohomology theories’, Preprint, 2018, arXiv:1710.00594v3.
[5]	Bachmann, T. and Hoyois, M., ‘Norms in motivic homotopy theory’, to appear in Astérisque (2020). Preprint at arXiv:1711.03061.
[6]	Bhatt, B. and Halpern-Leistner, D., ‘Tannaka duality revisited’, Adv. Math.316 (2017), 576-612. · Zbl 1401.14013
[7]	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. · Zbl 1357.19004
[8]	Clausen, D., Mathew, A., Naumann, N. and Noel, J., ‘Descent in algebraic \(K\)-theory and a conjecture of Ausoni-Rognes’, J. Eur. Math. Soc. (JEMS)22(4) (2020), 1149-1200. · Zbl 1453.18011
[9]	Déglise, F., ‘Orientation theory in arithmetic geometry’, in \(K\)-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.
[10]	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 arXiv:1805.05920.
[11]	Druzhinin, A., ‘Framed motives of smooth affine pairs’, Preprint, 2020, arXiv:1803.11388v8.
[12]	Elmanto, E., Hoyois, M., Khan, A. A., Sosnilo, V. and Yakerson, M., ‘Motivic infinite loop spaces’, Preprint, 2019, arXiv:1711.05248v5. · Zbl 1444.14050
[13]	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. · Zbl 1444.14050
[14]	Elmanto, E. and Kolderup, H., ‘On modules over motivic ring spectra’, Ann. K-Theory5(2) (2020), 327-355. · Zbl 1440.14120
[15]	Galatius, S., Madsen, I., Tillmann, U. and Weiss, M., ‘The homotopy type of the cobordism category’, Acta Math.202(2) (2009), 195-239. · Zbl 1221.57039
[16]	Garkusha, G. and Neshitov, A., ‘Fibrant resolutions for motivic Thom spectra’, Preprint, 2018, arXiv:1804.07621v1.
[17]	Garkusha, G., Neshitov, A. and Panin, I., ‘Framed motives of relative motivic spheres’, Preprint, 2018, arXiv:1604.02732v3.
[18]	Garkusha, G. and Panin, I., ‘Framed motives of algebraic varieties (after V. Voevodsky)’, to appear in J. Amer. Math. Soc. (2020). Preprint at arXiv:1409.4372.
[19]	Garkusha, G. and Panin, I., ‘Homotopy invariant presheaves with framed transfers’, Camb. J. Math.8(1) (2020), 1-94. · Zbl 1453.14066
[20]	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. · Zbl 0153.22301
[21]	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.
[22]	Haugseng, R., ‘Iterated spans and classical topological field theories’, Math. Z.289(3) (2018), 1427-1488. · Zbl 1400.18006
[23]	Hoyois, M., ‘The localization theorem for framed motivic spaces’, to appear in Compos. Math. (2020). Preprint at arXiv:1807.04253.
[24]	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.
[25]	Khan, A. A. and Rydh, D., ‘Virtual Cartier divisors and blow-ups’, Preprint, 2019, arXiv:1802.05702v2.
[26]	Levine, M. and Morel, F., Algebraic Cobordism (Springer, Location, 2007). · Zbl 1188.14015
[27]	Lowrey, P. and Schürg, T., ‘Derived algebraic cobordism’, J. Inst. Math. Jussieu15 (2016), 407-443. · Zbl 1375.14087
[28]	Lurie, J., ‘Derived Algebraic Geometry’, Ph.D. thesis, Massachusetts Institute of Technology, 2004. Available at https://www.math.ias.edu/ lurie/papers/DAG.pdf.
[29]	Lurie, J., ‘\( \left(\infty, 2\right)\) -categories and the Goodwillie calculus I’, unpublished paper (2009). URL: https://www.math.ias.edu/ lurie/papers/GoodwillieI.pdf.
[30]	Lurie, J., ‘Higher algebra’, unpublished paper (2017). URL: https://www.math.ias.edu/ lurie/papers/HA.pdf.
[31]	Lurie, J., ‘Higher topos theory’, unpublished paper (2017). URL: https://www.math.ias.edu/ lurie/papers/HTT.pdf. · Zbl 1175.18001
[32]	Lurie, J., ‘Spectral algebraic geometry’, unpublished paper (2018). URL: https://www.math.ias.edu/ lurie/papers/SAG-rootfile.pdf.
[33]	Navarro, A., ‘Riemann-Roch for homotopy invariant \(K\)-theory and Gysin morphisms’, Preprint, 2016, arXiv:1605.00980v1. · Zbl 1391.14017
[34]	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.
[35]	Panin, I., ‘Oriented cohomology theories of algebraic varieties II’, Homology Homotopy Appl.11(1) (2009), 349-405. · Zbl 1169.14016
[36]	Quillen, D., ‘Elementary proofs of some results of cobordism theory using Steenrod operations’, Adv. Math.7(1) (1971), 29-56. · Zbl 0214.50502
[37]	Raptis, G., ‘Some characterizations of acyclic maps’, J. Homotopy Relat. Struct.14 (2019), 773-785. · Zbl 1427.55013
[38]	Röndigs, O. and Østvær, P. A., ‘Modules over motivic cohomology’, Adv. Math.219(2) (2008), 689-727. · Zbl 1180.14015
[39]	Randal-Williams, O., ‘“Group-completion”, local coefficient systems and perfection’, Q. J. Math.64(3) (2013), 795-803. · Zbl 1280.55002
[40]	Rydh, D., ‘Noetherian approximation of algebraic spaces and stacks’, J. Algebra422 (2015), 105-147. · Zbl 1308.14006
[41]	Spitzweck, M., ‘A commutative \(\mathbf{P}^1\)-spectrum representing motivic cohomology over Dedekind domains’, Mém. Soc. Math. Fr.157 (2018). · Zbl 1408.14081
[42]	, ‘The Stacks Project’, URL: http://stacks.math.columbia.edu.
[43]	Toën, B. and Vezzosi, G., ‘Homotopical Algebraic Geometry. II: Geometric stacks and applications’, Mem. Amer. Math. Soc.193(902) (2008). · Zbl 1145.14003
[44]	Voevodsky, V., ‘Notes on framed correspondences’, unpublished paper (2001). URL: http://www.math.ias.edu/vladimir/files/framed.pdf.
[45]	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.