On the rational motivic homotopy category

Frédéric Déglise; Jean Fasel; Fangzhou Jin; Adeel A. Khan

doi:10.5802/jep.153

On the rational motivic homotopy category
[Sur la catégorie

𝔸^{1}

-homotopique rationnelle]

Frédéric Déglise ¹ ; Jean Fasel ² ; Fangzhou Jin ³ ; Adeel A. Khan ⁴

¹ ENS de Lyon, UMPA, UMR 5669 46 allée d’Italie, 69364 Lyon Cedex 07, France
² Institut Fourier - UMR 5582, Université Grenoble-Alpes CS 40700, 38058 Grenoble Cedex 9, France
³ School of Mathematical Sciences, Tongji University Siping Road 1239, 200092 Shanghai, China
⁴ Institut des Hautes Études Scientifiques 35 route de Chartres, 91440 Bures-sur-Yvette, France and Institute of Mathematics, Academia Sinica Taipei 10617, Taiwan

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 533-583.

Résumé
Abstract

Dans ce travail, nous étudions la structure de la catégorie $𝔸^{1}$ -homotopique stable rationnelle sur une base arbitraire. Notre première famille de résultats concerne les six opérations : nous prouvons la pureté absolue, la stabilité des objets constructibles et la dualité de Grothendieck-Verdier pour cette catégorie. Dans un deuxième temps, nous prouvons que la catégorie $𝔸^{1}$ -homotopique stable rationnelle est canoniquement SL-orientée et la comparons à la catégorie des motifs rationnels de Milnor-Witt. Cela permet de calculer les groupes d’ $𝔸^{1}$ -homotopie stable bivariants en termes des groupes de Chow-Witt supérieurs. Ces résultats s’obtiennent à partir d’énoncés analogues pour la partie négative de la catégorie $𝔸^{1}$ -homotopique stable 2-localisée.

We study the structure of the rational motivic stable homotopy category over general base schemes. Our first class of results concerns the six operations: we prove absolute purity, stability of constructible objects, and Grothendieck–Verdier duality for ${SH}_{ℚ}$ . Next, we prove that ${SH}_{ℚ}$ is canonically $SL$ -oriented; we compare ${SH}_{ℚ}$ with the category of rational Milnor–Witt motives; and we relate the rational bivariant $𝔸^{1}$ -theory to Chow–Witt groups. These results are derived from analogous statements for the minus part of $SH [1 / 2]$ .

Reçu le : 2020-07-14
Accepté le : 2021-01-18
Publié le : 2021-02-23

DOI : 10.5802/jep.153

Classification : 14F42, 19E15, 19G12, 11E81, 14C25, 14C35
Keywords: Motivic homotopy, motivic cohomology, six operations, Chow-Witt groups, K-theory, hermitian K-theory
Mots-clés : Théorie $\mathbb{A}^1$-homotopique, cohomologie motivique, six opérations, groupes de Chow-Witt, K-théorie, K-théorie hermitienne

Affiliations des auteurs :

Frédéric Déglise ¹ ; Jean Fasel ² ; Fangzhou Jin ³ ; Adeel A. Khan ⁴

¹ ENS de Lyon, UMPA, UMR 5669 46 allée d’Italie, 69364 Lyon Cedex 07, France
² Institut Fourier - UMR 5582, Université Grenoble-Alpes CS 40700, 38058 Grenoble Cedex 9, France
³ School of Mathematical Sciences, Tongji University Siping Road 1239, 200092 Shanghai, China
⁴ Institut des Hautes Études Scientifiques 35 route de Chartres, 91440 Bures-sur-Yvette, France and Institute of Mathematics, Academia Sinica Taipei 10617, Taiwan

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Fr\'ed\'eric D\'eglise and Jean Fasel and Fangzhou Jin and Adeel A. Khan},
     title = {On the rational motivic homotopy category},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {533--583},
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     year = {2021},
     doi = {10.5802/jep.153},
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Frédéric Déglise; Jean Fasel; Fangzhou Jin; Adeel A. Khan. On the rational motivic homotopy category. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 533-583. doi : 10.5802/jep.153. https://jep.centre-mersenne.org/articles/10.5802/jep.153/

Bibliographie
Cité par

[AGV73] M. Artin, A. Grothendieck & J.-L. Verdier - Théorie des topos et cohomologie étale des schémas, Lect. Notes in Math., vol. 269, 270, 305, Springer-Verlag, 1972–1973, Séminaire de Géométrie Algébrique du Bois–Marie 1963–64 (SGA 4)

[ALP17] A. Ananyevskiy, M. Levine & I. Panin - “Witt sheaves and the $η$ -inverted sphere spectrum”, J. Topology 10 (2017) no. 2, p. 370-385 | DOI | MR | Zbl

[Ana19] A. Ananyevskiy - “ $SL$ -oriented cohomology theories”, 2019 | arXiv

[Ayo07] J. Ayoub - Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, Astérisque, vol. 314-315, Société Mathématique de France, Paris, 2007 | Numdam | Zbl

[Ayo14] J. Ayoub - “La réalisation étale et les opérations de Grothendieck”, Ann. Sci. École Norm. Sup. (4) 47 (2014) no. 1, p. 1-145 | DOI | Zbl

[Bac18] T. Bachmann - “Motivic and real étale stable homotopy theory”, Compositio Math. 154 (2018) no. 5, p. 883-917 | DOI | MR | Zbl

[Bal01] P. Balmer - “Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture”, $K$ -Theory 23 (2001) no. 1, p. 15-30 | DOI | MR | Zbl

[Bal05] P. Balmer - “Witt groups”, in Handbook of $K$ -theory. Vol. 1, 2, Springer, Berlin, 2005, p. 539-576 | DOI | MR | Zbl

[BBD82] A. A. Beĭlinson, J. Bernstein & P. Deligne - “Faisceaux pervers”, in Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Société Mathématique de France, Paris, 1982, p. 5-171 | MR | Zbl

[BCD + 20] T. Bachmann, B. Calmès, F. Déglise, J. Fasel & P. A. Østvær - “Milnor-Witt motives”, 2020 | arXiv

[BD17] M. Bondarko & F. Déglise - “Dimensional homotopy t-structures in motivic homotopy theory”, Adv. Math. 311 (2017), p. 91-189 | DOI | MR | Zbl

[BF18] T. Bachmann & J. Fasel - “On the effectivity of spectra representing motivic cohomology theories”, 2018 | arXiv

[BGPW02] P. Balmer, S. Gille, I. Panin & C. Walter - “The Gersten conjecture for Witt groups in the equicharacteristic case”, Doc. Math. 7 (2002), p. 203-217 | MR | Zbl

[BH21] T. Bachmann & M. Hoyois - Norms in motivic homotopy theory, Astérisque, Société Mathématique de France, Paris, 2021, to appear

[BO74] S. Bloch & A. Ogus - “Gersten’s conjecture and the homology of schemes”, Ann. Sci. École Norm. Sup. (4) 7 (1974) no. 4, p. 181-201 | DOI | MR | Zbl

[Bon14] M. Bondarko - “Weights for relative motives: relation with mixed complexes of sheaves”, Internat. Math. Res. Notices (2014) no. 17, p. 4715-4767 | DOI | MR | Zbl

[BW02] P. Balmer & C. Walter - “A Gersten-Witt spectral sequence for regular schemes”, Ann. Sci. École Norm. Sup. (4) 35 (2002) no. 1, p. 127-152 | DOI | Numdam | MR | Zbl

[CD15] D.-C. Cisinski & F. Déglise - “Integral mixed motives in equal characteristics”, Doc. Math. (2015), p. 145-194, Extra volume: Alexander S. Merkurjev’s sixtieth birthday | MR | Zbl

[CD16] D.-C. Cisinski & F. Déglise - “Étale motives”, Compositio Math. 152 (2016) no. 3, p. 556-666 | DOI | Zbl

[CD19] D.-C. Cisinski & F. Déglise - Triangulated categories of mixed motives, Springer Monographs in Math., Springer, Cham, 2019 | DOI | Zbl

[CDH + 20a] B. Calmès, E. Dotto, J. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus & W. Steimle - “Hermitian $K$ -theory for stable $\infty$ -categories I: Foundations”, 2020 | arXiv

[CDH + 20b] B. Calmès, E. Dotto, J. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus & W. Steimle - “Hermitian $K$ -theory for stable $\infty$ -categories II: Cobordism categories and additivity”, 2020 | arXiv

[CDH + 20c] B. Calmès, E. Dotto, J. Harpaz, F. Hebestreit, M. Land, K. Moi, D. Nardin, T. Nikolaus & W. Steimle - “Hermitian $K$ -theory for stable $\infty$ -categories III: Grothendieck-Witt groups of rings”, 2020 | arXiv

[CF14] B. Calmès & J. Fasel - “Finite Chow-Witt correspondences”, 2014 | arXiv

[Cis19] D.-C. Cisinski - “Cohomological methods in intersection theory” (2019), arXiv:1905.03478

[CTHK97] J.-L. Colliot-Thélène, R. Hoobler & B. Kahn - “The Bloch-Ogus-Gabber theorem”, in Algebraic $K$ -theory (Toronto, ON, 1996), Fields Inst. Commun., vol. 16, American Mathematical Society, Proovidence, RI, 1997, p. 31-94 | MR | Zbl

[Del77] P. Deligne - Cohomologie étale, Lect. Notes in Math., vol. 569, Springer-Verlag, 1977, Séminaire de Géométrie Algébrique du Bois–Marie SGA $4 \frac{1}{2}$

[Del87] P. Deligne - “Le déterminant de la cohomologie”, in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol. 67, American Mathematical Society, Providence, RI, 1987, p. 93-177 | DOI | Zbl

[DF20] F. Déglise & J. Fasel - “The Borel character”, 2020 | arXiv

[DFJK19] F. Déglise, J. Fasel, F. Jin & A. A. Khan - “Borel isomorphism and absolute purity”, 2019 | arXiv

[DJK21] F. Déglise, F. Jin & A. A. Khan - “Fundamental classes in motivic homotopy theory”, J. Eur. Math. Soc. (JEMS) (2021), to appear

[Dég18a] F. Déglise - “Bivariant theories in motivic stable homotopy”, Doc. Math. 23 (2018), p. 997-1076 | MR | Zbl

[Dég18b] F. Déglise - “Orientation theory in arithmetic geometry”, in $K$ -Theory—Proceedings of the International Colloquium (Mumbai, 2016), Hindustan Book Agency, New Delhi, 2018, p. 239-347 | Zbl

[EHK + 20] E. Elmanto, M. Hoyois, A. A. Khan, V. Sosnilo & M. Yakerson - “Modules over algebraic cobordism”, Forum Math. Pi 8 (2020), article ID e14, 44 pages | DOI | MR

[EK20a] E. Elmanto & A. A. Khan - “Perfection in motivic homotopy theory”, Proc. London Math. Soc. (3) 120 (2020) no. 1, p. 28-38 | DOI | MR | Zbl

[EK20b] E. Elmanto & H. Kolderup - “On modules over motivic ring spectra”, Ann. $K$ -Theory 5 (2020) no. 2, p. 327-355 | DOI | MR | Zbl

[EKM08] R. Elman, N. Karpenko & A. Merkurjev - The algebraic and geometric theory of quadratic forms, AMS Colloquium Publications, vol. 56, American Mathematical Society, Providence, RI, 2008 | MR | Zbl

[Fas08] J. Fasel - Groupes de Chow-Witt, Mém. Soc. Math. France (N.S.), vol. 113, Société Mathématique de France, Paris, 2008 | Numdam | MR | Zbl

[Fel19] N. Feld - “Morel homotopy modules and Milnor-Witt cycle modules”, 2019 | arXiv

[Fel20] N. Feld - “Milnor-Witt cycle modules”, J. Pure Appl. Algebra 224 (2020) no. 7, p. 41 | DOI | MR | Zbl

[FS09] J. Fasel & V. Srinivas - “Chow-Witt groups and Grothendieck-Witt groups of regular schemes”, Adv. Math. 221 (2009) no. 1, p. 302-329 | DOI | MR | Zbl

[Fuj02] K. Fujiwara - “A proof of the absolute purity conjecture (after Gabber)”, in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, p. 153-183 | DOI | MR | Zbl

[Ful98] W. Fulton - Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1998 | MR | Zbl

[Gar19] G. Garkusha - “Reconstructing rational stable motivic homotopy theory”, Compositio Math. 155 (2019) no. 7, p. 1424-1443 | DOI | MR | Zbl

[Gil07] S. Gille - “A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group”, J. Pure Appl. Algebra 208 (2007) no. 2, p. 391-419 | DOI | MR | Zbl

[Gro64] A. Grothendieck - “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I”, Publ. Math. Inst. Hautes Études Sci. 20 (1964), p. 5-259 | DOI

[Gro77] A. Grothendieck - Cohomologie $ℓ$ -adique et fonctions $L$ , Lect. Notes in Math., vol. 589, Springer-Verlag, 1977, Séminaire de Géométrie Algébrique du Bois–Marie 1965–66 (SGA 5)

[Har66] R. Hartshorne - Residues and duality, Lect. Notes in Math., vol. 20, Springer-Verlag, Berlin-New York, 1966 | MR | Zbl

[Hoy14] M. Hoyois - “A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula”, Algebraic Geom. Topol. 14 (2014) no. 6, p. 3603-3658 | DOI | MR | Zbl

[Héb11] D. Hébert - “Structure de poids à la Bondarko sur les motifs de Beilinson”, Compositio Math. 147 (2011) no. 5, p. 1447-1462 | DOI | MR | Zbl

[ILO14] - Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (L. Illusie, Y. Laszlo & F. Orgogozo, eds.), Astérisque, vol. 363-364, Société Mathématique de France, Paris, 2014 | Zbl

[Jac17] J. Jacobson - “Real cohomology and the powers of the fundamental ideal in the Witt ring”, Ann. $K$ -Theory 2 (2017) no. 3, p. 357-385 | DOI | MR | Zbl

[Jin16] F. Jin - “Borel–Moore motivic homology and weight structure on mixed motives”, Math. Z. 283 (2016) no. 3, p. 1149-1183 | DOI | MR | Zbl

[Kha16] A. A. Khan - Motivic homotopy theory in derived algebraic geometry, Ph. D. Thesis, Universität Duisburg-Essen, 2016

[Kha19] A. A. Khan - “Virtual fundamental classes of derived stacks I”, 2019 | arXiv

[Kha21] A. A. Khan - “Voevodsky’s criterion for constructible categories of coefficients” (2021), Preprint, available at https://www.preschema.com/papers/six.pdf

[Kne77] M. Knebusch - “Symmetric bilinear forms over algebraic varieties”, in Conference on Quadratic Forms—1976 (Kingston, Ont., 1976), Queen’s Papers in Pure and Appl. Math., vol. 46, 1977, p. 103-283 | Zbl

[Lam05] T. Y. Lam - Introduction to quadratic forms over fields, Graduate Studies in Math., vol. 67, American Mathematical Society, Providence, RI, 2005 | MR | Zbl

[Lur09] J. Lurie - Higher topos theory, Annals of Math. Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009 | DOI | MR | Zbl

[Lur12] J. Lurie - “Higher algebra” (2012), Preprint, available at https://www.math.ias.edu/~lurie/papers/HigherAlgebra.pdf

[Lur18] J. Lurie - “Spectral algebraic geometry” (2018), Preprint, available at https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf

[Mor04] F. Morel - “On the motivic $π_{0}$ of the sphere spectrum”, in Axiomatic, enriched and motivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., 2004, p. 219-260 | DOI | MR

[Mor06] F. Morel - “Rational stable splitting of Grassmannians and rational motivic sphere spectrum”, 2006

[Mor12] F. Morel - $𝔸^{1}$ -algebraic topology over a field, Lect. Notes in Math., vol. 2052, Springer, Heidelberg, 2012

[MV99] F. Morel & V. Voevodsky - “ $𝔸^{1}$ -homotopy theory of schemes”, Publ. Math. Inst. Hautes Études Sci. (1999) no. 90, p. 45-143 | DOI | MR

[Pan10] I. Panin - “Homotopy invariance of the sheaf $W_{Nis}$ and of its cohomology”, in Quadratic forms, linear algebraic groups, and cohomology, Dev. Math., vol. 18, Springer, New York, 2010, p. 325-335 | DOI | MR | Zbl

[PW19] I. Panin & C. Walter - “On the motivic commutative ring spectrum BO”, St. Petersburg Math. J. 30 (2019) no. 6, p. 933–972 | MR | Zbl

[Rob15] M. Robalo - “ $K$ -theory and the bridge from motives to noncommutative motives”, Adv. Math. 269 (2015), p. 399-550 | DOI | MR | Zbl

[RØ08] O. Röndigs & P. A. Østvær - “On modules over motivic ring spectra”, Adv. Math. 219 (2008) no. 2, p. 689–727 | Zbl

[Sch94] C. Scheiderer - Real and étale cohomology, Lect. Notes in Math., vol. 1588, Springer-Verlag, Berlin, 1994 | Zbl

[Sch00] A. Scholl - “Integral elements in $K$ -theory and products of modular curves”, in The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C Math. Phys. Sci., vol. 548, Kluwer Acad. Publ., 2000, p. 467-489 | MR | Zbl

[Sch17] M. Schlichting - “Hermitian $K$ -theory, derived equivalences and Karoubi’s fundamental theorem”, J. Pure Appl. Algebra 221 (2017) no. 7, p. 1729-1844 | DOI | MR | Zbl

[Spi99] M. Spivakovsky - “A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms”, J. Amer. Math. Soc. 12 (1999) no. 2, p. 381-444 | DOI | MR | Zbl

[Spi18] M. Spitzweck - A commutative $ℙ^{1}$ -spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. France (N.S.), vol. 157, Société Mathématique de France, Paris, 2018 | DOI | MR | Zbl

[ST15] M. Schlichting & G. S. Tripathi - “Geometric models for higher Grothendieck-Witt groups in $𝔸^{1}$ -homotopy theory”, Math. Ann. 362 (2015) no. 3-4, p. 1143-1167 | DOI | MR | Zbl

[Sta21] Stacks project authors - “The Stacks project”, https://stacks.math.columbia.edu, 2021

[Tho84] R. W. Thomason - “Absolute cohomological purity”, Bull. Soc. math. France 112 (1984) no. 3, p. 397-406 | DOI | Numdam | MR | Zbl

[TT90] R. W. Thomason & T. Trobaugh - “Higher algebraic $K$ -theory of schemes and of derived categories”, in The Grothendieck Festschrift, Vol. III, Progress in Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, p. 247-435 | DOI | MR

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