On the rational motivic homotopy category
[Sur la catégorie 𝔸 1 -homotopique rationnelle]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 533-583.

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 . Next, we prove that SH is canonically SL-oriented; we compare SH 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 :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.153
Classification : 14F42,  19E15,  19G12,  11E81,  14C25,  14C35
Mots clés : Théorie 𝔸 1 -homotopique, cohomologie motivique, six opérations, groupes de Chow-Witt, K-théorie, K-théorie hermitienne
@article{JEP_2021__8__533_0,
     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},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.153},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.153/}
}
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/

[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 | Article | MR 3653315 | Zbl 1378.14021

[Ana19] A. Ananyevskiy - “SL-oriented cohomology theories”, 2019 | arXiv:1901.01597

[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 1153.14001

[Ayo14] J. Ayoub - “La réalisation étale et les opérations de Grothendieck”, Ann. Sci. École Norm. Sup. (4) 47 (2014) no. 1, p. 1-145 | Article | Zbl 1354.18016

[Bac18] T. Bachmann - “Motivic and real étale stable homotopy theory”, Compositio Math. 154 (2018) no. 5, p. 883-917 | Article | MR 3781990 | Zbl 06855294

[Bal01] P. Balmer - “Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture”, K-Theory 23 (2001) no. 1, p. 15-30 | Article | MR 1852452 | Zbl 0987.19002

[Bal05] P. Balmer - “Witt groups”, in Handbook of K-theory. Vol. 1, 2, Springer, Berlin, 2005, p. 539-576 | Article | MR 2181829 | Zbl 1115.19004

[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 751966 | Zbl 0536.14011

[BCD + 20] T. Bachmann, B. Calmès, F. Déglise, J. Fasel & P. A. Østvær - “Milnor-Witt motives”, 2020 | arXiv:2004.06634v1

[BD17] M. Bondarko & F. Déglise - “Dimensional homotopy t-structures in motivic homotopy theory”, Adv. Math. 311 (2017), p. 91-189 | Article | MR 3628213 | Zbl 1403.14053

[BF18] T. Bachmann & J. Fasel - “On the effectivity of spectra representing motivic cohomology theories”, 2018 | arXiv:1710.00594

[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 1934649 | Zbl 1015.19002

[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 | Article | MR 412191 | Zbl 0307.14008

[Bon14] M. Bondarko - “Weights for relative motives: relation with mixed complexes of sheaves”, Internat. Math. Res. Notices (2014) no. 17, p. 4715-4767 | Article | MR 3257549 | Zbl 1400.14062

[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 | Article | Numdam | MR 1886007 | Zbl 1012.19003

[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 3404379 | Zbl 1357.19004

[CD16] D.-C. Cisinski & F. Déglise - “Étale motives”, Compositio Math. 152 (2016) no. 3, p. 556-666 | Article | Zbl 06578150

[CD19] D.-C. Cisinski & F. Déglise - Triangulated categories of mixed motives, Springer Monographs in Math., Springer, Cham, 2019 | Article | Zbl 07138952

[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 -categories I: Foundations”, 2020 | arXiv:2009.07223

[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 -categories II: Cobordism categories and additivity”, 2020 | arXiv:2009.07224

[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 -categories III: Grothendieck-Witt groups of rings”, 2020 | arXiv:2009.07225

[CF14] B. Calmès & J. Fasel - “Finite Chow-Witt correspondences”, 2014 | arXiv:1412.2989

[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 1466971 | Zbl 0911.14004

[Del77] P. Deligne - Cohomologie étale, Lect. Notes in Math., vol. 569, Springer-Verlag, 1977, Séminaire de Géométrie Algébrique du Bois–Marie SGA 41 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 | Article | Zbl 0629.14008

[DF20] F. Déglise & J. Fasel - “The Borel character”, 2020 | arXiv:1903.11679

[DFJK19] F. Déglise, J. Fasel, F. Jin & A. A. Khan - “Borel isomorphism and absolute purity”, 2019 | arXiv:1902.02055

[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 3874952 | Zbl 1423.14152

[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 1451.14067

[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 | Article | MR 4190058

[EK20a] E. Elmanto & A. A. Khan - “Perfection in motivic homotopy theory”, Proc. London Math. Soc. (3) 120 (2020) no. 1, p. 28-38 | Article | MR 3999675 | Zbl 1440.14123

[EK20b] E. Elmanto & H. Kolderup - “On modules over motivic ring spectra”, Ann. K-Theory 5 (2020) no. 2, p. 327-355 | Article | MR 4113773 | Zbl 1440.14120

[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 2427530 | Zbl 1165.11042

[Fas08] J. Fasel - Groupes de Chow-Witt, Mém. Soc. Math. France (N.S.), vol. 113, Société Mathématique de France, Paris, 2008 | Numdam | MR 2542148 | Zbl 1190.14001

[Fel19] N. Feld - “Morel homotopy modules and Milnor-Witt cycle modules”, 2019 | arXiv:1912.12680

[Fel20] N. Feld - “Milnor-Witt cycle modules”, J. Pure Appl. Algebra 224 (2020) no. 7, p. 41 | Article | MR 4058234 | Zbl 1442.14026

[FS09] J. Fasel & V. Srinivas - “Chow-Witt groups and Grothendieck-Witt groups of regular schemes”, Adv. Math. 221 (2009) no. 1, p. 302-329 | Article | MR 2509328 | Zbl 1167.13006

[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 | Article | MR 1971516 | Zbl 1059.14026

[Ful98] W. Fulton - Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1998 | MR 1644323 | Zbl 0885.14002

[Gar19] G. Garkusha - “Reconstructing rational stable motivic homotopy theory”, Compositio Math. 155 (2019) no. 7, p. 1424-1443 | Article | MR 3975501 | Zbl 07077742

[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 | Article | MR 2277683 | Zbl 1127.19005

[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 | Article

[Gro77] A. Grothendieck - Cohomologie -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 222093 | Zbl 0212.26101

[Hoy14] M. Hoyois - “A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula”, Algebraic Geom. Topol. 14 (2014) no. 6, p. 3603-3658 | Article | MR 3302973 | Zbl 1351.14013

[Héb11] D. Hébert - “Structure de poids à la Bondarko sur les motifs de Beilinson”, Compositio Math. 147 (2011) no. 5, p. 1447-1462 | Article | MR 2834728 | Zbl 1233.14017

[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 1297.14003

[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 | Article | MR 3658988 | Zbl 1427.14047

[Jin16] F. Jin - “Borel–Moore motivic homology and weight structure on mixed motives”, Math. Z. 283 (2016) no. 3, p. 1149-1183 | Article | MR 3519998 | Zbl 1375.14023

[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:1909.01332

[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 0408.15019

[Lam05] T. Y. Lam - Introduction to quadratic forms over fields, Graduate Studies in Math., vol. 67, American Mathematical Society, Providence, RI, 2005 | MR 2104929 | Zbl 1068.11023

[Lur09] J. Lurie - Higher topos theory, Annals of Math. Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009 | Article | MR 2522659 | Zbl 1175.18001

[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 | Article | MR 2061856

[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 | Article | MR 1813224

[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 | Article | MR 2648736 | Zbl 1216.18014

[PW19] I. Panin & C. Walter - “On the motivic commutative ring spectrum BO”, St. Petersburg Math. J. 30 (2019) no. 6, p. 933–972 | MR 3882540 | Zbl 1428.14011

[Rob15] M. Robalo - “K-theory and the bridge from motives to noncommutative motives”, Adv. Math. 269 (2015), p. 399-550 | Article | MR 3281141 | Zbl 1315.14030

[RØ08] O. Röndigs & P. A. Østvær - “On modules over motivic ring spectra”, Adv. Math. 219 (2008) no. 2, p. 689–727 | Zbl 1180.14015

[Sch94] C. Scheiderer - Real and étale cohomology, Lect. Notes in Math., vol. 1588, Springer-Verlag, Berlin, 1994 | Zbl 0852.14003

[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 1744957 | Zbl 0982.14009

[Sch17] M. Schlichting - “Hermitian K-theory, derived equivalences and Karoubi’s fundamental theorem”, J. Pure Appl. Algebra 221 (2017) no. 7, p. 1729-1844 | Article | MR 3614976 | Zbl 1360.19008

[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 | Article | MR 1647069 | Zbl 0919.13009

[Spi18] M. Spitzweck - A commutative 1 -spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. France (N.S.), vol. 157, Société Mathématique de France, Paris, 2018 | Article | MR 3865569 | Zbl 1408.14081

[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 | Article | MR 3368095 | Zbl 1331.14028

[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 | Article | Numdam | MR 794741 | Zbl 0584.14007

[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 | Article | MR 1106918