Dans ce travail, nous étudions la structure de la catégorie -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 -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’-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 -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 . Next, we prove that is canonically -oriented; we compare with the category of rational Milnor–Witt motives; and we relate the rational bivariant -theory to Chow–Witt groups. These results are derived from analogous statements for the minus part of .
Accepté le :
Publié le :
Keywords: Motivic homotopy, motivic cohomology, six operations, Chow-Witt groups, K-theory, hermitian K-theory
Mot clés : Théorie $\mathbb{A}^1$-homotopique, cohomologie motivique, six opérations, groupes de Chow-Witt, K-théorie, K-théorie hermitienne
@article{JEP_2021__8__533_0, author = {D\'eglise, Fr\'ed\'eric and Fasel, Jean and Jin, Fangzhou and Khan, Adeel A.}, 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 = {http://www.numdam.org/articles/10.5802/jep.153/} }
TY - JOUR AU - Déglise, Frédéric AU - Fasel, Jean AU - Jin, Fangzhou AU - Khan, Adeel A. TI - On the rational motivic homotopy category JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 533 EP - 583 VL - 8 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.153/ DO - 10.5802/jep.153 LA - en ID - JEP_2021__8__533_0 ER -
%0 Journal Article %A Déglise, Frédéric %A Fasel, Jean %A Jin, Fangzhou %A Khan, Adeel A. %T On the rational motivic homotopy category %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 533-583 %V 8 %I Ecole polytechnique %U http://www.numdam.org/articles/10.5802/jep.153/ %R 10.5802/jep.153 %G en %F JEP_2021__8__533_0
Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel A. On the rational motivic homotopy category. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 533-583. doi : 10.5802/jep.153. http://www.numdam.org/articles/10.5802/jep.153/
[AGV73] Théorie des topos et cohomologie étale des schémas, Lect. Notes in Math., 269, 270, 305, Springer-Verlag, 1972–1973 Séminaire de Géométrie Algébrique du Bois–Marie 1963–64 (SGA 4)
[ALP17] Witt sheaves and the -inverted sphere spectrum, J. Topology, Volume 10 (2017) no. 2, pp. 370-385 | DOI | MR | Zbl
[Ana19] -oriented cohomology theories, 2019 | arXiv
[Ayo07] Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, Astérisque, 314-315, Société Mathématique de France, Paris, 2007 | Numdam | Zbl
[Ayo14] La réalisation étale et les opérations de Grothendieck, Ann. Sci. École Norm. Sup. (4), Volume 47 (2014) no. 1, pp. 1-145 | DOI | Zbl
[Bac18] Motivic and real étale stable homotopy theory, Compositio Math., Volume 154 (2018) no. 5, pp. 883-917 | DOI | MR | Zbl
[Bal01] Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture, -Theory, Volume 23 (2001) no. 1, pp. 15-30 | DOI | MR | Zbl
[Bal05] Witt groups, Handbook of -theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 539-576 | DOI | MR | Zbl
[BBD82] Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) (Astérisque), Volume 100, Société Mathématique de France, Paris, 1982, pp. 5-171 | MR | Zbl
[BCD + 20] Milnor-Witt motives, 2020 | arXiv
[BD17] Dimensional homotopy t-structures in motivic homotopy theory, Adv. Math., Volume 311 (2017), pp. 91-189 | DOI | MR | Zbl
[BF18] On the effectivity of spectra representing motivic cohomology theories, 2018 | arXiv
[BGPW02] The Gersten conjecture for Witt groups in the equicharacteristic case, Doc. Math., Volume 7 (2002), pp. 203-217 | MR | Zbl
[BH21] Norms in motivic homotopy theory, Astérisque, Société Mathématique de France, Paris, 2021 (to appear)
[BO74] Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4), Volume 7 (1974) no. 4, pp. 181-201 | DOI | MR | Zbl
[Bon14] Weights for relative motives: relation with mixed complexes of sheaves, Internat. Math. Res. Notices (2014) no. 17, pp. 4715-4767 | DOI | MR | Zbl
[BW02] A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4), Volume 35 (2002) no. 1, pp. 127-152 | DOI | Numdam | MR | Zbl
[CD15] Integral mixed motives in equal characteristics, Doc. Math. (2015), pp. 145-194 (Extra volume: Alexander S. Merkurjev’s sixtieth birthday) | MR | Zbl
[CD16] Étale motives, Compositio Math., Volume 152 (2016) no. 3, pp. 556-666 | DOI | Zbl
[CD19] Triangulated categories of mixed motives, Springer Monographs in Math., Springer, Cham, 2019 | DOI | Zbl
[CDH + 20a] Hermitian -theory for stable -categories I: Foundations, 2020 | arXiv
[CDH + 20b] Hermitian -theory for stable -categories II: Cobordism categories and additivity, 2020 | arXiv
[CDH + 20c] Hermitian -theory for stable -categories III: Grothendieck-Witt groups of rings, 2020 | arXiv
[CF14] Finite Chow-Witt correspondences, 2014 | arXiv
[Cis19] Cohomological methods in intersection theory (2019) (arXiv:1905.03478)
[CTHK97] The Bloch-Ogus-Gabber theorem, Algebraic -theory (Toronto, ON, 1996) (Fields Inst. Commun.), Volume 16, American Mathematical Society, Proovidence, RI, 1997, pp. 31-94 | MR | Zbl
[Del77] Cohomologie étale, Lect. Notes in Math., 569, Springer-Verlag, 1977 (Séminaire de Géométrie Algébrique du Bois–Marie SGA )
[Del87] Le déterminant de la cohomologie, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) (Contemp. Math.), Volume 67, American Mathematical Society, Providence, RI, 1987, pp. 93-177 | DOI | Zbl
[DF20] The Borel character, 2020 | arXiv
[DFJK19] Borel isomorphism and absolute purity, 2019 | arXiv
[DJK21] Fundamental classes in motivic homotopy theory, J. Eur. Math. Soc. (JEMS) (2021) (to appear)
[Dég18a] Bivariant theories in motivic stable homotopy, Doc. Math., Volume 23 (2018), pp. 997-1076 | MR | Zbl
[Dég18b] Orientation theory in arithmetic geometry, -Theory—Proceedings of the International Colloquium (Mumbai, 2016), Hindustan Book Agency, New Delhi, 2018, pp. 239-347 | Zbl
[EHK + 20] Modules over algebraic cobordism, Forum Math. Pi, Volume 8 (2020), e14, 44 pages | DOI | MR
[EK20a] Perfection in motivic homotopy theory, Proc. London Math. Soc. (3), Volume 120 (2020) no. 1, pp. 28-38 | DOI | MR | Zbl
[EK20b] On modules over motivic ring spectra, Ann. -Theory, Volume 5 (2020) no. 2, pp. 327-355 | DOI | MR | Zbl
[EKM08] The algebraic and geometric theory of quadratic forms, AMS Colloquium Publications, 56, American Mathematical Society, Providence, RI, 2008 | MR | Zbl
[Fas08] Groupes de Chow-Witt, Mém. Soc. Math. France (N.S.), 113, Société Mathématique de France, Paris, 2008 | Numdam | MR | Zbl
[Fel19] Morel homotopy modules and Milnor-Witt cycle modules, 2019 | arXiv
[Fel20] Milnor-Witt cycle modules, J. Pure Appl. Algebra, Volume 224 (2020) no. 7, p. 41 | DOI | MR | Zbl
[FS09] Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math., Volume 221 (2009) no. 1, pp. 302-329 | DOI | MR | Zbl
[Fuj02] A proof of the absolute purity conjecture (after Gabber), Algebraic geometry 2000, Azumino (Hotaka) (Adv. Stud. Pure Math.), Volume 36, Math. Soc. Japan, Tokyo, 2002, pp. 153-183 | DOI | MR | Zbl
[Ful98] Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1998 | MR | Zbl
[Gar19] Reconstructing rational stable motivic homotopy theory, Compositio Math., Volume 155 (2019) no. 7, pp. 1424-1443 | DOI | MR | Zbl
[Gil07] A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group, J. Pure Appl. Algebra, Volume 208 (2007) no. 2, pp. 391-419 | DOI | MR | Zbl
[Gro64] Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci., Volume 20 (1964), pp. 5-259 | DOI
[Gro77] Cohomologie -adique et fonctions , Lect. Notes in Math., 589, Springer-Verlag, 1977 Séminaire de Géométrie Algébrique du Bois–Marie 1965–66 (SGA 5)
[Har66] Residues and duality, Lect. Notes in Math., 20, Springer-Verlag, Berlin-New York, 1966 | MR | Zbl
[Hoy14] A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula, Algebraic Geom. Topol., Volume 14 (2014) no. 6, pp. 3603-3658 | DOI | MR | Zbl
[Héb11] Structure de poids à la Bondarko sur les motifs de Beilinson, Compositio Math., Volume 147 (2011) no. 5, pp. 1447-1462 | DOI | MR | Zbl
[ILO14] Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents (Illusie, L.; Laszlo, Y.; Orgogozo, F., eds.), Astérisque, 363-364, Société Mathématique de France, Paris, 2014 | Zbl
[Jac17] Real cohomology and the powers of the fundamental ideal in the Witt ring, Ann. -Theory, Volume 2 (2017) no. 3, pp. 357-385 | DOI | MR | Zbl
[Jin16] Borel–Moore motivic homology and weight structure on mixed motives, Math. Z., Volume 283 (2016) no. 3, pp. 1149-1183 | DOI | MR | Zbl
[Kha16] Motivic homotopy theory in derived algebraic geometry, Ph. D. Thesis, Universität Duisburg-Essen (2016) https://www.preschema.com/thesis/thesis.pdf
[Kha19] Virtual fundamental classes of derived stacks I, 2019 | arXiv
[Kha21] Voevodsky’s criterion for constructible categories of coefficients (2021) (Preprint, available at https://www.preschema.com/papers/six.pdf)
[Kne77] Symmetric bilinear forms over algebraic varieties, Conference on Quadratic Forms—1976 (Kingston, Ont., 1976) (Queen’s Papers in Pure and Appl. Math.), Volume 46, 1977, pp. 103-283 | Zbl
[Lam05] Introduction to quadratic forms over fields, Graduate Studies in Math., 67, American Mathematical Society, Providence, RI, 2005 | MR | Zbl
[Lur09] Higher topos theory, Annals of Math. Studies, 170, Princeton University Press, Princeton, NJ, 2009 | DOI | MR | Zbl
[Lur12] Higher algebra (2012) (Preprint, available at https://www.math.ias.edu/~lurie/papers/HigherAlgebra.pdf)
[Lur18] Spectral algebraic geometry (2018) (Preprint, available at https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf)
[Mor04] On the motivic of the sphere spectrum, Axiomatic, enriched and motivic homotopy theory (NATO Sci. Ser. II Math. Phys. Chem.), Volume 131, Kluwer Acad. Publ., 2004, pp. 219-260 | DOI | MR
[Mor06] Rational stable splitting of Grassmannians and rational motivic sphere spectrum, 2006
[Mor12] -algebraic topology over a field, Lect. Notes in Math., 2052, Springer, Heidelberg, 2012
[MV99] -homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci. (1999) no. 90, pp. 45-143 | DOI | MR
[Pan10] Homotopy invariance of the sheaf and of its cohomology, Quadratic forms, linear algebraic groups, and cohomology (Dev. Math.), Volume 18, Springer, New York, 2010, pp. 325-335 | DOI | MR | Zbl
[PW19] On the motivic commutative ring spectrum BO, St. Petersburg Math. J., Volume 30 (2019) no. 6, p. 933–972 | MR | Zbl
[Rob15] -theory and the bridge from motives to noncommutative motives, Adv. Math., Volume 269 (2015), pp. 399-550 | DOI | MR | Zbl
[RØ08] On modules over motivic ring spectra, Adv. Math., Volume 219 (2008) no. 2, p. 689–727 | Zbl
[Sch94] Real and étale cohomology, Lect. Notes in Math., 1588, Springer-Verlag, Berlin, 1994 | Zbl
[Sch00] Integral elements in -theory and products of modular curves, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998) (NATO Sci. Ser. C Math. Phys. Sci.), Volume 548, Kluwer Acad. Publ., 2000, pp. 467-489 | MR | Zbl
[Sch17] Hermitian -theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra, Volume 221 (2017) no. 7, pp. 1729-1844 | DOI | MR | Zbl
[Spi99] A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Amer. Math. Soc., Volume 12 (1999) no. 2, pp. 381-444 | DOI | MR | Zbl
[Spi18] A commutative -spectrum representing motivic cohomology over Dedekind domains, Mém. Soc. Math. France (N.S.), 157, Société Mathématique de France, Paris, 2018 | DOI | MR | Zbl
[ST15] Geometric models for higher Grothendieck-Witt groups in -homotopy theory, Math. Ann., Volume 362 (2015) no. 3-4, pp. 1143-1167 | DOI | MR | Zbl
[Sta21] The Stacks project, https://stacks.math.columbia.edu, 2021
[Tho84] Absolute cohomological purity, Bull. Soc. math. France, Volume 112 (1984) no. 3, pp. 397-406 | DOI | Numdam | MR | Zbl
[TT90] Higher algebraic -theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III (Progress in Math.), Volume 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435 | DOI | MR
Cité par Sources :