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On the rational motivic homotopy category. (Sur la catégorie $$\mathbb{A}^1$$-homotopique rationnelle.) (English. French summary) Zbl 1471.14052
The authors study the rational stable motivic homotopy category $$\mathrm{SH}(S)_{\mathbb{Q}}$$ over a base scheme $$S$$ and prove the following results: (I) Weak orientation, i.e. Thom isomorphisms for SL-oriented vector bundles. (II) Absolute purity. (III) Finiteness, i.e. the six operations preserve constructible objects. (IV) Grothendieck-Verdier Duality. (V) Comparison, i.e. $$\mathrm{SH}_{\mathbb{Q}}$$ is equivalent to the category of modules over the rational Milnor-Witt motivic cohomology spectrum. (VI) Bivariant theory: over a regular base the bivariant theory associated with $$\mathrm{SH}_{\mathbb{Q}}$$ coincides in degree $$(2n,n)$$ with the Chow-Witt group of quadratic cyclces of $$\delta$$-dimension $$n$$.
The category $$\mathrm{SH}(S)_{\mathbb{Q}}$$ admits a decomposition $$\mathrm{SH}(S)_{\mathbb{Q},+} \times \mathrm{SH}(S)_{\mathbb{Q},-}$$ into a plus part and a minus part (already after inverting $$2$$). Similar results for the plus part $$\mathrm{SH}(S)_{\mathbb{Q},+}$$ had been proven by D.-C. Cisinski and F. Déglise [Triangulated categories of mixed motives. Cham: Springer (2019; Zbl 07138952)] and the authors show the corresponding statements for the minus part $$\mathrm{SH}(S)_{\mathbb{Q},-}$$.
The Key Lemma (Lemma 2.2) of the paper states that, for any scheme $$S$$, the minus part $$\mathrm{SH}(S)_{\mathbb{Q},-}$$ is equivalent to the minus part of its characteristic zero fibre $$S_\mathbb{Q} = S \times_{\mathrm{Spec}(\mathbb{Z})}\mathrm{Spec}(\mathbb{Q})$$ which is used subsequently to prove finiteness (Proposition 3.3), Grothendieck-Verdier duality (Proposition 3.4), and absolute purity (Theorem 3.6).
The next result of the paper is the comparison of the minus part with modules over the $$\Lambda$$-linear homotopical Witt spectrum for a coefficient ring $$\Lambda\subseteq\mathbb{Q}$$ such that $$2\in\Lambda^\times$$. First, it is shown for regular $$S$$ that the Witt sheaf $$\underline{\mathrm{W}}_S^\Lambda$$ is an infinite loop space in $$\mathrm{D}_{\mathbb{A}^1}(S,\Lambda)$$ (Theorem 5.7) and hence defining a spectrum $$\mathbf{H}\underline{\mathrm{W}}_S^\Lambda$$ which turns out to be isomorphic, as a ring spectrum in $$\mathrm{D}_{\mathbb{A}^1}(S,\Lambda)$$, to the minus part $$\Lambda_{S,-}$$ of the unit $$\Lambda_S$$ (Theorem 5.13) so that there is an equivalence $$\mathrm{D}_{\mathbb{A}^1}(S,\Lambda)_- \simeq \mathbf{H}\underline{\mathrm{W}}_S^\Lambda\text{-mod}$$ of symmetric monoidal $$\infty$$-categories (Corollary 5.16). Secondly, this extends to arbitrary $$S$$ via pullback along the canonical morphism $$S\to\mathrm{Spec}(\mathbb{Z})$$ (Definition 5.18 and 5.19). Finally, by combining with the analogous result for the plus part, the equivalence between $$\mathrm{SH}(S)_{\mathbb{Q}}$$ and the category of modules over the Milnor-Witt rational motivic cohomology spectrum $$\mathbf{H}_{\mathrm{MW}}\mathbb{Q}_S$$ (Definition 6.1) is deduced (Corollary 6.3); this equivalence is compatible with the six operations.
In the remainder of the paper, the authors introduce the notion of a SL-orientation for motivic $$\infty$$-categories (Definition 7.3) and show that $$\mathrm{D}_{\mathbb{A}^1}(-,\mathbb{Z})_-$$ (Theorem 7.8) and $$\mathrm{SH}_{\mathbb{Q}}$$ (Corollary 7.9) admit canonical SL-orientations, and ultimately the associated bivariant theories are computed explicitly in terms of rational twisted Chow-Witt groups (Theorem 8.7). The paper concludes with several appendices.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects) 19G12 Witt groups of rings 11E81 Algebraic theory of quadratic forms; Witt groups and rings 14C25 Algebraic cycles 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry
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##### References:
 [1] Artin, M.; Grothendieck, A.; Verdier, J.-L., Théorie des topos et cohomologie étale des schémas, 269, 270, 305 (19721973), Springer-Verlag · Zbl 0234.00007 [2] Ananyevskiy, A.; Levine, M.; Panin, I., Witt sheaves and the $$\eta$$-inverted sphere spectrum, J. Topology, 10, 2, 370-385 (2017) · Zbl 1378.14021 [3] Ananyevskiy, A., $$\text{SL}$$-oriented cohomology theories (2019) [4] Ayoub, J., Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, 314-315 (2007), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 1153.14001 [5] Ayoub, J., La réalisation étale et les opérations de Grothendieck, Ann. Sci. École Norm. Sup. (4), 47, 1, 1-145 (2014) · Zbl 1354.18016 [6] Bachmann, T., Motivic and real étale stable homotopy theory, Compositio Math., 154, 5, 883-917 (2018) · Zbl 06855294 [7] Balmer, P., Witt cohomology, Mayer-Vietoris, homotopy invariance and the Gersten conjecture, -Theory, 23, 1, 15-30 (2001) · Zbl 0987.19002 [8] Balmer, P., Handbook of $$K$$-theory. Vol. 1, 2, Witt groups, 539-576 (2005), Springer: Springer, Berlin · Zbl 1115.19004 [9] Beĭlinson, A. A.; Bernstein, J.; Deligne, Pierre, Analysis and topology on singular spaces, I (Luminy, 1981), 100, Faisceaux pervers, 5-171 (1982), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 0536.14011 [10] Bachmann, T.; Calmès, B.; Déglise, F.; Fasel, J.; Østvær, P. A., Milnor-Witt motives (2020) [11] Bondarko, Mikhail; Déglise, Frédéric, Dimensional homotopy t-structures in motivic homotopy theory, Adv. Math., 311, 91-189 (2017) · Zbl 1403.14053 [12] Bachmann, Tom; Fasel, Jean, On the effectivity of spectra representing motivic cohomology theories (2018) [13] Balmer, Paul; Gille, Stefan; Panin, Ivan; Walter, Charles, The Gersten conjecture for Witt groups in the equicharacteristic case, Doc. Math., 7, 203-217 (2002) · Zbl 1015.19002 [14] Bachmann, Tom; Hoyois, Marc, Norms in motivic homotopy theory (2021), Société Mathématique de France: Société Mathématique de France, Paris [15] Bloch, S.; Ogus, A., Gersten’s conjecture and the homology of schemes, Ann. Sci. École Norm. Sup. (4), 7, 4, 181-201 (1974) · Zbl 0307.14008 [16] Bondarko, Mikhail, Weights for relative motives: relation with mixed complexes of sheaves, Internat. Math. Res. Notices, 17, 4715-4767 (2014) · Zbl 1400.14062 [17] Balmer, P.; Walter, C., A Gersten-Witt spectral sequence for regular schemes, Ann. Sci. École Norm. Sup. (4), 35, 1, 127-152 (2002) · Zbl 1012.19003 [18] Cisinski, D.-C.; Déglise, Frédéric, Integral mixed motives in equal characteristics, Doc. Math., 145-194 (2015) · Zbl 1357.19004 [19] Cisinski, D.-C.; Déglise, Frédéric, Étale motives, Compositio Math., 152, 3, 556-666 (2016) · Zbl 1453.14059 [20] Cisinski, D.-C.; Déglise, Frédéric, Triangulated categories of mixed motives (2019), Springer: Springer, Cham · Zbl 07138952 [21] Calmès, B.; Dotto, E.; Harpaz, J.; Hebestreit, F.; Land, M.; Moi, K.; Nardin, D.; Nikolaus, T.; Steimle, W., Hermitian $$K$$-theory for stable $$\infty$$-categories I: Foundations (2020) [22] Calmès, B.; Dotto, E.; Harpaz, J.; Hebestreit, F.; Land, M.; Moi, K.; Nardin, D.; Nikolaus, T.; Steimle, W., Hermitian $$K$$-theory for stable $$\infty$$-categories II: Cobordism categories and additivity (2020) [23] Calmès, B.; Dotto, E.; Harpaz, J.; Hebestreit, F.; Land, M.; Moi, K.; Nardin, D.; Nikolaus, T.; Steimle, W., Hermitian $$K$$-theory for stable $$\infty$$-categories III: Grothendieck-Witt groups of rings (2020) [24] Calmès, B.; Fasel, Jean, Finite Chow-Witt correspondences (2014) [25] Cisinski, D.-C., Cohomological methods in intersection theory (2019) [26] Colliot-Thélène, J.-L.; Hoobler, R.; Kahn, B., Algebraic $$K$$-theory (Toronto, ON, 1996), 16, The Bloch-Ogus-Gabber theorem, 31-94 (1997), American Mathematical Society: American Mathematical Society, Proovidence, RI · Zbl 0911.14004 [27] Deligne, Pierre, Cohomologie étale, 569 (1977), Springer-Verlag [28] Deligne, Pierre, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), 67, Le déterminant de la cohomologie, 93-177 (1987), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 0629.14008 [29] Déglise, Frédéric; Fasel, Jean, The Borel character (2020) [30] Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel A., Borel isomorphism and absolute purity (2019) [31] Déglise, Frédéric; Jin, Fangzhou; Khan, Adeel A., Fundamental classes in motivic homotopy theory, J. Eur. Math. Soc. (JEMS) (2021) [32] Déglise, Frédéric, Bivariant theories in motivic stable homotopy, Doc. Math., 23, 997-1076 (2018) · Zbl 1423.14152 [33] Déglise, Frédéric, $$K$$-Theory—Proceedings of the International Colloquium (Mumbai, 2016), Orientation theory in arithmetic geometry, 239-347 (2018), Hindustan Book Agency: Hindustan Book Agency, New Delhi · Zbl 1451.14067 [34] Elmanto, Elden; Hoyois, Marc; Khan, Adeel A.; Sosnilo, Vladimir; Yakerson, Maria, Modules over algebraic cobordism, Forum Math. Pi, 8, 44 p. pp. (2020) · Zbl 1458.14027 [35] Elmanto, Elden; Khan, Adeel A., Perfection in motivic homotopy theory, Proc. London Math. Soc. (3), 120, 1, 28-38 (2020) · Zbl 1440.14123 [36] Elmanto, Elden; Kolderup, Håkon, On modules over motivic ring spectra, Ann. -Theory, 5, 2, 327-355 (2020) · Zbl 1440.14120 [37] Elman, R.; Karpenko, N.; Merkurjev, A., The algebraic and geometric theory of quadratic forms, 56 (2008), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1165.11042 [38] Fasel, Jean, Groupes de Chow-Witt, 113 (2008), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 1190.14001 [39] Feld, N., Morel homotopy modules and Milnor-Witt cycle modules (2019) [40] Feld, N., Milnor-Witt cycle modules, J. Pure Appl. Algebra, 224, 7, 41 (2020) · Zbl 1442.14026 [41] Fasel, Jean; Srinivas, V., Chow-Witt groups and Grothendieck-Witt groups of regular schemes, Adv. Math., 221, 1, 302-329 (2009) · Zbl 1167.13006 [42] Fujiwara, K., Algebraic geometry 2000, Azumino (Hotaka), 36, A proof of the absolute purity conjecture (after Gabber), 153-183 (2002), Math. Soc. Japan: Math. Soc. Japan, Tokyo · Zbl 1059.14026 [43] Fulton, W., Intersection theory, 2 (1998), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0885.14002 [44] Garkusha, Grigory, Reconstructing rational stable motivic homotopy theory, Compositio Math., 155, 7, 1424-1443 (2019) · Zbl 07077742 [45] Gille, S., A graded Gersten-Witt complex for schemes with a dualizing complex and the Chow group, J. Pure Appl. Algebra, 208, 2, 391-419 (2007) · Zbl 1127.19005 [46] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci., 20, 5-259 (1964) · Zbl 0136.15901 [47] Grothendieck, A., Cohomologie $$\ell$$-adique et fonctions $${L}, 589 (1977)$$, Springer-Verlag [48] Hartshorne, Robin, Residues and duality, 20 (1966), Springer-Verlag: Springer-Verlag, Berlin-New York · Zbl 0212.26101 [49] Hoyois, Marc, A quadratic refinement of the Grothendieck-Lefschetz-Verdier trace formula, Algebraic Geom. Topol., 14, 6, 3603-3658 (2014) · Zbl 1351.14013 [50] Hébert, D., Structure de poids à la Bondarko sur les motifs de Beilinson, Compositio Math., 147, 5, 1447-1462 (2011) · Zbl 1233.14017 [51] Illusie, L.; Laszlo, Y.; Orgogozo, F., Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, 363-364 (2014), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 1297.14003 [52] Jacobson, J., Real cohomology and the powers of the fundamental ideal in the Witt ring, Ann. -Theory, 2, 3, 357-385 (2017) · Zbl 1427.14047 [53] Jin, Fangzhou, Borel-Moore motivic homology and weight structure on mixed motives, Math. Z., 283, 3, 1149-1183 (2016) · Zbl 1375.14023 [54] Khan, Adeel A., Motivic homotopy theory in derived algebraic geometry (2016) [55] Khan, Adeel A., Virtual fundamental classes of derived stacks I (2019) [56] Khan, Adeel A., Voevodsky’s criterion for constructible categories of coefficients (2021) [57] Knebusch, M., Conference on Quadratic Forms—1976 (Kingston, Ont., 1976), 46, Symmetric bilinear forms over algebraic varieties, 103-283 (1977) · Zbl 0408.15019 [58] Lam, T. Y., Introduction to quadratic forms over fields, 67 (2005), American Mathematical Society: American Mathematical Society, Providence, RI · Zbl 1068.11023 [59] Lurie, Jacob, Higher topos theory, 170 (2009), Princeton University Press: Princeton University Press, Princeton, NJ · Zbl 1175.18001 [60] Lurie, Jacob, Higher algebra (2012) · Zbl 1175.18001 [61] Lurie, Jacob, Spectral algebraic geometry (2018) [62] Morel, F., Axiomatic, enriched and motivic homotopy theory, 131, On the motivic $$\pi_0$$ of the sphere spectrum, 219-260 (2004), Kluwer Acad. Publ. [63] Morel, F., Rational stable splitting of Grassmannians and rational motivic sphere spectrum (2006) [64] Morel, F., $$\mathbb{A}^1$$-algebraic topology over a field, 2052 (2012), Springer: Springer, Heidelberg [65] Morel, F.; Voevodsky, V., $${\mathbb{A}}^1$$-homotopy theory of schemes, Publ. Math. Inst. Hautes Études Sci., 90, 45-143 (1999) · Zbl 0983.14007 [66] Panin, I., Quadratic forms, linear algebraic groups, and cohomology, 18, Homotopy invariance of the sheaf $$W_{\text{Nis}}$$ and of its cohomology, 325-335 (2010), Springer: Springer, New York · Zbl 1216.18014 [67] Panin, I.; Walter, C., On the motivic commutative ring spectrum BO, St. Petersburg Math. J., 30, 6, 933-972 (2019) · Zbl 1428.14011 [68] Robalo, Marco, $$K$$-theory and the bridge from motives to noncommutative motives, Adv. Math., 269, 399-550 (2015) · Zbl 1315.14030 [69] Röndigs, Oliver; Østvær, Paul Arne, On modules over motivic ring spectra, Adv. Math., 219, 2, 689-727 (2008) · Zbl 1180.14015 [70] Scheiderer, Claus, Real and étale cohomology, 1588 (1994), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0852.14003 [71] Scholl, A., The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), 548, Integral elements in $$K$$-theory and products of modular curves, 467-489 (2000), Kluwer Acad. Publ. · Zbl 0982.14009 [72] Schlichting, M., Hermitian $$K$$-theory, derived equivalences and Karoubi’s fundamental theorem, J. Pure Appl. Algebra, 221, 7, 1729-1844 (2017) · Zbl 1360.19008 [73] Spivakovsky, M., A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Amer. Math. Soc., 12, 2, 381-444 (1999) · Zbl 0919.13009 [74] Spitzweck, Markus, A commutative $$\mathbb{P}^1$$-spectrum representing motivic cohomology over Dedekind domains, 157 (2018), Société Mathématique de France: Société Mathématique de France, Paris · Zbl 1408.14081 [75] Schlichting, M.; Tripathi, G. S., Geometric models for higher Grothendieck-Witt groups in $$\mathbb{A}^1$$-homotopy theory, Math. Ann., 362, 3-4, 1143-1167 (2015) · Zbl 1331.14028 [76] Stacks project authors, The Stacks project (2021) [77] Thomason, R. W., Absolute cohomological purity, Bull. Soc. math. France, 112, 3, 397-406 (1984) · Zbl 0584.14007 [78] Thomason, R. W.; Trobaugh, T., The Grothendieck Festschrift, Vol. III, 88, Higher algebraic $$K$$-theory of schemes and of derived categories, 247-435 (1990), Birkhäuser Boston: Birkhäuser Boston, Boston, MA
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