On the rational motivic homotopy category
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 533-583.

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].

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.

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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
Mot clés : Théorie $\mathbb{A}^1$-homotopique, cohomologie motivique, six opérations, groupes de Chow-Witt, K-théorie, K-théorie hermitienne
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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, Volume 8 (2021), pp. 533-583. doi : 10.5802/jep.153. https://jep.centre-mersenne.org/articles/10.5802/jep.153/

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