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
Mot 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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     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/

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