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 .
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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
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@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/} }
TY - JOUR AU - Frédéric Déglise AU - Jean Fasel AU - Fangzhou Jin AU - Adeel A. Khan TI - On the rational motivic homotopy category JO - Journal de l’École polytechnique — Mathématiques PY - 2021 SP - 533 EP - 583 VL - 8 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.153/ DO - 10.5802/jep.153 LA - en ID - JEP_2021__8__533_0 ER -
%0 Journal Article %A Frédéric Déglise %A Jean Fasel %A Fangzhou Jin %A Adeel A. Khan %T On the rational motivic homotopy category %J Journal de l’École polytechnique — Mathématiques %D 2021 %P 533-583 %V 8 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.153/ %R 10.5802/jep.153 %G en %F JEP_2021__8__533_0
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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