On the rational motivic homotopy category

Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel A.

Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 533-583.

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 ${\mathrm{SH}}_{\mathbb{Q}}$. Next, we prove that ${\mathrm{SH}}_{\mathbb{Q}}$ is canonically $\mathrm{SL}$-oriented; we compare ${\mathrm{SH}}_{\mathbb{Q}}$ 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 $\mathrm{SH}[1/2]$.

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

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