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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.

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
Full Text: DOI
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