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Stable $$\mathbb A^1$$-homotopy and $$R$$-equivalence. (English) Zbl 1244.14016
In this beautiful short paper, the authors analyze the existence of rational points by means of stable motivic homotopy theory. For a smooth proper variety $$X$$ over a field $$k$$, a $$k$$-rational point on $$X$$ induces a splitting of the induced map $$X \to \mathrm{Spec}\,k$$ in the unstable motivic homotopy category. It is a fundamental question in which way the motivic homotopy category can detect rational points. Morel and Voevodsky have shown that the existence of a $$k$$-rational point is invariant under unstable $$\mathbb{A}^1$$-homotopy. The authors of the paper under review show that the existence of a $$k$$-rational point can be detected by the stable $$\mathbb{A}^1$$-homotopy theory of $$S^1$$-spectra. More precisely, they show that the existence of a $$k$$-rational point is equivalent to the existence of a splitting of the induced morphism $$\pi_0^s(X_+) \to \pi_0^s(\mathrm{Spec}\,k_+)$$ of $$S^1$$-stable zeroth homotopy group sheaves. Moreover, the authors show that the rational points can even be detected by a splitting of zeroth $$\mathbb{A}^1$$-homology sheaves in Morel’s $$\mathbb{A}^1$$-derived category with $$\mathbb{Q}$$-coefficients, a rationalized version of motivic homotopy theory. This result is important, since the stable homotopy category and the rational derived category are a priori easier to analyze.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14G05 Rational points 55P42 Stable homotopy theory, spectra
##### Keywords:
stable motivic homotopy; R-equivalence; rational points
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##### References:
 [1] A. Asok, C. Haesemeyer, Stable $$\mathbb{A}^1$$-homotopy and quadratic $$0$$-cycles, 2010 (in preparation). [2] A. Asok, F. Morel, Smooth varieties up to $$\mathbb{A}^1$$-homotopy and algebraic $$h$$-cobordisms, 2009, Preprint. Available at: http://arxiv.org/abs/0810.0324. · Zbl 1255.14018 [3] A. Asok, Birational invariants and $$\mathbb{A}^1$$-connectedness, 2010, Preprint. Available at: http://arxiv.org/abs/1001.4574. · Zbl 1328.14036 [4] Cisinski, D.-C.; Déglise, F., Local and stable homological algebra in Grothendieck abelian categories, Homology, homotopy appl., 11, 1, 219-260, (2009) · Zbl 1175.18007 [5] Jardine, J.F., Motivic symmetric spectra, Doc. math., 5, 445-553, (2000), (electronic) · Zbl 0969.19004 [6] Levine, M., Slices and transfers, Doc. math., 393-443, (2010), Extra Volume: Andrei A. Suslin’s Sixtieth Birthday · Zbl 1210.14024 [7] Manin, Yu.I., Cubic forms, (), Algebra, geometry, arithmetic, Translated from the Russian by M. Hazewinkel · Zbl 0582.14010 [8] Morel, F., The stable $$\mathbb{A}^1$$-connectivity theorems, $$K$$-theory, 35, 1-2, 1-68, (2005) · Zbl 1117.14023 [9] F. Morel, $$\mathbb{A}^1$$-algebraic topology over a field, 2006. Preprint. Available at: http://www.mathematik.uni-muenchen.de/ morel/preprint.html. [10] Morel, F.; Voevodsky, V., $$\mathbb{A}^1$$-homotopy theory of schemes, Inst. hautes études sci. publ. math., 90, 45-143, (2001), 1999 · Zbl 0983.14007 [11] Nishimura, H., Some remark[s] on rational points, Mem. coll. sci. univ. Kyoto. ser. A. math., 29, 189-192, (1955) · Zbl 0068.14802 [12] Nisnevich, Ye.A., The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory, (), 241-342
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