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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
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References:
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