The first stable homotopy groups of motivic spheres
The homotopy groups of the motivic sphere spectrum \(\pi_{p,q}\mathbb{S}\) are fundamental invariants in motivic homotopy theory, which are analogs of the homotopy groups in algebraic topology that in addition carry arithmetic information about fields. A famous result of \textit{F. Morel} [in: Proceedings of the international congress of mathematicians (ICM). Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035--1059 (2006; Zbl 1097.14014)] says that \(\pi_{p,q}\mathbb{S}=0\) when \(p<q\) and \(\pi_{n,n}\mathbb{S}=K^{MW}_n(F)\) is the Milnor-Witt \(K\)-theory, but very few is known on the \(r\)-line \(\pi_{n+r,n}\mathbb{S}\) for \(r>0\). This paper is a solid step towards the understanding of these groups, where the authors are able to compute the \(1\)-line (Theorem 5.5): if \(S\) is a scheme and \(\Lambda\) is a ring such that \((S,\Lambda)\) is a \textit{compatible pair} (Definition 2.1), for example if \(S\) is smooth over a field whose exponential characteristic is invertible in \(\Lambda\), or if \(S\) is regular and \(\Lambda=\mathbb{Q}\), then there is an exact sequence of Nisnevich sheaves over smooth \(S\)-schemes \[ 0\to K^M_{2-n}\otimes\Lambda/24 \to \pi_{n+1,n}\mathbb{S}_\Lambda \to \pi_{n+1,n}f_0(\mathbf{KQ}_\Lambda) \] where \(f_0(\mathbf{KQ}_\Lambda)\) is the effective cover of the hermitian \(K\)-theory spectrum, and the sequence is also right exact for \(n\geq-4\). The main tool is Voevodsky's slice spectral sequence [\textit{V. Voevodsky}, in: Motives, polylogarithms and Hodge theory. Part I: Motives and polylogarithms. Papers from the International Press conference, Irvine, CA, USA, June 1998. Somerville, MA: International Press. 3--34 (2002; Zbl 1047.14012)], which is an analog of the Atiyah-Hirzebruch spectral sequence. Using some convergence results on the slice spectral sequence, the basic idea is to identify the slices of the sphere spectrum (Theorem 2.12) and compute the first differentials in terms of motivic Steenrod operations (Proposition 4.18).
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