The Steinberg relation in \(\mathbb{A}^1\)-stable homotopy (Q2764649)

There are several different notations for the stable homotopy groups of a \(T\)-spectrum \(X\) over a field \(k\). The authors write \(\pi_{p + q\alpha}X = [\Sigma^{\infty}(S^{p}_{s} \wedge S^{q}_{t}),X]\) for morphisms in the motivic stable category, suitably defined, and where \(S^{1}_{s}\) is the simplicial circle and \(S^{1}_{t} = \mathbb{A}^{1}- \{ 0 \}\) is the geometric circle. Other topologists would write \(\pi_{p,q}X\) for this invariant, while algebraists denote it by \(H^{-q,p-q}(k,X)\) according to an indexing convention which reflects the indexing in motivic cohomology theories. NEWLINENEWLINENEWLINEIn this paper, the authors show that the function \(k^{\ast} \to \pi_{-\alpha}S = \pi_{0,-1}S = H^{1,1}(k,S)\) defined by sending \(a \in k^{\ast}\) to the stable homotopy class of the pointed map \((a): S^{0} \to \mathbb{G}_{m}\) which picks out \(a \in \mathbb{G}_{m}(k)\) satisfies the Steinberg relation \((a)(1-a) = 0\) in \(\pi_{-2\alpha}X\).