Remarks on the motivic sphere without $\mathbb A^1$-invariance

We generalize several basic facts about the motivic sphere spectrum in $\mathbb A^1$-homotopy theory to the category $\mathrm{MS}$ of non-$\mathbb A^1$-invariant motivic spectra over a derived scheme. On the one hand, we show that all the Milnor-Witt K-theory relations hold in the graded endomorphism ring of the motivic sphere. On the other hand, we show that the positive eigenspace $\mathbf 1_\mathbb Q^+$ of the rational motivic sphere is the rational motivic cohomology spectrum $\mathrm H\mathbb Q$, which represents the eigenspaces of the Adams operations on rational algebraic K-theory. We deduce several familiar characterizations of $\mathrm H\mathbb Q$-modules in $\mathrm{MS}$: a rational motivic spectrum is an $\mathrm H\mathbb Q$-module iff it is orientable, iff the involution $\langle -1\rangle$ is the identity, iff the Hopf map $\eta$ is zero, iff it satisfies \'etale descent. Moreover, these conditions are automatic in many cases, for example over non-orderable fields and over $\mathbb Z[\zeta_n]$ for any $n\geq 3$.