${\mathbb{A}}^1$-invariance of localizing invariants

Weibel proved that $p$-inverted K-theory is ${\mathbb{A}}^1$-invariant on ${\mathbb{F}}_p$-schemes and K-theory with $\mathbb{Z}\mathrm{/}p$-coefficients is ${\mathbb{A}}^1$-invariant on $\mathbb{Z}[\frac{1}{p}]$-schemes. We extend this result to all finitary localizing invariants of small stable $\mathrm{\infty }$-categories. Along the way we study the Frobenius and Verschiebung endofunctors defined by Tabuada and provide a categorical version of Stienstra’s projection formula.