On $q$-de Rham cohomology via $\mathrm{\Lambda }$-rings

We show that Aomoto’s $q$-deformation of de Rham cohomology arises as a natural cohomology theory for $\mathrm{\Lambda }$-rings. Moreover, Scholze’s $(q-1)$-adic completion of $q$-de Rham cohomology depends only on the Adams operations at each residue characteristic. This gives a fully functorial cohomology theory, including a lift of the Cartier isomorphism, for smooth formal schemes in mixed characteristic equipped with a suitable lift of Frobenius. If we attach $p$-power roots of $q$, the resulting theory is independent even of these lifts of Frobenius, refining a comparison by Bhatt, Morrow and Scholze.