On q-de Rham cohomology via Λ-rings

We show that Aomoto's q-deformation of de Rham cohomology arises as a natural cohomology theory for Λ-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.