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.