Aomoto's $q$-deformation of de Rham cohomology for polynomial rings involves the Gaussian $q$-analogues $[n]_q = 1+q+...+q^{n-1}$ of the integers. Scholze showed how to extend this definition to smooth rings, and conjectured independence from the choice of co-ordinates. I will explain how this theory arises naturally for lambda-rings, and in mixed characteristic depends only on a lift of Frobenius. I will also discuss the possibility of a $q$-analogue of the de Rham--Witt complex for smooth schemes, not relying on any additional structure, but involving arbitrary pth roots of $q$.