Differentials over differential fields

Given an algebra $A$ over a differential field $K$, we study derivations on $A$ that are compatible with the derivation on $K$. There is a universal object, which is a twisted version of the usual module of differentials, and we establish some of its basic properties. In the context of differential algebraic geometry, one gets a sheaf of these $\tau$-differentials which can be interpreted as certain natural functions on the prolongation of a variety, as studied by Buium. This sheaf corresponds to the Kodaira-Spencer class of the variety.