Over a field $ k $ of characteristic zero, actions of $ \widehat{\mathbb{G}_{a}} $ on a variety $ \operatorname{Spec}(A) $ are obtained from $ A $-derivations $ \delta $ of $ A $ over $ k $ via the co-action below: \begin{equation*} a \mapsto \sum_{j=0}^{\infty} \frac{\delta^{j}(a)t^{j}}{j!}. \end{equation*} The sheaf $ \Omega_{A/k} $ parameterizes $ A $-derivations of $ A $ over $ k $.

If $ k $ is a field of positive characteristic then actions of $ \widehat{\mathbb{G}_{a}} $ are parameterized by iterative, higher derivations. A collection of $ k $-homomorphisms $ \phi_{i}: A \to A $ is a higher derivation if $ \phi_{0} = \operatorname{id}_{A} $ and $ \phi_{n}(a_{1}a_{2}) = \sum_{(i,j) \in \mathbb{N}_{0}^{2}} \phi_{i}(a_{1}) \phi_{j}(a_{2}) $. A higher derivation is called iterative if $ \phi_{i} \circ \phi_{j} = \binom{i+j}{i} \phi_{i+j} $. One obtains an action of $ \widehat{\mathbb{G}_{a}} $ on $ \operatorname{Spec}(A) $ from an iterative, higher derivation $ \{\phi_{i}\}_{i \in \mathbb{N}_{0}} $ via the co-action below: \begin{equation*} a \mapsto \sum_{j=0}^{\infty} \phi_{j}(a)t^{j}. \end{equation*} Has anyone tried to generalize the notion of higher differentials so that one could create a sheaf $ \Xi_{A/k} $ with a similar universal property?