Differential Characters and $D$-Group Schemes

Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\mathrm{Spec}~ K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$. Our proof relies entirely on understanding the structure of jet spaces. Our method also allows us togive a classification of the module of differential characters $\mathbf{X}_\infty(G)$ in terms of primitive characters as a $K\{\partial\}$-module.