The problem is however to control the singular behaviour. A useful idea is to restrict singularities to a closed subscheme given by some ideal sheaf. The simplest case are distributions on a $k$-affine scheme$X$ over a field, supported at a point $p$. For a group scheme, one assumes that the point is the neutral element of the group and the algebra of distributions becomes a Hopf algebra.

Definition

Let $X$ be a $k$-affine scheme $X$ (where $k$ is a field). For $n$ a nonnegative integer distributions on $X$ supported at $p$ of order $\leq n$ are linear functionals $k[X]\to k$ on the ring of regular functions $k[X]$ which vanish on $(n+1)$-st power of the ideal $I_p = \{ f\in k[X] | f(p) = 0\}$ of regular functions vanishing at $p$.

If $X = G$ is a $k$-affine group scheme by the Hopf algebra of distributions we mean the algebra of distributions supported at the unit of the group with its natural Hopf algebra structure.