Classification of group representations on a local k-algebra

Let be a local and complete noetherian k-algebra and let be the natural morphism between the group of automorphisms of and the group of automorphisms of its tangent space. The main result of this paper is to show that Ker φ is an inverse limit of unipotent algebraic groups, that is, Ker φ=lim←Kj, . As a consequence: (1) we can describe the structure of the set of classes of representations in terms of the groups Hℓ(G,Ni), ℓ=1,2, where Ni are the additive factor groups of a natural resolution of Kj; (2) theorems are obtained for classifying representations valid for any dimension, any characteristic and even when is not regular; (3) we obtain the complete classification of the representations of on k[[z]] when p=char(k)>0; and (4) an algorithm Σ is obtained which allows us to calculate successive approximations of the set of classes of representations of on k[[u,v]], p=char(k)>0.