Classification of group representations on a local [formula omitted]-algebra

Abstract

Let O be a local and complete noetherian k-algebra and let φ : Aut .5ptk( O) → Aut .5ptk( G m ̄ O) be the natural morphism between the group of automorphisms of O 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 ←K j , j ∈ N . As a consequence: (1) we can describe the structure of the set of classes of representations ρ : G → Aut .5ptk( O) in terms of the groups H ℓ(G,N i) , ℓ=1,2, where N i are the additive factor groups of a natural resolution of K j ; (2) theorems are obtained for classifying representations valid for any dimension, any characteristic and even when O is not regular; (3) we obtain the complete classification of the representations of Z/p 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 Z/p on k[[u,v]], p=char(k)>0.