Higher local skew fields

Abstract

n-dimensional local skew fields are a natural generalization of n-dimensional local fields. The latter have numerous applications to problems of algebraic geometry, both arithmetical and geometrical, as it is shown in this volume. From this viewpoint, it would be reasonable to restrict oneself to commutative fields only. Nevertheless, already in class field theory one meets non-commutative rings which are skew fields finite-dimensional over their center K . For example, K is a (commutative) local field and the skew field represents elements of the Brauer group of the field K (see also an example below). In [Pa] A.N. Parshin pointed out another class of non-commutative local fields arising in differential equations and showed that these skew fields possess many features of commutative fields. He defined a skew field of formal pseudodifferential operators in n variables and studied some of their properties. He raised a problem of classifying non-commutative local skew fields.In this section we treat the case of n = 2 and list a number of results, in particular a classification of certain types of 2-dimensional local skew fields. Basic definitionsDefinition. A skew field K is called a complete discrete valuation skew field if K is complete with respect to a discrete valuation (the residue skew field is not necessarily commutative). A field K is called an n-dimensional local skew field if there are skew fields K = K n , K n−1 , . . . , K 0 such that each K i for i > 0 is a complete discrete valuation skew field with residue skew field K i−1 . Examples.(1) Let k be a field. Formal pseudo-differential operators over k((X)) form a 2dimensional local skew field K = k((X))((∂ −1 X )), ∂ X X = X∂ X + 1. If char (k) = 0 we get an example of a skew field which is an infinite dimensional vector space over its centre.