Algebraic Transformation Groups and the Similarity Problem

Abstract

1. Algebraic geometry in KV. Let be an infinite field. K is given the so-called Zariski topology by defining a subset F to be closed if it is the set of common solutions of a collection of polynomial equations in n variables. Closed sets in K' are just the finite subsets. In K2, closed sets are finite unions of points and algebraic curves (e.g., the parabola y = x2 is an algebraic curve as it is the solutions of the polynomial equation y x2 = 0). The most unusual feature of this topology is that the open sets are all large. To be precise, each open set is dense, or equivalently, any two open sets have a non-empty intersection. This topology is not a T2-topology, although it is T1. In any topology, the subsets constructed from the open sets (or closed sets) by lattice operations (union, intersection, and complementation) are known as the constructible subsets. For the Zariski topology on , this Boolean algebra has a particularly nice interpretation, since if S is a constructible subset of Kn, then to decide the membership of a point x = (xI, ., xn) in S, one must simply check a finite number of polynomial equalities and inequalities involving the coordinates xi of x. For example, identify Mn (K) (n by n matrices over K) with Kn2 and let S be the set of all matrices having some fixed rank r. Let x = (xi1) and let mk,a(x) be a k th order minor determinant of x determined by the multi-index a. Then x belongs to S if and only if all mr+i,a(x) = 0 and some m,,,3(x) 4 0. Thus the somewhat unusual looking second condition of the theorem mentioned above is connected with the constructibility of a certain set of matrices. On K with the Zariski topology, the polynomial ring K[X,.. .,Xn] (denoted simply by K[X] most often) acts as a ring of continuous K-valued functions. This topological space, together with the ring of functions, is called affine n-space. More generally, a closed subset V in K is an affine variety if it is given the induced topology and a ring of continuous functions by restricting the polynomial functions on K to V. Each point of V has coordinates, and certain continuous functions are computed algebraically in terms of those coordinates. The ring of continuous functions is denoted by K[ V] and is called the coordinate ring of V. Certain open subsets of K are affine varieties by the following process. If f(X) belongs to K[X] then it defines a principal open subset n= {x in KI f(x) # 0}. K7n is homeomorphic to the closed set in n+ defined by the polynomial equation 1 Xn+1f(X1,...,Xn) = 0, under the mapping