Singular locus of an infinite integral extension

Abstract

By adjoining the square roots or higher roots of a certain infinite number of polynomials in two variables with rational coefficients, i.e. by constructing suitable infinite abelian coverings of the rational plane, we obtain examples of two-dimensional normal noetherian pseudogeometric surfaces which are locally excellent but which are not globally excellent because they have infinitely many singular points. The branch loci of these coverings are grids, i.e., infinite families of horizontal and vertical lines in the plane. For studying such coverings we establish a normality criterion and discuss the theory of excellent rings. Involved in the construction of these coverings is the total residuation of prime ideals in cofinal infinite families of field extensions and the complete splitting of prime numbers in finite number fields.