Local Analysis of Grauert-Remmert-Type Normalization Algorithms

Normalization is a fundamental ring-theoretic operation; geometrically it resolves singularities in codimension one. Existing algorithmic methods for computing the normalization rely on a common recipe: successively enlarge the given ring in form of an endomorphism ring of a certain (fractional) ideal until the process becomes stationary. While Vasconcelos' method uses the dual Jacobian ideal, Grauert Remmert-type algorithms rely on so-called test ideals. For algebraic varieties, one can apply such normalization algorithms globally, locally, or formal analytically at all points of the variety. In this paper, we relate the number of iterations for global Grauert Remmert-type normalization algorithms to that of its local descendants. We complement our results by a study of ADE singularities. All intermediate singularities occuring in the normalization process are determined explicitly. Besides ADE singularities the process yields simple space curve singularities from the list of Frühbis-Krüger.