Derivativewise unramified infinite integral extensions

Let A be a normal noetherian domain with quotient field K and let B be a localization of the integral closure of A in an infinite algebraic fieid extension of K. Two obvious necessary conditions in order that B be noetherian are finite splitting and finite ramification of prime ideals of A in B. We consider various situations in which these conditions are also sufficient. As an important ingredient of this we give conditions for a prime ideal in B to be the extension of its contraction in A. When B is noetherian we investigate preservation from A to B of the property of being pseudogeometric. We do all this by introducing the concept of compositumwise unramifiedness and various related notions. We also present structure theorems for compositumwise unramified extensions in the local case. This local theory ensures finite splitting of all prime ideals by assuming it only for maximal ideals