Nonfinite unramified extensions
Suppose A and B are commutative rings, f: \(A\to B\) is a homomorphism, \(p\subset B\) is a prime ideal and \(q=f^{-1}(p)\). Then B (or f) is said to be unramified over A at p in case \(f(q)B_ p=p_ p\) and \((B/p)_ p\) is a finite separable field extension of \((A/q)_ q\). Here \(B_ p\) is the localization of B at p. B is said to be unramified if it is unramified at each prime. Suppose A is a normal Noetherian domain containing \({\mathbb{Q}}\) and \(C\supset A\) is a domain which is affine and unramified over A. Let B be the integral closure of A in the field of quotients of C. Now C is a normal domain, so \(A\subset B\subset C\) and this gives rise to four radical ideals, the finiteness and discriminant ideals of A, and the Zariski complement and branch locus ideals of B. The purpose of the paper under review is to analyze these ideals and their behavior under Galois closure and tensor product. The authors also give a topological proof of a theorem of Zariski on the purity of the discriminant ideal.
scientific article; zbMATH DE number 4110822
scientific article