Nonfinite unramified extensions
type: Item
instance of: scholarly article
title: Nonfinite unramified extensions
zbMATH Open document ID: 0678.13005
DOI: 10.1016/0001-8708(87)90056-9
author: J. Drost
author: Edwin H. Connell
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 13A10
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 4110822
zbMATH Keywords: unramified extensions
zbMATH Keywords: radical ideals
zbMATH Keywords: Galois closure
zbMATH Keywords: tensor product
zbMATH Keywords: purity of the discriminant ideal
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90056-9
OpenAlex ID: W1968519034
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description: scientific article; zbMATH DE number 4110822