Derivativewise unramified infinite integral extensions
type: Item
instance of: scholarly article
title: Derivativewise unramified infinite integral extensions
zbMATH Open document ID: 0815.13004
DOI: 10.1016/0021-8693(91)90075-J
author: William J. Heinzer
author: Shreeram S. Abhyankar
published in: Journal of Algebra
publication date: 1992-06-25T00:00:00Z
review text: Let \(A\) be a normal noetherian domain with quotient field \(K\) and let \(B\) be the integral closure of \(A\) in an algebraic field extension \(L\) of \(K\) which is not necessarily finite. We are interested in the following general questions: (1) Under what conditions does it follow that \(B\) is noetherian? Locally noetherian? (2) When does \(A\) pseudometric imply \(B\) is pseudometric? (3) When does \(A\) regular imply \(B\) is regular? (4) When does the singular locus of \(A\) closed imply that the singular locus of \(B\) is closed? (5) When does \(A\) locally analytically normal (or analytically irreducible) imply \(B\) is locally analytically normal (or analytically irreducible)? (6) When does \(A\) excellent imply \(B\) is excellent? (7) When does \(A\) a UFD imply \(B\) is a UFD? The present paper deals with questions (1) and (2). In forthcoming papers we plan to discuss the remaining questions. The three papers in this sequence being: ``Singular locus of an infinite integral extension'', J. Algebra 143, No. 2, 436-469 (1991; Zbl 0755.14015); ``Ramification in infinite integral extensions'', ibid. 170, No. 3, 861-879 (1994); ``Examples and counterexamples in commutative ring theory''. Given any prime ideal \(Q\) in \(B\), an element \(x\) of \(B\) is said to be derivativewise unramified over \(A\) at \(Q\) if there exists a monic polynomial \(f(X)\) with coefficients in \(A\) such that \(f(x) = 0\) but \(f'(x) \notin Q\); moreover, \(B\) is said to be compositumwise unramified over \(A\) at \(Q\) if there exists a subset \(W\) of \(B\) such that every element of \(W\) is derivativewise unramified over \(A\) at \(Q\), and \(B\) is the localization of \(A[W]\) at some multiplicative set in \(A[W]\). Certain weaker versions of this are called almost compositumwise unramifiedness and weakly almost compositumwise unramifiedness. When these properties hold ``uniformly'' on a set of primes \(Z\) of \(B\), we respectively say that \(B\) is uniformly almost compo…
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 12F15
zbMATH DE Number: 7303
zbMATH Keywords: infinite integral extensions
zbMATH Keywords: noetherianness
zbMATH Keywords: integral closure of noetherian domain
zbMATH Keywords: derivativewise unramified element
zbMATH Keywords: compositumwise unramified integral closure
zbMATH Keywords: pseudometric
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7303