Derivativewise unramified infinite integral extensions
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 compositumwise unramified over \(A\) at \(Z\), or \(B\) is uniformly weakly almost compositumwise unramified over \(A\) at \(Z\). -- In section 6 we prove that if \(A\) is a normal noetherian domain and \(B\) is a domain which is almost compositumwise unramified over \(A\) at each prime ideal of \(B\) and for every nonzero prime ideal \(P\) of \(A\) we have that \(B\) is uniformly almost compositumwise unramified over \(A\) at the set of all prime ideals of \(B\) containing \(P\), then: \(B\) is noetherian if and only if every prime ideal in \(A\) is finitely split in \(B\), i.e., if and only if at most a finite number of prime ideals in \(B\) contract to any given prime ideal in \(A\). In relation to question (2), we consider the preservation of the pseudogeometric property from \(A\) to \(B\). Likewise, we consider the preservation of normality from \(A\) to \(B\). In section 4 we establish certain local versions of theorems of section 6 and we give structure theorems for compositumwise unramified extensions in the local case.
compositumwise unramified integral closure
scientific article; zbMATH DE number 7303
scientific article