On weak normality and symmetric algebras. (English) Zbl 0534.13002
A symmetric R-algebra of rank one is by definition isomorphic to a symmetric algebra \(S_ R[L]\) where L is an invertible R-module, hence it is locally a polynomial algebra in one variable. In this paper the author proves that if \(R\subset S\) is a finite extension of reduced noetherian rings, then R is weakly normal in S if and only if every R-algebra A such that \(S\otimes_ RA\) is a symmetric algebra of rank one is itself a symmetric R-algebra of rank one. In the case where R is a field this is already well-known.
Reviewer: A.Verschoren
MSC:
| 13B02 | Extension theory of commutative rings |
| 13E05 | Commutative Noetherian rings and modules |
| 14C22 | Picard groups |
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