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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Pureté, rigidité, et morphismes entiers


Author: Gabriel Picavet
Journal: Trans. Amer. Math. Soc. 323 (1991), 283-313
MSC: Primary 14E40; Secondary 13B20, 13F05, 14A15
DOI: https://doi.org/10.1090/S0002-9947-1991-1013336-X
MathSciNet review: 1013336
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Abstract: Bousfield and Kan have shown that a ring morphism with domain $ {\mathbf{Z}}$ is rigid; we say that a ring morphism is rigid if it admits a factorization by an epimorphism, followed by a pure morphism. A ring $ A$ is said to be rigid if every morphism with domain $ A$ is a rigid one. Our principal results are: the rigid domains are the Prüferian rings $ A$, with $ \operatorname{Dim} (A) \leq 1$, and the Noetherian rigid rings are the Z.P.I. rings. The quasi-compact open sets of an affine rigid scheme, having as underlying ring a domain or a Noetherian ring, are affine and schematically dense if they contain the assassin of the ring. Every injective integral ring morphism with rigid domain is a pure morphism. We give two criteria of purity for integral injective morphisms. As a consequence of these results we obtain the following properties: if $ A$ is a normal ring, containing the field of rationals, or is a regular ring, containing a field, every injective integral morphism with domain $ A$ is a pure one. For a reduced ring, we define the category of reduced modules and show that any injective integral morphism is pure with respect to the category of the reduced modules.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1013336-X
Article copyright: © Copyright 1991 American Mathematical Society