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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Flat or open implies going down


Authors: David E. Dobbs and Ira J. Papick
Journal: Proc. Amer. Math. Soc. 65 (1977), 370-371
MSC: Primary 13C10
DOI: https://doi.org/10.1090/S0002-9939-1977-0441948-9
MathSciNet review: 0441948
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Abstract | References | Similar Articles | Additional Information

Abstract: Let R, T be commutative rings with identity, and $ f:R \to T$ a unital ring homomorphism. We give an elementary, unified proof of the fact that f has the going down property, if T is flat as an R-module or if the induced map $ F:{\text{Spec}}(T) \to {\text{Spec}}(R)$ is open.


References [Enhancements On Off] (What's this?)

  • [1] Revêtements étales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1); Dirigé par Alexandre Grothendieck. Augmenté de deux exposés de M. Raynaud. MR 0354651
  • [2] Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
  • [3] Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0441948-9
Keywords: Going down, flat, open
Article copyright: © Copyright 1977 American Mathematical Society