Flat or open implies going down
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- by David E. Dobbs and Ira J. Papick PDF
- Proc. Amer. Math. Soc. 65 (1977), 370-371 Request permission
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
- 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
- Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
Additional Information
- © Copyright 1977 American Mathematical Society
- 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