A note on flat algebras
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- by Augusto Nobile PDF
- Proc. Amer. Math. Soc. 64 (1977), 206-208 Request permission
Abstract:
The following results are proved. If $f:A \to B$ is a homomorphism of rings, with B noetherian, I is an ideal of B contained in the Jacobson radical, and $B/{I^n}$ is A-flat, for all n, then f is flat. If, using similar notations and assumptions, I is generated by a regular sequence, then the flatness of B/I implies the flatness of f. A simple geometric application is given.References
- Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 238860
- Michel Raynaud and Laurent Gruson, Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math. 13 (1971), 1–89 (French). MR 308104, DOI 10.1007/BF01390094
Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 64 (1977), 206-208
- MSC: Primary 14C20; Secondary 13A15
- DOI: https://doi.org/10.1090/S0002-9939-1977-0498548-4
- MathSciNet review: 0498548