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Epimorphic flat maps*

Published online by Cambridge University Press:  20 January 2009

Frederick W. Call
Frederick W. Call
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH
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In this note, we derive a necessary and sufficient condition for a flat map of (commutative) rings to be a flat epimorphism. Flat epimorphisms φ:A → B(i.e.φ is an epimorphism in the category of rings, and the ring B is flat as an A-module) have been studied by several authors in different forms. Flat epimorphisms generalize many of the results that hold for localizations with respect to a multiplicatively closed set (see, for example [6]).In a geometric formulation, D. Lazard [3, Chapitre IV, Proposition 2.5] has shown that isomorphism classes of flat epimorphisms from a ring A are in 1-1 correspondence with those subsets of Spec A such that the sheaf structure induced from the canonical sheaf structure of Spec A yields an affine scheme. N. Popescu and T. Spircu [4, Théorème 2.7] have given a characterization for a ring homomorphism to be a flat epimorphism, but our characterization, under the assumption of flatness is easier to apply. For corollaries, we can obtain known results due to D. Lazard, T. Akiba, and M. F. Jones, and generalize a geometric theorem of D. Ferrand.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 57 - 59
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

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