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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some counterexamples concerning a differential criterion for flatness


Authors: William C. Brown and Sarah Glaz
Journal: Proc. Amer. Math. Soc. 81 (1981), 505-510
MSC: Primary 13C11; Secondary 13B10
MathSciNet review: 601717
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Abstract: Let $ A$ denote a commutative ring with identity. We suppose $ A$ contains a field $ k$ of characteristic zero. Let $ \Omega _k^1(A)$ and $ d:A \to \Omega _k^1(A)$ denote the $ A$-module of first-order $ k$-differentials on $ A$ and the canonical derivation of $ A$ into $ \Omega _k^1(A)$ respectively. If $ \mathfrak{A}$ is an ideal of $ A$ which is flat as an $ A$-module, then $ xdy - ydx \in {\mathfrak{A}^2}\Omega _k^1(A)$ for all $ x,y$ in $ \mathfrak{A}$. We give examples in this paper which show that the converse of this statement is false. We also show that if $ \mathfrak{A}$ is a maximal ideal of a Noetherian ring $ A$, then $ xdy - ydx \in {\mathfrak{A}^2}\Omega _k^1(A)$ for all $ x,y$ in $ \mathfrak{A}$ does imply $ \mathfrak{A}$ is flat.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0601717-4
PII: S 0002-9939(1981)0601717-4
Keywords: Condition $ (D)$, monomial ring, differentials, flatness
Article copyright: © Copyright 1981 American Mathematical Society