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

 

Higher derivations on finitely generated integral domains. II


Author: William C. Brown
Journal: Proc. Amer. Math. Soc. 51 (1975), 8-14
MSC: Primary 13B10
MathSciNet review: 0376644
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Abstract: We prove

Theorem. Let $ A = k[{x_1}, \ldots ,{x_m}]$ be a finitely generated integral domain over a field $ k$ of characteristic zero. Then $ A$ regular, i.e. the local ring $ {A_q}$ is regular for every prime ideal $ q \subseteq A$, is equivalent to the following two conditions: (1) no prime of $ A$ of height greater than one is differential, and (2) for all $ \phi \in {\operatorname{Hom} _k}(A,A),\phi \in \operatorname{Der} _k^n(A)$ if and only if $ \Delta \phi \in \Sigma _{i = 1}^{n - 1}\operatorname{Der} _k^i(A) \cup \operatorname{Der} _k^{n - i}(A)(n = 1,2, \ldots )$.

Here $ \Delta $ denotes the Hochschild coboundary operator, $ \cup $ denotes the cup product, and $ \operatorname{Der} _k^n(R)$ is the module of higher derivations of rank $ n$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0376644-8
PII: S 0002-9939(1975)0376644-8
Keywords: $ n$th order derivation, $ \operatorname{der} _k^n(A)$
Article copyright: © Copyright 1975 American Mathematical Society