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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0376644-8
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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Keywords: <IMG WIDTH="18" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$n$">th order derivation, <!– MATH $\operatorname {der} _k^n(A)$ –> <IMG WIDTH="75" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="$\operatorname {der} _k^n(A)$">
Article copyright: © Copyright 1975 American Mathematical Society