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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Higher derivations on finitely generated integral domains. II
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by William C. Brown PDF
Proc. Amer. Math. Soc. 51 (1975), 8-14 Request permission

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
  • © Copyright 1975 American Mathematical Society
  • 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