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Higher derivations on finitely generated integral domains


Author: W. C. Brown
Journal: Proc. Amer. Math. Soc. 42 (1974), 23-27
MSC: Primary 13B10
DOI: https://doi.org/10.1090/S0002-9939-1974-0337923-2
MathSciNet review: 0337923
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Abstract: In this paper, we prove the following theorem: Let $ A = k[{x_1}, \cdots ,{x_t}]$ 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 all primes $ q \subseteq A$, is equivalent to the following two conditions: (1) No nonminimal prime of A is differential, and (2) $ \operatorname{der}^n (A/k) = \mathrm{Der}^n (A/k)$ for all n. Here $ \operatorname{Der}^n (A/k)$ denotes the A-module of all nth order derivations of A into A which are zero or k, and $ \operatorname{der}^n(A/k)$ denotes the A-submodule of $ \operatorname{Der}^n(A/k)$ generated by composites $ {\delta _1} \circ \cdots \circ {\delta _j}(1 \leqq j \leqq n)$ of first order derivations $ {\delta _i}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0337923-2
Keywords: nth order derivation, $ \operatorname{der}^n(A/k)$
Article copyright: © Copyright 1974 American Mathematical Society

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