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Transactions of the American Mathematical Society

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Derivations into the integral closure

Authors: Richard Draper and Klaus Fischer
Journal: Trans. Amer. Math. Soc. 271 (1982), 283-298
MSC: Primary 32B05; Secondary 13B10, 13B20, 32B30
MathSciNet review: 648093
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Abstract: Let $ A$ be a reduced analytical $ k$-algebra of dimension $ r$ and $ A' $ its integral closure in the full ring of quotients of $ A$. We investigate the condition on $ A$ that there exist $ r$ elements $ {x_1}, \ldots ,{x_r}$ in $ A$ and $ k$-derivations $ {d_1}, \ldots ,{d_r}$ from $ A$ into $ A' $ so that $ {d_i}({x_j})$ is the $ r \times r$ identity matrix and so that $ {d_1}, \ldots ,{d_r}$ freely generate $ {\operatorname{Der} _k}(A,\,A' )$. We show this is equivalent to a number of other conditions. If $ A$ is a complete intersection, then the above is equivalent to the Jacobian ideal $ J$ becoming principal in $ A' $. If $ A / \sqrt J $ is regular of dimension $ r - 1$ and satisfies the above condition, then $ A' $ is regular.

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Keywords: Derivations, integral closure, reduced analytical $ k$-algebra, Jacobian ideal, complete intersection, ramified primes
Article copyright: © Copyright 1982 American Mathematical Society

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