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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Derivations into the integral closure
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by Richard Draper and Klaus Fischer PDF
Trans. Amer. Math. Soc. 271 (1982), 283-298 Request permission

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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 271 (1982), 283-298
  • MSC: Primary 32B05; Secondary 13B10, 13B20, 32B30
  • DOI: https://doi.org/10.1090/S0002-9947-1982-0648093-4
  • MathSciNet review: 648093