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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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A Frobenius characterization of rational singularity in $2$-dimensional graded rings
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by Richard Fedder PDF
Trans. Amer. Math. Soc. 340 (1993), 655-668 Request permission


A ring $R$ is said to be $F$-rational if, for every prime $P$ in $R$, the local ring ${R_P}$ has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if $R$ is a $2$-dimensional graded ring with an isolated singularity at the irrelevant maximal ideal $m$, then the following are equivalent: (1) $R$ has a rational singularity at $m$. (2) $R$ is $F$-rational. (3) $a(R) < 0$. Here $a(R)$ (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module ${H_m}(R)$. The proof of this result relies heavily on the properties of derivations of $R$, and suggests further questions in that direction; paradigmatically, if one knows that $D(a)$ satisfies a certain property for every derivation $D$, what can one conclude about the original ring element $a$?
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 340 (1993), 655-668
  • MSC: Primary 13A35; Secondary 13D45, 13N05, 14H20
  • DOI:
  • MathSciNet review: 1116312