A Frobenius characterization of rational singularity in -dimensional graded rings

Author:
Richard Fedder

Journal:
Trans. Amer. Math. Soc. **340** (1993), 655-668

MSC:
Primary 13A35; Secondary 13D45, 13N05, 14H20

MathSciNet review:
1116312

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Abstract: A ring is said to be -rational if, for every prime in , the local ring has the property that every system of parameters ideal is tightly closed (as defined by Hochster-Huneke). It is proved that if is a -dimensional graded ring with an isolated singularity at the irrelevant maximal ideal , then the following are equivalent:

(1) has a rational singularity at .

(2) is -rational.

(3) .

Here (as defined by Goto-Watanabe) denotes the least nonvanishing graded piece of the local cohomology module .

The proof of this result relies heavily on the properties of derivations of , and suggests further questions in that direction; paradigmatically, if one knows that satisfies a certain property for every derivation , what can one conclude about the original ring element ?

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DOI:
https://doi.org/10.1090/S0002-9947-1993-1116312-3

Article copyright:
© Copyright 1993
American Mathematical Society