-purity and rational singularity
Author: Richard Fedder
Journal: Trans. Amer. Math. Soc. 278 (1983), 461-480
MSC: Primary 13H10; Secondary 13D03, 14B05
MathSciNet review: 701505
Abstract: We investigate singularities which are -pure (respectively -pure type). A ring of characteristic is -pure if for every -module is exact where denotes the -algebra structure induced on via the Frobenius map (if and , then in ). -pure type is defined in characteristic 0 by reducing to characteristic .
It is proven that when is the quotient of a regular local ring , is -pure at the prime ideal if and only if . Here, denotes the ideal . Several theorems result from this criterion. If is a quasihomogeneous hypersurface having weights and an isolated singularity at the origin:
(1) implies has -pure type at .
(2) implies does not have -pure type at .
(3) remains unsolved, but does connect with a problem that number theorists have studied for many years.
This theorem parallels known results about rational singularities. It is also proven that classifying -pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the -pure locus in the maximal spectrum of , where is a perfect field of characteristic , is Zariski open.
An important conjecture is that is -pure (type) should imply is -pure (type) whenever is a Cohen-Macauley, normal local ring. It is proven that is a sufficient, though not necessary, condition.
A local ring of characteristic is -injective if the Frobenius map induces an injection on the local cohomology modules . An example is constructed which is -injective but not -pure. From this a counterexample to the conjecture that is -pure implies is -pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic 0 version of the conjecture.
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