-purity and rational singularity
Trans. Amer. Math. Soc. 278 (1983), 461-480
Primary 13H10; Secondary 13D03, 14B05
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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.
Hochster and Joel
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Advances in Math. 21 (1976), no. 2, 117–172. MR 0417172
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0252389 (40 #5609)
- M. Hochster and J. L. Roberts, The purity of the Frobenius and local cohomology, Adv. in Math. 21 (1976), 117-172. MR 0417172 (54:5230)
- K. Watanabe, On plurigenera of normal isolated singularities, Math. Ann. 250 (1980), 65-94. MR 581632 (82f:32025)
- M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. in Math. 13 (1974), 115-175. MR 0347810 (50:311)
- R. Elkik, Singularities rationelles et deformations, Invent. Math. 47 (1978), 139-147. MR 501926 (80c:14004)
- C. Huneke, The theory of -sequences and powers of ideals, Adv. in Math. 46 (1982), 249-280. MR 683201 (84g:13021)
- M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463-488. MR 0463152 (57:3111)
- K. Watanabe, T. Ishikawa, S. Tachibana and K. Otsuka, On tensor products of Gorenstein rings, J. Math. Kyoto Univ. 9 (1969), 413-423. MR 0257062 (41:1716)
- R. Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
- E. Kunz, Characterization of regular local rings of characteristic , Amer. J. Math. 91 (1969), 772-784. MR 0252389 (40:5609)
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