-purity and rational singularity in graded complete intersection rings

Author:
Richard Fedder

Journal:
Trans. Amer. Math. Soc. **301** (1987), 47-62

MSC:
Primary 14B05; Secondary 13H10, 14M10

MathSciNet review:
879562

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Abstract: A simple criterion is given for determining ``almost completely'' whether the positively graded complete intersection ring , of dimension , has an -pure type singularity at . Specifically, if for and for , then there exists an integer determined by the singular locus of such that:

(1) has -pure type if .

(2) does not have -pure type if .

The characterization given by this theorem is particularly effective if the singularity of at is isolated. In that case, so that only the condition is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if has an isolated rational singularity, then has -pure type. The converse is also ``almost true'' with the only exception being the case where .

In proving this criterion, a weak but more stable form of -purity, called -contractedness, is defined and explored. is -contracted (in characteristic ) if every system of parameters for is contracted with respect to the Frobenius map . Just as for -purity, the notion of -contracted type is defined in characteristic 0 by reduction to characteristic . The two notions of -pure (type) and -contracted (type) coincide when is Gorenstein; whence, in particular, when is a complete intersection ring.

**[1]**John A. Eagon and M. Hochster,*𝑅-sequences and indeterminates*, Quart. J. Math. Oxford Ser. (2)**25**(1974), 61–71. MR**0337934****[2]**Richard Fedder,*𝐹-purity and rational singularity*, Trans. Amer. Math. Soc.**278**(1983), no. 2, 461–480. MR**701505**, 10.1090/S0002-9947-1983-0701505-0**[3]**Shiro Goto and Keiichi Watanabe,*The structure of one-dimensional 𝐹-pure rings*, J. Algebra**49**(1977), no. 2, 415–421. MR**0453729****[4]**Melvin Hochster and Joel L. Roberts,*Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay*, Advances in Math.**13**(1974), 115–175. MR**0347810****[5]**Melvin Hochster and Joel L. Roberts,*The purity of the Frobenius and local cohomology*, Advances in Math.**21**(1976), no. 2, 117–172. MR**0417172****[6]**Melvin Hochster,*Cyclic purity versus purity in excellent Noetherian rings*, Trans. Amer. Math. Soc.**231**(1977), no. 2, 463–488. MR**0463152**, 10.1090/S0002-9947-1977-0463152-5**[7]**Eben Matlis,*Injective modules over Noetherian rings*, Pacific J. Math.**8**(1958), 511–528. MR**0099360****[8]**Keiichi Watanabe,*Rational singularities with 𝑘*-action*, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 339–351. MR**686954**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1987-0879562-4

Article copyright:
© Copyright 1987
American Mathematical Society