purity and rational singularity in graded complete intersection rings
Author:
Richard Fedder
Journal:
Trans. Amer. Math. Soc. 301 (1987), 4762
MSC:
Primary 14B05; Secondary 13H10, 14M10
MathSciNet review:
879562
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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 Keiichi 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, 𝑅sequences and indeterminates, Quart. J.
Math. Oxford Ser. (2) 25 (1974), 61–71. MR 0337934
(49 #2703)
 [2]
Richard
Fedder, 𝐹purity and rational
singularity, Trans. Amer. Math. Soc.
278 (1983), no. 2,
461–480. MR
701505 (84h:13031), http://dx.doi.org/10.1090/S00029947198307015050
 [3]
Shiro
Goto and Keiichi
Watanabe, The structure of onedimensional 𝐹pure
rings, J. Algebra 49 (1977), no. 2,
415–421. MR 0453729
(56 #11989)
 [4]
Melvin
Hochster and Joel
L. Roberts, Rings of invariants of reductive groups acting on
regular rings are CohenMacaulay, Advances in Math.
13 (1974), 115–175. MR 0347810
(50 #311)
 [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
(54 #5230)
 [6]
Melvin
Hochster, Cyclic purity versus purity in
excellent Noetherian rings, Trans. Amer. Math.
Soc. 231 (1977), no. 2, 463–488. MR 0463152
(57 #3111), http://dx.doi.org/10.1090/S00029947197704631525
 [7]
Eben
Matlis, Injective modules over Noetherian rings, Pacific J.
Math. 8 (1958), 511–528. MR 0099360
(20 #5800)
 [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
(84e:14005)
 [1]
 J. A. Eagon and M. Hochster, sequences and indeterminates, Quart. J. Math Oxford Ser. 25 (1974), 6171. MR 0337934 (49:2703)
 [2]
 R. Fedder, purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461480. MR 701505 (84h:13031)
 [3]
 S. Goto and K.i. Watanabe, The structure of dimensional pure rings, J. Algebra 49 (1977), 415421. MR 0453729 (56:11989)
 [4]
 M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are CohenMacaulay, Adv. in Math. 13 (1974), 115175. MR 0347810 (50:311)
 [5]
 , The purity of the Frobenius and local cohomology, Adv. in Math. 21 (1976), 117172. MR 0417172 (54:5230)
 [6]
 M. Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), 463488. MR 0463152 (57:3111)
 [7]
 E. Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511528. MR 20 # 5800. MR 0099360 (20:5800)
 [8]
 K.i. Watanabe, Rational singularities with action, Proc. Trente Conf., Lecture Notes in Pure and Appl. Math. 84, Dekker, New York, 1983. MR 686954 (84e:14005)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
14B05,
13H10,
14M10
Retrieve articles in all journals
with MSC:
14B05,
13H10,
14M10
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708795624
PII:
S 00029947(1987)08795624
Article copyright:
© Copyright 1987 American Mathematical Society
