$F$-purity and rational singularity in graded complete intersection rings
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- by Richard Fedder PDF
- Trans. Amer. Math. Soc. 301 (1987), 47-62 Request permission
Abstract:
A simple criterion is given for determining “almost completely” whether the positively graded complete intersection ring $R = K[{X_1}, \ldots , {X_{n + t}}]/({G_i}, \ldots , {G_t})$, of dimension $n$, has an $F$-pure type singularity at $m = ({X_1}, \ldots , {X_{n + t}})$. Specifically, if $\operatorname {deg} ({X_i}) = {\alpha _i} > 0$ for $1 \leq i \leq n + t$ and $\operatorname {deg} ({G_i}) = {d_i} > 0$ for $i \leq i \leq t$, then there exists an integer $\delta$ determined by the singular locus of $R$ such that: (1) $R$ has $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} < \delta$. (2) $R$ does not have $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0$. The characterization given by this theorem is particularly effective if the singularity of $R$ at $m$ is isolated. In that case, $\delta = 0$ so that only the condition $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$ is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if $R$ has an isolated rational singularity, then $R$ has $F$-pure type. The converse is also “almost true” with the only exception being the case where $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$. In proving this criterion, a weak but more stable form of $F$-purity, called $F$-contractedness, is defined and explored. $R$ is $F$-contracted (in characteristic $p > 0$) if every system of parameters for $m$ is contracted with respect to the Frobenius map $F: R \to R$. Just as for $F$-purity, the notion of $F$-contracted type is defined in characteristic 0 by reduction to characteristic $p$. The two notions of $F$-pure (type) and $F$-contracted (type) coincide when $R$ is Gorenstein; whence, in particular, when $R$ is a complete intersection ring.References
- John A. Eagon and M. Hochster, $R$-sequences and indeterminates, Quart. J. Math. Oxford Ser. (2) 25 (1974), 61–71. MR 337934, DOI 10.1093/qmath/25.1.61
- Richard Fedder, $F$-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), no. 2, 461–480. MR 701505, DOI 10.1090/S0002-9947-1983-0701505-0
- Shiro Goto and Keiichi Watanabe, The structure of one-dimensional $F$-pure rings, J. Algebra 49 (1977), no. 2, 415–421. MR 453729, DOI 10.1016/0021-8693(77)90250-2
- 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 347810, DOI 10.1016/0001-8708(74)90067-X
- Melvin Hochster and Joel L. Roberts, The purity of the Frobenius and local cohomology, Advances in Math. 21 (1976), no. 2, 117–172. MR 417172, DOI 10.1016/0001-8708(76)90073-6
- Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, DOI 10.1090/S0002-9947-1977-0463152-5
- Eben Matlis, Injective modules over Noetherian rings, Pacific J. Math. 8 (1958), 511–528. MR 99360, DOI 10.2140/pjm.1958.8.511
- Keiichi Watanabe, Rational singularities with $k^{\ast }$-action, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 339–351. MR 686954
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 301 (1987), 47-62
- MSC: Primary 14B05; Secondary 13H10, 14M10
- DOI: https://doi.org/10.1090/S0002-9947-1987-0879562-4
- MathSciNet review: 879562