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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$ F$-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 $ 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.

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Article copyright: © Copyright 1987 American Mathematical Society