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Transactions of the American Mathematical Society

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

 

 

$ F$-purity and rational singularity


Author: Richard Fedder
Journal: Trans. Amer. Math. Soc. 278 (1983), 461-480
MSC: Primary 13H10; Secondary 13D03, 14B05
DOI: https://doi.org/10.1090/S0002-9947-1983-0701505-0
MathSciNet review: 701505
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Abstract: We investigate singularities which are $ F$-pure (respectively $ F$-pure type). A ring $ R$ of characteristic $ p$ is $ F$-pure if for every $ R$-module $ M,0 \to M \otimes R \to M \otimes\, ^1R$ is exact where $ ^1R$ denotes the $ R$-algebra structure induced on $ R$ via the Frobenius map (if $ r \in R$ and $ s \in \, ^{1}R$, then $ r \cdot s = {r^p}s$ in $ ^1R$). $ F$-pure type is defined in characteristic 0 by reducing to characteristic $ p$.

It is proven that when $ R = S/I$ is the quotient of a regular local ring $ S$, $ R$ is $ F$-pure at the prime ideal $ Q$ if and only if $ ({I^{[p]}}:I) \not\subset {Q^{[p]}}$. Here, $ {J^{[p]}}$ denotes the ideal $ \{ {a^p}\vert a \in J\} $. Several theorems result from this criterion. If $ f$ is a quasihomogeneous hypersurface having weights $ ({r_1},\ldots,{r_n})$ and an isolated singularity at the origin:

(1) $ \sum\nolimits_{i = 1}^n {{r_i} > 1} $ implies $ K[{X_1},\ldots,{X_n}]/(f)$ has $ F$-pure type at $ m = ({X_1},\ldots,{X_n})$.

(2) $ \sum\nolimits_{i = 1}^n {{r_i} < 1} $ implies $ K[{X_1},\ldots,{X_n}]/(f)$ does not have $ F$-pure type at $ m$.

(3) $ \sum\nolimits_{i = 1}^n {{r_i} = 1} $ 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 $ F$-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the $ F$-pure locus in the maximal spectrum of $ K[{X_1},\ldots,{X_n}]/I$, where $ K$ is a perfect field of characteristic $ P$, is Zariski open.

An important conjecture is that $ R/fR$ is $ F$-pure (type) should imply $ R$ is $ F$-pure (type) whenever $ R$ is a Cohen-Macauley, normal local ring. It is proven that $ \operatorname{Ext}^1{(^1}R,R) = 0$ is a sufficient, though not necessary, condition.

A local ring $ (R,m)$ of characteristic $ p$ is $ F$-injective if the Frobenius map induces an injection on the local cohomology modules $ H_m^i(R) \to H_m^i{(^1}R)$. An example is constructed which is $ F$-injective but not $ F$-pure. From this a counterexample to the conjecture that $ R/fR$ is $ F$-pure implies $ R$ is $ F$-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.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0701505-0
Article copyright: © Copyright 1983 American Mathematical Society