Deformation of F-purity and F-regularity
At the time this paper was written, it was an open question whether F-regularity deforms. Since then it has been answered in the negative by \textit{A. G. Singh} [Am. J. Math. 121 No. 4, 919-929 (1999; see the following review Zbl 0946.13002)]. However, the paper under review proves that F-regularity deforms for normal local rings whenever for any ideal \(I\) of pure height one representing the inverse of the canonical module in the class group, the symbolic powers \(I^{(i)}\) all satisfy Serre's condition (S\(_3\)) and the anti-canonical cover \(\bigoplus_{i \geq 0} I^{(i)}\) is a Noetherian ring. The deformation of F-regularity had previously been shown for more restrictive classes by \textit{I. Aberbach, M. Katzman} and \textit{B. MacCrimmon} [J. Algebra 204, No. 1, 281-285 (1998; Zbl 0915.13003)], \textit{M. Hochster} and \textit{C. Huneke} [Trans. Am. Math. Soc. 346, No. 1, 1-62 (1994; Zbl 0844.13002)]\ and \textit{K. E. Smith} [``F-regularity deforms for \(\mathbb Q\)-Gorenstein rings'' (preprint)]. The author also shows examples for which F-purity does not deform. Earlier examples had been constructed by \textit{R. Fedder} [Trans. Am. Math. Soc. 278, 461-480 (1983; Zbl 0519.13017)]. Unlike Fedder's, Singh's examples are integral domains.
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