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\(F\)-purity of hypersurfaces. (English) Zbl 1291.13009

Suppose that \(R\) is a ring of characteristic \(p > 0\). \(R\) is called \(F\)-pure if the Frobenius map is a pure map of rings (in the case that Frobenius is finite, this is the same as saying that Frobenius splits). In [J. Algebr. Geom. 11, No. 2, 363–392 (2002; Zbl 1013.13004)], N. Hara and K.- Watanabe introduced the notion of F-singularities of pairs and in particular of \(F\)-pure pairs. In [Math. Res. Lett. 15, No. 5–6, 1251–1261 (2008; Zbl 1185.13010)], the reviewer introduced the notion of sharply \(F\)-pure pairs, a more specialized notion of \(F\)-purity that behaved more like \(F\)-purity for rings. In this review, we consider only pairs of the form \((R, f^t)\) where \(R\) is an \(F\)-pure ring and \(f \in R\) (in particular, we do not consider non-principal ideals or divisors).
Unfortunately if \(t\) is the \(F\)-pure threshold, then \((R, f^t)\) is not always sharply \(F\)-pure. An argument of M. Mustaţă found in the paper of the reviewer above shows that it is in the case that the \(F\)-pure threshold does not have a \(p\) dividing its denominator, and the ambient ring is regular. The first main result of the paper under review shows the following.
{ Theorem.} If \(t\) is the \(F\)-pure threshold of \((R, f)\), and if there is no \(p\) dividing the denominator of \(t\), then \((R, f^t)\) is sharply \(F\)-pure. Even if the denominator of \(t\) is divisible by \(p\), then the \((R, f^t)\) is \(F\)-pure in the sense of Hara-Watanabe.
In another result, the author shows that there can never be an \(F\)-pure threshold strictly between the values \({b \over p^e}\) and \({b \over p^e - 1}\) for any \(e> 0\). This generalizes a result in [M. Blickle et al., Trans. Am. Math. Soc. 361, No. 12, 6549–6565 (2009; Zbl 1193.13003)] from the case of an ambient regular ring.

MSC:

13A35 Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure
14F18 Multiplier ideals
14B05 Singularities in algebraic geometry
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