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An excellent ring of prime characteristic for which the Frobenius map is pure is also Frobenius split in many commonly occurring situations in positive characteristic commutative algebra and algebraic geometry. However, using a fundamental construction from rigid geometry, we show that excellent $$F$$-pure rings of prime characteristic are not Frobenius split in general, even for Euclidean domains. Our construction uses the existence of a complete non-Archimedean field $$k$$ of characteristic $$p$$ with no nonzero continuous $$k$$-linear maps $$k^{1/p} \to k$$. An explicit example of such a field is given based on ideas of Gabber, and may be of independent interest. Our examples settle a long-standing open question in the theory of $$F$$-singularities whose origin can be traced back to when Hochster and Roberts introduced the notion of $$F$$-purity. The excellent Euclidean domains we construct also admit no nonzero $$R$$-linear maps $$R^{1/p} \rightarrow R$$. These are the first examples that illustrate that $$F$$-purity and Frobenius splitting define different classes of singularities for excellent domains, and are also the first examples of excellent domains with no nonzero $$p^{-1}$$-linear maps. The latter is particularly interesting from the perspective of the theory of test ideals.
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Frobenius powers
This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals.
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- Award ID(s):
- 1945611
- PAR ID:
- 10212976
- Date Published:
- Journal Name:
- Mathematische Zeitschrift
- Volume:
- 296
- Issue:
- no. 1-2
- ISSN:
- 0025-5874
- Page Range / eLocation ID:
- 541–572
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s00209-019-02442-2
