skip to main content


Title: 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.  more » « less
Award ID(s):
1945611
PAR ID:
10212976
Author(s) / Creator(s):
; ;
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
More Like this
  1. 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. 
    more » « less
  2. Abstract

    We study the symbolic powers of square-free monomial ideals via symbolic Rees algebras and methods in prime characteristic. In particular, we prove that the symbolic Rees algebra and the symbolic associated graded algebra are split with respect to a morphism that resembles the Frobenius map and that exists in all characteristics. Using these methods, we recover a result by Hoa and Trung that states that the normalized $a$-invariants and the Castelnuovo–Mumford regularity of the symbolic powers converge. In addition, we give a sufficient condition for the equality of the ordinary and symbolic powers of this family of ideals and relate it to Conforti–Cornuéjols conjecture. Finally, we interpret this condition in the context of linear optimization.

     
    more » « less
  3. Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence and thus satisfy the stable Harbourne conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided that its symbolic powers are given by saturations with the maximal ideal. Although this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is Noetherian. 
    more » « less
  4. We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely F F -regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic p > 5 p>5 , which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5 p>5 . In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5 p > 5 . Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze. 
    more » « less
  5. The main goal of this paper is to prove, in positive characteristicpp, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that ofpp. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of anc\mathfrak {c}-compressed Gorenstein Artinian graded algebra from its socle degree, wherec\mathfrak {c}is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal.

     
    more » « less