Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the F F -jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to F F -jumping numbers.

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We develop a formalism of unit F -modules in the style of Lyubeznik and Emerton-Kisin for rings which have finite F -representation type after localization and completion at every prime ideal. As applications, we show that if R is such a ring then the iterated local cohomology modulesIs (R) have finitely many associated primes, and that all local cohomology modules H n I (R/gR) have closed support when g is a nonzerodivisor on R. Background The Frobenius endomorphismLet p be a prime number and R be a commutative ring of characteristic p > 0. Then the function F : R → R given by F (x) = x p for all x ∈ R is additive, i.e. satisfies F (x + y) = F (x) + F (y) for all x, y ∈ R. Since it is clearly multiplicative and maps 1 to 1, F is a ring homomorphism. This ring homomorphism F is called the Frobenius endomorphism of R. Given an integer e ≥ 0 we denote by F e the e-th iterate of Frobenius, which is given by F e (x) = x p e for all x ∈ R; in particular, F 0 is the identity map on R. Since F e : R → R is a ring homomorphism, its image is a subring of R, which we denote by R p e := {x p e | x ∈ R}.Associated to any ring homomorphism one always has a restriction of scalars functor, and the same is true for F . However, the fact that F has the same target and source can make this very confusing, and special notation is introduced to deal with this situation. Here we will follow the notation from algebraic geometry, but other notations are also common in the literature; most notably some authors write M 1/p e for what we denote F e * M . Given an R-module M and an integer e ≥ 0 we denote by F e * M the R-module obtained via restriction of scalars via F e : R → R. In particular, F e * M is equal to M as an abelian group. Given an element u ∈ M we will usually write F e * u when we want to emphasize that we view u as an element of F e * M and not as an element of M . With this notation the R-module structure of F e * M is given byfor all x ∈ R and u ∈ M . Remark 2.1. Here we are making a slight departure from standard terminology in that, for us, F e * M = M as sets, whereas most authors would define F e * M as the set {F e * u | u ∈ M }. Our definition has the pleasant consequence that, given integers i, j ≥ 0 the set Hom R (F i * M, F j * M ) is actually the subset of Hom Z (M, M ) given by Homwhereas in the other terminology Hom R (F i * M, F j * M ) is only naturally identified with this set. This means, for example, that given α ∈ Hom R (F i * M, F j * M ) and β ∈ Hom R (F a * M, F b * M ) the composition β • α is well defined as an element of End Z (M, M ); one can then think about what the linearity of β • α is if necessary. Definition 2.2. Let R be a noetherian ring of characteristic p > 0. We say that R is F -finite whenever the Frobenius endomorphism F : R → R is module-finite, i.e. whenever F * R is a finitely generated R-module.Recall that a composition of module-finite ring homomorphisms is again module-finite. In particular, if R is F -finite then F e * R is a finitely generated R-module for every e ≥ 0. Every...

Following work of Mustaţȃ and Bitoun we recently developed a notion of Bernstein-Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein-Sato polynomial.Here we prove that for monomial ideals the roots of the Bernstein-Sato polynomial (over C) agree with the Bernstein-Sato roots of the mod-p reductions of the ideal for p large enough. We regard this as evidence that the characteristic-p notion of Bernstein-Sato root is reasonable.

The Bernstein-Sato polynomial is an important invariant of an element or an ideal in a polynomial ring or power series ring of characteristic zero, with interesting connections to various algebraic and topological aspects of the singularities of the vanishing locus. Work of Mustaţă, later extended by Bitoun and the third author, provides an analogous Bernstein-Sato theory for regular rings of positive characteristic. In this paper, we extend this theory to singular ambient rings in positive characteristic. We establish finiteness and rationality results for Bernstein-Sato roots for large classes of singular rings, and relate these roots to other classes of numerical invariants defined via the Frobenius map. We also obtain a number of new results and simplified arguments in the regular case.