F-jumping Numbers Research Articles

An estimate for 𝐹-jumping numbers via the roots of the Bernstein-Sato polynomial

An estimate for 𝐹-jumping numbers via the roots of the Bernstein-Sato polynomial

The associated graded module of the test module filtration

We show that each direct summand of the associated graded module of the test module filtration admits a natural Cartier structure. If λ is an F-jumping number, then this Cartier structure is nilpotent on if and only if the denominator of λ is divisible by p. We also show that these Cartier structures coincide with certain Cartier structures that are obtained by considering certain -modules associated to M that were used to construct Bernstein-Sato polynomials. Moreover, we point out that the zeros of the Bernstein-Sato polynomial attached to an F-regular Cartier module correspond to its F-jumping numbers. This generalizes [6, Theorem 5.4] where a stronger version of F-regularity was used. Finally, we develop a basic theory of non-F-pure modules and prove a weaker connection between Bernstein-Sato polynomials and Cartier modules for which Mf is F-regular and certain jumping numbers attached to M.

D-modules, Bernstein–Sato polynomials and F-invariants of direct summands

We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization Rf and the local cohomology module HIi(R) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein–Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I⊆R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein–Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.

Discreteness of 𝐹-jumping numbers at isolated non-ℚ-Gorenstein points

We show that the F F -jumping numbers of a pair ( X , a ) (X, \mathfrak {a}) in positive characteristic have no limit points whenever the symbolic Rees algebra of − K X -K_X is finitely generated outside an isolated collection of points. We also give a characteristic zero version of this result, as well as a generalization of the Hartshorne–Speiser–Lyubeznik–Gabber stabilization theorem describing the non- F F -pure locus of a variety.

Local 𝔪-adic constancy of F-pure thresholds and test ideals

AbstractIn this paper, we consider a corollary of the ACC conjecture for F-pure thresholds. Specifically, we show that the F-pure threshold (and more generally, the test ideals) associated to a polynomial with an isolated singularity are locally constant in the 𝔪-adic topology of the corresponding local ring. As a by-product of our methods, we also describe a simple algorithm for computing all of the F-jumping numbers and test ideals associated to an arbitrary polynomial over an F-finite field.

F-jumping and F-Jacobian ideals for hypersurfaces

We introduce two families of ideals, F-jumping ideals and F-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the D-modules Mα that were introduced by Blickle, Mustaţă and Smith. Using strong connections between F-jumping ideals and generalized test ideals, we give a characterization of F-jumping numbers for hypersurfaces via D-modules and F-modules. In addition, we use F-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for F-regularity. Moreover, we prove several properties of F-Jacobian ideals that resemble those of Jacobian ideals of polynomials.

Rings of Frobenius operators

AbstractLet R be a local ring of prime characteristic. We study the ring of Frobenius operators ${\mathcal F}(E)$, where E is the injective hull of the residue field of R. In particular, we examine the finite generation of ${\mathcal F}(E)$ over its degree zero component ${\mathcal F}^0(E)$, and show that ${\mathcal F}(E)$ need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of ${\mathcal F}(E)$ in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Given an ideal a⊆R in a (log) Q-Gorenstein F-finite ring of characteristic p>0, we study and provide a new perspective on the test ideal τ(R,at) for a real number t>0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ(R,at) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F-jumping numbers of τ(R,at) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

Estimates for F-jumping numbers and bounds for Hartshorne–Speiser–Lyubeznik numbers

AbstractGiven an ideal a on a smooth variety in characteristic zero, we estimate theF-jumping numbers of the reductions of a to positive characteristic in terms of the jumping numbers of a and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.

Frobenius and Cartier algebras of Stanley–Reisner rings

We study the generation of the Frobenius algebra of the injective hull of a complete Stanley–Reisner ring over a field with positive characteristic. In particular, by extending the ideas used by M. Katzman to give a counterexample to a question raised by G. Lyubeznik and K.E. Smith about the finite generation of Frobenius algebras, we prove that the Frobenius algebra of the injective hull of a complete Stanley–Reisner ring can be only principally generated or infinitely generated. Also, by using our explicit description of the generators of such algebra and applying the recent work by M. Blickle about Cartier algebras and generalized test ideals, we are able to show that the set of F-jumping numbers of generalized test ideals associated to complete Stanley–Reisner rings form a discrete subset inside the non-negative real numbers.

Test ideals via a single alteration and discreteness and rationality of $F$-jumping numbers

Suppose (X,�) is a log-Q-Gorenstein pair. Recent work of M. Blickle and the first two authors gives a uniform description of the multiplier ideal J(X;�) (in characteristic zero) and the test ideal �(X;�) (in characteristic p > 0) via regular alterations. While in general the alteration required depends heavily on �, for a fixed Cartier divisor D on X it is straightforward to find a single alteration (e.g. a log resolution) computing J(X;� + �D) for all � � 0. In this paper, we show the analogous statement in positive characteristic: there exists a single regular alteration computing �(X;� + �D) for all � � 0. Along the way, we also prove the discreteness and rationality for the F-jumping numbers of �(X;� + �D) for � � 0 where the index of KX + � is arbitrary (and may be divisible by the characteristic).

A note on discreteness of 𝐹-jumping numbers

Suppose that R R is a ring essentially of finite type over a perfect field of characteristic p > 0 p > 0 and that a ⊆ R \mathfrak {a} \subseteq R is an ideal. We prove that the set of F F -jumping numbers of τ b ( R ; a t ) \tau _b(R; \mathfrak {a}^t) has no limit points under the assumption that R R is normal and Q \mathbb {Q} -Gorenstein – we make no assumption as to whether the Q \mathbb {Q} -Gorenstein index is divisible by p p . Furthermore, we also show that the F F -jumping numbers of τ b ( R ; Δ , a t ) \tau _b(R; \Delta , \mathfrak {a}^t) are discrete under the more general assumption that K R + Δ K_R + \Delta is R \mathbb {R} -Cartier.

Discreteness and rationality of F-jumping numbers on singular varieties

We prove that the $F$-jumping numbers of the test ideal $\tau(X; \Delta, \ba^t)$ are discrete and rational under the assumptions that $X$ is a normal and $F$-finite variety over a field of positive characteristic $p$, $K_X+\Delta$ is $\bQ$-Cartier of index not divisible $p$, and either $X$ is essentially of finite type over a field or the sheaf of ideals $\ba$ is locally principal. This is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero.

$F$-thresholds of hypersurfaces

′) for every λ ′ 0 that is F-finite, i.e. such that the Frobenius morphism F : R − → R is finite. If a is an ideal in R and if λ is a nonnegative real number, then the corresponding test ideal is denoted by τ(a � ). In this context we say that λ is an F-jumping exponent (or an F-threshold) if τ(a � ) 6 τ(a � ′ ) for every λ ′ < λ. The following is our main result about F-jumping exponents in positive characteristic. Theorem 1.1. If R is an F-finite regular ring, and if a = (f) is a principal ideal, then the F-jumping exponents of a are rational, and they form a discrete set. The discreteness and the rationality of F-jumping numbers has been proved in [BMS] for every ideal when the ring R is essentially of finite type over an F-finite field. We mention also that for R = k[[x, y]], with k a finite field, the above result has been proved in [Ha] using a completely different approach. We stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic. Even