1. Introduction 1.1. Background. Let R := C[x 1 ,...,x n ] be a polynomial ring over C, f ∈ R be a nonzero polynomial and D R be the ring of C-linear differential operators on R;t h a ti s ,D R is the Weyl algebra over C. Bernstein [Ber72] and Sato [SS72], independently and in different contexts, showed that there is a nonzero polynomial b f (s) and an element ξ(s) ∈ D R [s] satisfying the functional equation

The monic polynomial b f (s) of least degree satisfying the equation above for some operator ξ(s) ∈ D R [s] is called the Bernstein-Sato polynomial of f . This invariant measures the singularities of the zero-locus of f in very subtle ways. For example, work of Kollár [Kol96] and Ein, Lazarsfeld, Smith, and Varolin [ELSV04] gives that the log-canonical threshold lct(f )off is the smallest root of b f (-s), and that every jumping number in the interval [0, 1) is a root of b f (-s). Furthermore, Kashiwara [Kas83] and Malgrange [Mal83] proved that the eigenvalues of the monodromy action on the cohomology of the Milnor fiber of f are given by exp(2πiα)w h e r e α ranges through all the roots of the Bernstein-Sato polynomial of f . Kashiwara [Kas77] showed that the roots of b f (s) are rational and negative which, combined with the previous result, shows that the monodromy action is quasi-unipotent (see also [Mal75]). An alternative characterization of b f (s) due to Malgrange exhibits b f (s)a st h e minimal polynomial for the action of an operator s on a certain D-module N f . Budur, Mustat ¸ȃ, and Saito constructed an analogue of s and N f for the case of an arbitrary ideal a ⊆ R. Namely, even though N a is usually not finitely generated, there exists a minimal polynomial of s on N a ; the Bernstein-Sato polynomial of a is defined as this polynomial [BMS06]. One has that N a splits as a direct sum

where (N a ) λ is the λ-generalized eigenspace.

One can recover the minimal polynomial b a (s) from this decomposition, since the roots are given by

and the multiplicity of a root λ is given by

An extension of this rich theory has been proposed recently for the case where R is a possibly singular C-algebra. Whenever R is a direct summand of a polynomial ring over C, Àlvarez Montaner, Huneke, and the second author [ ÀMHNB17] showed that one can find b f (s)a n dξ(s) as in Equation (1.1) and thus define a Bernstein-Sato polynomial for elements f of R. We remark that to carry out this construction one must take D R to be the ring of C-linear differential operators of R in the sense of Grothendieck. This line of research has continued with explorations into connections with V -filtrations, multiplier ideals, and an extension of these constructions for the case of ideals [ ÀMHJ + 22]. Some aspects of the theory have also been developed in positive characteristic. This began with the work of Mustat ¸ȃ, who began this exploration in the case where R := K[x 1 ,...,x n ] is a polynomial ring over a perfect field K of characteristic p>0 (or, more generally, a regular F -finite ring) and a =( f ) is a principal ideal. Mustat ¸ȃ's notion was later refined by Bitoun [Bit18] and extended to the case of arbitrary ideals by the third author [QG21b].

The main goal in this paper is to explore the theory of Bernstein-Sato roots in positive characteristic after dropping the regularity assumption on R.O n ec a n therefore think of this paper as providing a characteristic p counterpart to some of the work on differential operators in singular rings [ ÀMHNB17, ÀMHJ + 22]. In order to explain our results, we need to elaborate on this notion of Bernstein-Sato invariants in positive characteristic.

Suppose R := K[x 1 ,...,x n ] is a polynomial ring over a perfect field K of characteristic p>0 and let a ⊆ R be an ideal. One starts, through mimicking the construction of Budur, Mustat ¸ȃ, and Saito, by defining a D R -module N a associated to a. In the characteristic p setting the action of the operator s on N a naturally extends to an action of the algebra C( Z (p) , F p ) of continuous functions from the p-adics Z (p) to F p . Note that we are explaining this construction using the algebra C( Z (p) , F p ) of continuous functions from the p-adics Z (p) to F p ,i nt h es t y l eo f Bitoun [Bit18], as opposed to the operators s p 0 ,s p 1 ,... in the style of Mustat ¸ȃand the third author. See Subsection 7.2 for the equivalence between these two points of view.

Given a p-adic integer α ∈ Z (p) ,w el e tm α be the maximal ideal of C( Z (p) , F p ) that consists of functions that vanish on α,a n dw el e t( N a ) α :=A n n m α N a .A careful analysis of the module N a allows one to show that there is a decomposition

for which only finitely many (N a ) α are nonzero [Bit18], [QG21b, Proposition 6.1]. A posteriori, we conclude that the (N a ) α can also be viewed as quotients of N a , namely (N a ) α ∼ = N a /m α N a .

In analogy with the situation over C, we want to obtain some invariant of a from this decomposition. A Bernstein-Sato root of a is thus defined to be a p-adic integer α ∈ Z (p) such that (N a ) (α) is nonzero (cf. (1.2)); we think of these as characteristic p analogues of the roots of the Bernstein-Sato polynomial. In this setting, however, we have m α = m 2 α for all α ∈ Z (p) a n di ti st h e r e f o r en o tc l e a rh o wt oa s s o c i a t ea multiplicity to each Bernstein-Sato root (cf. (1.3)), and thus there is no notion of Bernstein-Sato polynomial.

The Bernstein-Sato roots of a nonzero ideal in a regular F -finite ring are known to be rational and negative [Bit18,Corollary 2.4.3], [QG21b, Theorem 6.7], which gives a characteristic p analogue of Kashiwara's theorem, and they are also known to be intimately linked to the F -jumping numbers of a [Bit18, Theorem 2.4.1], [QG21b,Theorem 6.11].

1.2. Summary of results. In this paper, we pose and work with an elementary definition of Bernstein-Sato root in positive characteristic. Namely, we define a Bernstein-Sato root of an ideal a to be a p-adic integer that occurs as the p-adic limit of a sequence of the form (ν e ) such that a ν e is not contained in φ∈Hom R p e (R,R) φ(a ν e +1 ) (see Definitions 3.1 and 4.1). This elementary notion naturally extends the notion of Bernstein-Sato root in positive characteristic for regular rings described in the previous subsection. Namely, even after dropping the regularity assumption on R, we can still build the module N a , equip it with a C( Z (p) , F p )-action, and consider the modules (N a ) α = N a /m α N a for all α ∈ Z (p) . We have that α ∈ Z (p) is a Bernstein-Sato root of a if and only if the module (N a ) α Licensed to Mathematical Sciences Research Institute. Prepared on Sat May 11 18:06:43 EDT 2024 for download from IP 173.239.64.5.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use is nonzero (Theorem 7.3). In contrast with the regular case, when R is singular an ideal a ⊆ R may have infinitely many Bernstein-Sato roots (see Example 9.1) and the quotients N a → (N a ) α a priori need not split.

We begin by isolating the necessary assumptions on a so that these pathologies do not occur, and we encapsulate these in the notion of Bernstein-Sato admissible ideals (see Definition 4.10). It is implicit in the work of the third author that ideals of regular rings are always Bernstein-Sato admissible [QG21b, Theorem 6.1]. We show that these ideals also abound in other classes of rings.

Theorem A (Theorem 6.4, Theorem 6.7). Let R be a noetherian F -finite ring, and assume one of the following holds:

(a) The ring R is graded with finite F -representation type.

(b) The ring R is a direct summand of a regular ring. Then every ideal a ⊆ R is Bernstein-Sato admissible.

As mentioned, whenever an ideal a is Bernstein-Sato admissible we show that the module N a b e h a v e sa si nt h er e g u l a rc a s e .

Theorem B. Let R be a noetherian F -finite ring and a ⊆ R be a Bernstein-Sato admissible ideal. Then:

(i) (Theorem 4.13) The ideal a has only finitely many Bernstein-Sato roots.

(ii) (Corollary 7.4) The module N a splits as a direct sum N a = α∈ Z (p) (N a ) α .

If in addition R is F -split, then (iii) (Theorems 4.16 & 4.18) All of the Bernstein-Sato roots of a are rational and lie in the intervalfoot_1 [-r, 0],w h e r er is the number of generators of a.

We remark that, for nonprincipal ideals, the lower bound is new even in the case where R is regular. Combining this with a result of the third author on the behavior of Bernstein-Sato roots of monomial ideals under mod-p reduction [QG21a, Theorem 3.1], we are able to give the following characteristic zero result.

Corollary C (Corollary 4.20). Let R = C[x 1 ,...,x n ] be a polynomial ring over C and a ⊆ R be a monomial ideal generated by r elements. If λ is a root of the Bernstein-Sato polynomial of a then 1 -r ≤ λ.

To study the action of differential operators on ideals in the ring, we introduce a family of numerical invariants called differential thresholds. The collection of differential thresholds of an ideal contains several of its invariants defined via Frobenius, including all of its jumping numbers, F -thresholds, and Cartier thresholds (see Subsection 5). This unified approach allows us to obtain properties that were not known in certain cases. We show that if a ⊆ R is a Bernstein-Sato admissible ideal, then the set of differential thresholds for a is a discrete set of rational numbers (see Theorems 5.15 and 5.16). As a consequence, we obtain that the F -thresholds of rings with graded finite F -representation type are rational numbers (see Corollary 6.5). This extends previous results obtained for certain ideals in Stanley-Reisner rings [BC21, Theorems A & B]. We also exhibit a close relation between differential thresholds and Bernstein-Sato roots.

Theorem D (Theorem 5.17). Let R be F -split. Let a be an ideal with r generators. There is an equality of cosets in Z (p) /Z:

We are able to define a module R f f α for any p-adic number that is the positive characteristic analogue of the modules R f f α with α ∈ Q in characteristic zero [Wal15]. If R is regular and α ∈ Z (p) ∩ Q <0 ,t h e nR f f α = M -α for regular rings, where M -α is the F -module introduced in earlier work of Blickle, Mustat ¸ȃ, and Smith to study jumping numbers of principal ideals [BMS09] (see also [NBP16]). In Proposition 7.15, we show that in contrast to the situation in characteristic zero [Sai21], α is a Bernstein-Sato root if and only if

of the simplicity of R f f α in terms of Bernstein-Sato roots and differential thresholds.

Theorem E (Theorem 8.13). Suppose that R is a strongly F -regular domain. Let f ∈ R be a Bernstein-Sato admissible nonzerodivisor and α ∈ Z (p) ∩ [-1, 0).T h e n the following are equivalent:

Moreover, the finite generation or finite length R f f α as a D R -module provides information about the distribution of the Bernstein-Sato roots and differential thresholds (see Theorems 8.1 & 8.4).

Since in the regular case M α is an F -finite F -module, it has finite length as a D R -module [BMS09] (see also [Lyu97]). In Theorem 8.19, we show that R f f α is a holonomic D R -module for every Bernstein algebra, and so it has finite length as a D R -module. This is a recently defined class of singular algebras whose D R -modules satisfy the Bernstein inequality [ ÀMHJ + 21]. We stress that our results regarding R f f α do not use the theory of F -modules, which is not available for singular rings.

We point out that we prove some results that are new even in the case where R is regular (e.g., Corollary 4.19). In addition, we provide new proofs of previously known theorems (e.g., Theorem 4.18).

Axel Stäbler has pointed out to us that Bernstein-Sato polynomials for Cartier modules [BS16,Stä21] can also be used to give a notion of Bernstein-Sato polynomial for certain singular algebras. Namely, if R = S/I is strongly F -regular and Q-Gorenstein, and S is regular, one may consider R as a Cartier module over the ring S, and apply the theory of ibid. to obtain Bernstein-Sato polynomials. In contrast, our approach uses the operators on the singular ring itself and is developed for rings that are not necessary strongly F -regular. In particular, in our approach an ideal in a strongly F -regular ring may have more roots in the interval [-1, 0] than jumping numbers (see Example 9.3). 1.3. Notation. We fix a prime numb er p,a n d Z (p) denotes the ring of p-adic integers. Unless otherwise stated, all rings have characteristic p and are F -finite, meaning that the Frobenius endomorphism is module-finite.

Given an ideal a in a ring R,w es e ta 0 = R by convention (even when a = (0)). We use multi-index notation: given a tuple of integers a =( a 1 ,...,a n ) ∈ Z n and a tuple of elements g =(g 1 ,...,g n ) ∈ S n in a commutative ring S,w ed e n o t e g a := g a 1 1 •••g a n n .T h es y m b o l1denotes the tuple 1 :=(1, 1,...,1). Recall we have a multi-index binomial theorem: given a commutative ring S, tuples x,y ∈ S n and am u l t i -e x p o n e n ta ∈ (Z ≥0 ) n we have

2.1. Base p and p-adic expansions. Fix a prime number p.L e t Z (p) denote the completion with respect to (p)o fZ (p) , i.e., the ring of p-adic integers. Given α ∈ Z (p) , there exists a unique sequence of integers (α e ) e∈Z ≥0 such that: (i) 0 ≤ α e ≤ p -1, and

(ii) α = e≥0 p e α e as a series in Z (p) . We call α e the e-th p-adic digit of α. We reserve the notation α e for this notion. We define the e-th p-adic truncation of α to be the unique integer n with 0 ≤ n<p e such that α ≡ n mod p e ;e q u i v a l e n t l y ,

Recall that a p-adic integer α is rational if and only if α ∈ Z (p) ; this is equivalent to α admitting an eventually periodic sequence of p-adic digits.

A p-adic number α has a purely periodic sequence of p-adic digits if and only if α ∈ Z (p) ∩ [-1, 0]. In particular, the sequence of p-adic digits of α is periodic of period e if and only if (1-p e )α is an integer between 0 and p e -1, and in this case we have (1 -p e )α = α <e ; our convention is that the period is not necessarily minimal. In particular, if α ∈ Z (p) ∩ [-1, 0] with (1 -p e )α ∈ Z ≥0 ,t h e n( 1-p ae )α = α <ea for all a. Similarly, the sequence of p-adic digits of α is eventually periodic of period e if and only if (1 -p e )α ∈ Z.

We can also extract the p-adic truncations of an arbitrary element α ∈ Z (p) .F o r our purposes, it suffices to determine for any such α an infinite sequence of p-adic truncations.

Lemma 2.1. Let α ∈ Z (p) ,a n dl e te ∈ Z >0 such that (p e -1)α ∈ Z.T h e n ,f o ra l l a 0, we have

Proof. The claim is clear when α ∈ Z,s ot a k eα/ ∈ Z.W e fi r s t o b s e r v e t h a t (1

We have α -α < 0, so for a 0, we have -α ≤(1 -p ae )(α -α ) and thus (1 -p ae )(α -α )+ α ≥0.

(iii) Assume that R is a domain. We say that R is strongly F -regular if for every nonzero r ∈ R there exists e ∈ Z ≥0 such that the R-module homomorphism ϕ : R → F e * R given by 1 → F e * r splits.

Remark 2.4. Suppose that R is a ring of prime characteristic p.

In the definitions of F -finite, F -pure, and F -split, the conditions on F * R can be replaced by F e * R for some e ≥ 1, or by F e * R for every e ≥ 1.

Definition 2.5. Let R be an F -finite ring and e ≥ 0 be an integer. An additive map φ : R → R is a p -e -linear map if φ(r p e f )=rφ(f ) for all r, f ∈ R.W ed e n o t e by C e R the set of all p -e -linear maps. Then, we have

Test ideals were introduced by Hochster and Huneke, and they are a fundamental tool in the theory of tight closure [HH90, HH94, HH94]. Hara and Yoshida [HY03] extended the notion of test ideals, τ R (a λ ), to pairs (R, a λ ), where a ⊆ R is an ideal and λ ∈ R. One can approach the theory of test ideals using Cartier operators [BMS08,BMS09,Sch11,BB11,Bli13]. We now give the definition in terms of Cartier operators for strongly F -regular rings [TT08].

Definition 2.6. Let R be an F -finite strongly F -regular ring. Let a ⊆ R be an ideal, and λ ∈ R >0 . The test ideal of the pair (a,λ) is defined by

The notion of test ideal discussed here is sometimes called the big test ideal. We note that the chain of ideals {C e R • a p e λ } is increasing, and so, τ R (a λ )= C e R • a p e λ for e 0, because R is noetherian. We now recall well-known properties of test ideals. We refer to the work done specifically for strongly F -regular rings [TT08]. For a more general approach, we refer to Blickle's work on this subject [Bli13].

Every ideal a ⊆ R is associated to a family of test ideals τ (a λ ) parameterized by real numbers λ ∈ R >0 which forms a decreasing nested chain of ideals as λ increases.

Definition 2.8. Let R be an F -finite strongly F -regular ring and let a ⊆ R be an ideal. A real number λ ≥ 0i sa nF -jumping number of a if τ R (a λ ) = τ R (a λ-ε ) for every ε>0. Let R be a K-algebra, where K is a field. The ring of K-linear differential operators of R is the subring D R|K ⊆ Hom Z (R, R) whose elements are characterized inductively as follows. The differential operators of order zero are D 0 R|K = Hom R (R, R). A linear map δ ∈ Hom K (R, R)i sa no p e r a t o ro fo r d e rl e s st h a n or equal to if δr -rδ is an operator of order less than or equal to -1. We write D R|K for the collection of differential operators of order at most . We define D R|K = ∈Z ≥0 D R|K , which is a ring with composition as the multiplication.

Example 2.9. Let R be either the polynomial ring K[x 1 ,...,x n ]o rt h ef o r m a l power series ring K[[x 1 ,...,x n ]] with coefficients in a ring K. The ring of K-linear differential operators is:

that is, the free R-module generated by the differential operators

Furthermore, if K is a field of characteristic zero, we have

Example 2.10. Let S = K[x 1 ,...,x d ] be a polynomial ring over a field K of characteristic zero and, given an ideal I ⊆ S,s e tR = S/I. Then, the ring of K-linear differential operators of R is characterized in terms of the differential operators in S [MR01, Theorem 5.13]. Specifically,

Let R be an F -finite ring (not necessarily regular). We denote by D R the ring of F p -linear differential operators on R. In this context, we have that

where D (e) R =E n d R p e (R) and, if K is a perfect field contained in R, then the ring D R|K of K-linear differential operators on R agrees with D R [Yek92,SVdB97]. G i v e na ni n t e g e re ≥ 0, we call D We will also use a result of Smith that states that, whenever R is an F -split domain, R is simple as a D R -module if and only if R is strongly F -regular [Smi95].

Licensed to Mathematical Sciences Research Institute. Prepared on Sat May 11 18:06:43 EDT 2024 for download from IP 173.239.64.5.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2.4. Differential operators and V -filtrations. We introduce a few facts about the relationship between D R and D R[t] ,w h e r eR[t]=R[t 1 ,...,t r ]i sap o l y n o m i a l ring over R. However, these facts are only used in Section 7, where we show that the definition of Bernstein-Sato roots we give in Section 4 agrees with the definition that one arrives to by considering the D-module constructions of Bitoun, Mustat ¸ȃ, and the third author in the regular case [Bit18,Mus09,QG21b]. For this reason, we encourage the reader to skip the remaining of this section until they want to read Section 7.

If ξ ∈ D R is a differential operator on R,t h e nξ acts on R[t] by the formula ξ • (gt k )=( ξ • g)t k for g ∈ R and k ∈ (Z ≥0 ) r , and one checks that this exhibits ξ as a differential operator on R

; we observe that it respects the level filtration and it therefore induces maps

. We want to show that these maps are isomorphisms. Lemma 2.11. Let S be a commutative ring, G be a finite free S-module, and M be an arbitrary S-module. Then the natural map End S (M )

Proof. We have the following natural isomorphisms

where the last isomorphism comes from the tensor-hom adjunction. We then check that the composition of these isomorphisms is the morphism given in the statement.

Lemma 2.12. Let R be an F -finite ring. The morphisms

previously defined are isomorphisms.

Proof. Fix an e ≥ 0. We then have

)o fd e g r e ed. We also denote

) i .I np a r t i c ular, we have I n = R[t] ≥n for all n ∈ Z. Note that the previous isomorphisms respect the gradings, and they therefore induce isomorphisms

Lemma 2.13. Let e and i be integers with e ≥ 0.T h e n :

(i) We have

Proof. It is enough to prove the claims for D ) ≥i follows from the fact that I j = R[t] ≥j for all j ∈ Z. For the other inclusion, suppose that ξ ∈ V i D (e) R[t] , and therefore ξ • R[t] j ⊆ R[t] ≥j+i for all j ∈ Z.L e tξ = k∈Z ξ k where ξ k is the homogeneous component of degree k for ξ, and observe that if g ∈ R[t] is homogeneous of degree d,t h e nξ k • g is the degree k +d homogeneous component of ξ •g ∈ R[t] ≥i+d . We conclude that ξ k •g =0 whenever k<i , which proves the statement.

We now claim that (D

F p [t] ) 0 , we reduce to the case R = F p .L e t σ (e) ∈ D (e) F p [t] denote the unique operator of level e such that σ (e) • t a = 1f o ra =(p e -1,...,p e -1), 0o t h e r w i s e for all a ∈{ 0,...,p e -1} r ;o b s e r v eσ (e) is homogeneous of degree -r(p e -1). Then D (e)

F p [t] is spanned over F p by the operators of the form t b σ (e) t a ,w h e r eb ranges through (Z ≥0 ) r and a ranges through {0,...,p e -1} r , and therefore

is spanned by those for which |b|-r(p e -1) + |a| = i or, equivalently, |a|-i = r(p e -1) -|b|.I ft b σ (e) t a ∈ (D R[t] ) i , then there is a multi-exponent c ∈ N r 0 with |c| = i such that a j ≥ c j for every i, because r(p e -1) -|b|≥0. We can thus write t b σ (e) t a = t b σ (e) t a-c t c , which proves the claim.

Given an integer i ≥ 0, we denote by s p i the unique R-linear operator on R[t] with the property that

for all a ∈ (Z ≥0 ) r ,where(-) i denotes i-th p-adic digit. In Lemma 2.14 we aggregate some properties of these operators.

Lemma 2.14. We have: (i) For all integers i ≥ 0 and e>i,

(ii) For all integers i ≥ 0,

The operators s p i commute with each other.

(iv) For all integers i ≥ 0 we have (s p i ) p = s p i or, equivalently, p-1 j=0 (s p ij)=0. (v) If M is an F p -vector space equipped with an action of the operators s p 0 ,s p 1 ,...,s p e-1 ,t h e nM splits as a sum of multi-eigenspaces for the action of these operators; namely, M = α∈F e p M α where for all α = (α 0 ,...,α e-1 ) ∈ F e p we define M α := {u ∈ M :

Proof. For (i) we simply observe that ( -|(p e -1)1 -a|-r ) i =(-rp e + |a|) i = |a| i (recall that whenever α ≡ β mod p i Z p we have α i = β i ). For part (ii), we note that s p i has degree zero, that it is R-linear and that it commutes with multiplication by t p i+1 j for all j =1,...,r. Parts (iii) and (iv) follow because each s p i is R-linear and acts on monomials t a by an F p -scalar. Part (v) follows from (iii) and (iv).

Remark 2.15. It is possible to give a formula for the operators s p i in terms of partial derivatives:

where the ∂ [a] notation stands for divided power differential operators [QG21b, Proposition 3.3]. We remark that the transpose of these operators already appeared in work of Ma and Zhang [MZ14] as higher-order Euler operators.

2.5. The ring of continuous functions from Z (p) to F p .

Definition 2.16. Given a set X and an integer e ≥ 0 we denote by Cont e ( Z (p) ,X) the collection of all functions φ : Z (p) → X such that φ(α)=φ(β) whenever α ≡ β mod p e . W ed e n o t eC o n t ( Z (p) ,X)= ∞ e=0 Cont e ( Z (p) ,X), and we call Cont( Z (p) ,X) the set of continuous functions from Z (p) to X.

Note that these are indeed the continuous functions when X is endowed with the discrete topology, which is the only case we consider. When A is a ring, the sets Cont e ( Z (p) ,A)a n dC o n t ( Z (p) ,A)a c q u i r eA-algebra structures by pointwise addition and multiplication. A function in Cont e ( Z (p) ,X) is uniquely determined by its values on {0, 1,...,p e -1}. Consequently, a function in Cont( Z (p) ,X) is uniquely determined by its values in Z ≥0 and, given an F p -vector space V , we have canonical isomorphisms

Fix an integer e ≥ 0, and let us consider the algebra Cont e ( Z (p) , F p ).

Remark 2.17. The more geometrically-minded reader might like to think of the results in this section via the following remark, which was pointed out to the third author by Bhatt (see [QG21c, Remark III.2] for a more detailed discussion). There is a homeomorphism Spec(Cont( Z (p) , F p )) ∼ = Z (p) whichbecomesanisomorphismof ringed spaces when we equip Z (p) with the sheaf of rings F p associated to F p .I np a rticular, all local rings of Cont( Z (p) , F p ) are fields, and therefore all Cont( Z (p) , F p )modules are flat.

To every p-adic integer α ∈ Z (p) we associate the maximal ideal

β whenever α ≡ β mod p e . We have algebra isomorphisms

where the e i are orthogonal idempotents. In particular, every Cont e ( Z (p) , F p )module M splits as M = We now give a presentation of the algebra Cont e (Z p , F p ). Recall that, given a p-adic integer α ∈ Z (p) ,w ed e n o t eα i the i-th digit in the p-adic expansion of α (see Subsection 2.1). Given an integer e ≥ 0w ed e n o t eb y σ p e : Z (p) → F p the function σ p e (α)=α e ;n o t et h a tσ p e is in Cont e+1 ( Z (p) , F p ). The function σ p e can be expressed using binomial coefficients: σ p e (α)= α p e .T o see this, use Lucas' theorem to observe that whenever α = n ∈ Z ≥0 ,w eh a v e n p e ≡ α e modulo p, and that therefore σ p e is the unique continuous extension to Z (p) of the map n → n p e (considered as a map from Z to F p ).

Lemma 2.19. The ring Cont e ( Z (p) , F p ) is generated by the operators σ p 0 , σ p 1 ,...,σ p e-1 as an F p -algebra. Moreover, the assignment x i → σ p i induces an F p -algebra isomorphism

Proof. Every function in Cont e ( Z (p) , F p ) is determined uniquely by its values on {0, 1,...,p e -1} and, conversely, any F p -valued function on {0, 1,...,p e -1} extends uniquely to an element of Cont e ( Z (p) , F p ). By identifying every element of {0,...,p e -1} with its base-p expansion we obtain a bijection {0,...,p e -1} ∼ = F e p . We therefore have F p -algebra isomorphisms Cont e ( Z (p) , F p ) ∼ = Fun({0,...,p e -1}, F p ) ∼ = Fun(F e p , F p ), and one checks that, under these identifications, the functions σ p e are sent to the coordinate functions on F e p .S i n c eF e p is a finite set every F p -valued function on it is a polynomial on the coordinate functions. We conclude that Cont e ( Z (p) , F p )i s aq u o t i e n to fF p [x 0 ,...,x e-1 ]/(x p i -x i ) and, since both of these algebras have the same number of elements, the result follows.

We conclude that Cont( Z (p) , F p ) is generated by the operators σ p i (i ∈ Z ≥0 ), and that we have an algebra isomorphism

In particular, given an F -finite ring R a n da ne l e m e n tf ∈ R,w ec a ni d e n t i f y Cont(

Let us now turn our attention to the algebra Cont( Z (p) , F p ) and its modules. Once again we associate to every α ∈ Z (p) a maximal ideal

Every maximal ideal of Cont( Z (p) , F p )i so ft h ef o r mm α for some α ∈ Z (p) [Bit18].

Given a Cont( Z (p) , F p )-module M and a p-adic integer α ∈ Z (p) ,w ed e n o t eb y M α the quotient

If N ⊆ M is a submodule, N α is naturally a submodule of M α by the following result.

Lemma 2.20. The module Cont( α ϕ = 0 for a sufficiently large e; indeed, it suffices to take e large enough so that ϕ ∈ Cont e ( Z (p) , F p ). We conclude that, given an element u ∈ M and a p-adic integer α ∈ Z (p) there exists some large e α such that χ

) forms an open cover of Z (p) which, by the compactness of Z (p) , admits a finite subcover

Proposition 2.22. Let M be a Cont( Z (p) , F p )-module. Suppose that there are only finitely many α ∈ Z (p) such that M α =0,s a yα (1) ,...,α (n) . Then the natural map

We claim that we have

Indeed, if β = α (i) for any i, then applying the functor (-) β to the exact sequence above yields 0

Finally, we illustrate how the algebras Cont e ( Z (p) , F p )a n dC o n t ( Z (p) , F p )a r i s e naturally in the context of differential operators.

We consider the map

) 0 (see Lemma 2.12), and the morphism respects these decompositions. It therefore suffices to prove the claims in the case where R = F p .

In this case, note that an operator δ ∈ Cont(

Now, let r =1 . I fδ is a differential operator of degree zero on F p [t] then there exists a unique function φ δ :

F p [t] ) 0 ,t h e nδ • t a+bp e =(δ • t a )t bp e for all b ∈ Z ≥0 ,s oφ δ (a)=φ δ (a + bp e ); therefore, φ δ extends to an element of Cont e ( Z (p) , F p ). The assignment δ → ξ δ , where ξ δ (a)=φ δ (1 -a) provides a two-sided inverse to the morphism given.

Remark 2.24. It may seem unnatural to take ξ(-r -|a|) in the definition of Δ, as opposed to ξ(|a|). This is a natural consequence of the convention of working with the operator s 1 = -r i=1 ∂ t i t i in characteristic zero, as opposed to the operator r i=1 t i ∂ t i . N o t et h a tw eh a v eΔ ( σ p e )=s p e for every e ≥ 0.

Definition 3.1. Let R be an F -finite ring and a ⊆ R be an ideal. We say that an integer n ≥ 0isdifferential jump of level e of a if the inclusion

) for the collection of all differential jumps of level e.

We note that n ∈B • a (p e ) if and only if

(i) If W ⊆ R is a multiplicative subset and a ⊆ R is an ideal, then we have

..,g k ∈ R are such that (g 1 ,...,g k ) = R and Max(R) denotes the collection of all maximal ideals of R,th enw e have

To compute differential jumps, we can also reduce to the case of an infinite residue field by Lemma 3.3. Lemma 3.3. Let (R, m,K) be an F -finite local ring and consider the extension (S, n,L) given by

Then:

(i) The extension (S, n,L) is faithfully flat, F -finite, and local.

(ii) For every ideal a ⊆ R we have (a)= (aS),w h e r e denotes analytic spread.

(iv) For every integer e ≥ 0 and every ideal a ⊆ R we have B • a (p e )=B • aS (p e ). Proof. The statements in (i), (ii), (iii) are standard [HS06, §8.5]. For (iv) observe that, by Lemma 2.12, we have

R -submodule of R,e q u i v a l e n t l y ,D

We record two natural families of D

For (ii), given f ∈ I e (a), and ψ ∈C e R ,w eh a v eψ

• a 0 for every e,s oB • a (p e ) = ∅ for every e.

Lemma 3.7. Let a, b ⊆ R be two ideals and e ≥ 0 be an integer. If

R -ideal, and n = B b a (p e ).

Lemma 3.11. Let a ⊆ R be an ideal and n, e, a ≥ 0 be integers. If

Lemma 3.12. Let a ⊆ R be an ideal generated by r elements. If n ≥ mp e +(r -1)(p e -1),t h e na n = a n-mp e (a [p e ] ) m .

Proof. The statement reduces to the case m = 1. The containment of the right-hand side in the left is clear. For the other, by the pigeonhole principle, any monomial of degree greater than r(p e -1) in the generators of a must be a multiple of a p e th power of a generator, from which the claim is clear.

Proposition 3.13. Let R be an F -finite ring and a ⊆ R be an ideal. If a is generated by r elements, n ≥ r(p e -1) + 1,a n dD

Proof. By Lemma 3.12, we have that

If f is a nonzerodivisor, a =(f ), and n ≥ p e ,w eh a v ef

Lemma 3.14. Let a ⊆ R be an ideal and fix integers n<m .T h e nD

Proof. The statement follows from the chain of ideals

Proof. It suffices to prove the lemma for a = 1. Fix a splitting σ of the Frobenius. Let a =(f 1 ,...,f t )a n db =(g 1 ,...,g s ).

For the forward implication, if

and so

and so

Proposition 3.16. Let R be an F -finite and F -split ring, a be an ideal with r generators, and m ≥ n ≥ 0 be integers. Then for all integers e, a ≥ 0 we have:

For part (i), we consider the chain of ideals

(see Lemma 3.12). Acting with D (e+a) R

, we obtain the chain

By Lemma 3.15, the two ideals in the middle differ, so the two outer ones must also differ. The statement then follows from Lemma 3.14.

Part (ii) follows similarly: we consider the chain

which gives

Lemma 3.14 and Lemma 3.15 give that the two outer ideals are equal, and hence the two in the middle must also be equal. Another application of Lemma 3.14 gives the statement.

We now begin the study of the first invariant of real interest: the Bernstein-Sato roots of an ideal. These provide a characteristic-p analogue of the roots of the Bernstein-Sato polynomial, although Definition 4.1 provides no indication of why that would be the case; such an explanation is given later in Section 7. However, using Definition 4.1 has many advantages: it does not involve any technicalities beyond those of differential jumps, it provides an easier way of computing Bernstein-Sato roots and it also is more useful for proving statements.

Definition 4.1. Let R be F -finite, and a be an ideal. We say that α ∈ Z (p) is a Bernstein-Sato root of a if there is a sequence (ν e ) ∞ e=0 with ν e ∈B • a (p e ) such that α is the p-adic limit of ν e .W ed e n o t eb yB S R ( a)t h es e to fB e r n s t e i n -S a t or oo t so fa.

We recall that a sequence of p-adic numbers (ν e )c o n v e r g e st oap-adic number α if and only if for every m ∈ Z ≥0 there is some N ∈ Z ≥0 such that p m | (α -ν e ) for all e ≥ N .

Remark 4.2. Note that, given a sequence (ν e ) ∞ e=0 ⊆ Z ≥0 , the condition that ν e ∈ B • a (p e ) for every e ≥ 0 passes to subsequences. Indeed, if (ν e i ) is a subsequence, then ν e i ∈B • a (p e i ) ⊆B • (p i ) for every i ≥ 0.

Proposition 4.3. Let R be F -finite, a be an ideal with r generators and α ∈ Z (p) be a p-adic integer. The following are equivalent: (a) α is a Bernstein-Sato root of a.

(b) For al l e ≥ 0 there is some s e ∈{ 0,...,r -1} such that α <e + s e p e ∈ B • a (p e ). (c) There is an infinite subset {e j }⊆Z ≥0 and differential jumps ν j ∈B • a (p e j ) such that (ν j ) converges to α.

Proof. For (a) implies (b), let α b eaB e r n s t e i n -S a t or o o to fa,a n dν e ∈B • a (p e ) such that α = lim ν e . For every a there is some e a such that p a |(α -ν j ) for all j ≥ e a ; without loss of generality, we can take e a ≥ a. Consider the sequence η a = ν e a . By Lemma 3.11, η a ∈B • a (p a ), and by construction, p a |(α -ν a ) for all a. Then, by Proposition 3.13, we may subtract a multiple of p a from η a to obtain another sequence μ a in which 0 ≤ μ a <r p a ,a n dp a |(α -μ a ) for all a. It follows that μ a = α <a + s a p a ∈B • a (p a ) with s a ∈{0,...,r-1} as required. The implication (b) implies (c) is clear. For (c) implies (a), it suffices to see that given a p-adically convergent sequence of the form ν e a ∈B • a (p e a )f o re a an infinite increasing sequence of integers that we can extend this to a sequence ν e ∈B • a (p e ) for all e ∈ Z ≥0 . This follows from Lemma 3.11.

Remark 4.4. It follows from the definition and from Remark 3.6 that if a = R,then BSR(a)=∅. On the other hand, if a R is a proper ideal, then by Remark 3.6 there is a differential jump of level e for every e, and by compactness of Z (p) condition (c) of Proposition 4.3 holds for some α,s oB S R ( a) = ∅.

In Section 7 we show that whether a p-adic integer is a Bernstein-Sato root or not is given in terms of the nonvanishing of a certain module, whose construction is compatible with localization. From this it follows that Bernstein-Sato roots are local invariants; however, we give a proof of this fact here that does not require the material of Section 7.

To begin, let a ⊆ R be an ideal and fix an integer r ≥ 0 such that a can be generated by r elements. Given a positive integer e ≥ 0a n dap-adic integer α we denote by J a (p e ,α)t h es e t J a (p e ,α):={α <e + sp e | s =0, 1,...,r-1}∩B • a (p e ). Lemma 4.5. The p-adic integer α is not a Bernstein-Sato root of a if and only if there is some e large enough so that J a (p e ,α)=∅,i nw h i c hc a s eJ a (p a ,α)=∅ for all a ≥ e.

Proof. The first statement follows from the equivalence of (a) and (b) in Proposition 4.3. For the second statement, it is enough to show that whenever J a (p e ,α)=∅ then J a (p e+1 ,α)=∅, which we prove by contradiction. Suppose that a is generated by r elements. If we had some n ∈J a (p e+1 ,α), then we have n ∈B • a (p e+1 ), and thus n ∈B • a (p e ) (see Lemma 3.11). From Proposition 3.13 we conclude that there is some integer k ≥ 0 such that n -kp e ∈B • a (p e )a n d0≤ n -kp e <r p e .S i n c e n ≡ α mod p e+1 ,w eha v en -kp e ≡ α mod p e , and therefore n -kp e = α <e + sp e for some s ∈{ 0, 1,...,r -1}. We conclude that n -kp e ∈J a (p e ,α), giving the desired contradiction. Proof. This follows from Remark 3.2.

Lemma 4.7. Let m ⊆ R be a maximal ideal, α ∈ Z (p) be a p-adic integer and e ≥ 0 be an integer. If J aR m (p e )=∅, then there exists some g ∈ R \ m such that J aR g (p e )=∅.

Proof. Let S denote the set S := {α <e + sp e : s =0, 1,...,r-1}. By Remark 3.2, J aR m (p e )=∅ precisely when (D

R • a n+1 .T h e result then follows from the fact that the support of a finitely-generated module is closed.

Proposition 4.8. Let R be a noetherian F -finite ring and a ⊆ R be an ideal. Let g 1 ,...,g k ∈ R be such that (g 1 ,...,g k )=( 1 )and let Max(R) denote the set of all maximal ideals of R. We then have:

Proof. Lemmas 4.5 and 4.6 give the first equality, and show that BSR(a) ⊇ m BSR(aR m ). To prove that BSR(a) ⊆ BSR(aR m ), suppose that α ∈ Z (p) is such that α/ ∈ BSR(aR m ) for all maximal ideals m. By Lemma 4.5 and Lemma 4.7 we conclude that for every m ∈ Max(R) there is some integer e m and some License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use element g m ∈ R \ m such that J R g m (p e m ,α)=∅. The elements (g m |m ∈ Max(R)) generate the unit ideal and therefore there is a finite subcollection of them, say g 1 = g m 1 ,...,g k = g m k , that still generates the unit ideal. If e =m a x {e m 1 ,...,e m k }, then, J aR g i (p e ,α)=∅ for all i =1,...,k by Lemma 4.5. Therefore J a (p e ,α)=∅ by Lemma 4.6. We conclude that α/ ∈ BSR(a) by Lemma 4.5. Proposition 4.9. Let R be a noetherian F -finite ring and a ⊆ R be an ideal.

(i) If R is positively graded with homogeneous maximal ideal m and a is homogeneous, then BSR(a)=BSR(aR m ).

(ii

Proof. Both facts follow from Remark 3.2.

4.2. Finiteness and rationality results. We now introduce a finiteness condition that has important consequences for Bernstein-Sato roots.

Definition 4.10. Let R be an F -finite ring and a ⊆ R be an ideal generated by r elements. We say that a is Bernstein-Sato admissible if there is a constant C>0 such that # B • a (p e ) ∩ [0,rp e ) ≤ C for every e ∈ Z ≥0 .W es a yt h a tR is a Bernstein-Sato admissible ring if all of its ideals are Bernstein-Sato admissible.

In our definition of Bernstein-Sato root we only require that the number of differential jumps of level e in the interval [0,rp e ) is bounded, but the subtraction property of differential jumps (Proposition 3.13) gives a stronger statement.

Proposition 4.11. Let R be an F -finite ring and a ⊆ R be an ideal. The ideal a is Bernstein-Sato admissible if and only if there are constants A, B > 0 such that for all integers e, s ≥ 0 we have

Proof. Suppose that a is generated by r elements. We note that if a is Bernstein-Sato admissible, then there exists A, B > 0 such that for all e, s ≥ 0w eh a v e Corollary 4.12. Whether a is Bernstein-Sato admissible or not does not depend on the choice of r.

Theorem 4.13. Let R be an F -finite ring and a ⊆ R a Bernstein-Sato admissible ideal. Then a has finitely many Bernstein-Sato roots.

Proof. Pick an integer C>0 such that # B • a (p e ) ∩ [0,rp e ) ≤ C for all e ≥ 0. We claim that there are at most C Bernstein-Sato roots, and we prove it by contradiction. Suppose that {α 1 ,...,α C+1 } are distinct Bernstein-Sato roots of a, and choose N large enough so that α i ≡ α j mod p N for all i = j. By Proposition 4.3 we know there is some e large enough and ν 1 ,...ν C+1 ∈B • a (p e ), with 0 ≤ ν i < rp e , such that ν i ≡ α i mod p N , and therefore ν 1 ,...,ν C+1 are distinct differential jumps. This gives the desired contradiction.

Lemma 4.14. Let R be F -split, a ⊆ R be an ideal generated by r elements, α ∈ Z (p) be a Bernstein-Sato root of a and a ≥ 0 be an integer. There exists some i ∈{0, 1,...,r(p a -1)} such that p a α + i is a Bernstein-Sato root of a.

Proof. Pick a sequence (ν e ) such that ν e ∈B • a (p e )w h o s ep-adic limit is α.B y Proposition 3.16, for every e there is some i e ∈{ 0, 1,...,r(p a -1)} such that p a ν e + i e ∈B • a (p e+a ). Since {0, 1,...,r(p a -1)} is a finite set, there is some i ∈ {0, 1,...,r(p a -1)} and an increasing sequence (e j ) such that p a ν e j +i ∈B • a (p e j +a ). The p-adic limit of p a ν e j + i is p a α + i, and the result follows from Proposition 4.3.

We recall that an ideal J is a reduction of an ideal I with reduction number n if JI n = I n+1 and JI n-1 = JI n .

R • a n+k+1 , and applying Lemma 3.14.

Let us now prove part (ii); we use the alternative characterization of Bernstein-Sato admissibility given in Proposition 4.11. Suppose that a is Bernstein-Sato admissible and pick constants A a , B a such that # B • a (p e ) ∩ [0,s) ≤ A a (s/p e )+B a for all e, s ≥ 0. By applying part (i) we conclude that

For the other direction, suppose b is Bernstein-Sato admissible and choose constants

= kA b s p e + kB b . We now tackle part (iii). Suppose that α ∈ BSR(a), and choose a sequence (ν e ) with ν e ∈B • a (p e ) such that α is the p-adic limit of ν e . By part (i), for every e ≥ 0 there is some i e ∈{ 0, 1,...,k} such that ν e -i e ∈B • b (p e ). We conclude there is some i ∈{ 0, 1,...,k} and a subsequence (ν e j ) such that ν e j -i ∈B • b (p e j ). The p-adic limit of this subsequence is α -i which, by Remark 4.2, is a Bernstein-Sato root of b. Therefore, α ∈ BSR(b)+i. The other statement follows similarly.

Theorem 4.16. Let R be an F -finite F -split ring. Let a be a Bernstein-Sato admissible ideal. Then every Bernstein-Sato root of a is rational.

Proof. Recall that BSR(a) denotes the set of Bernstein-Sato roots of a, and let BSR(a) ⊆ Z (p) /Z be its image under the quotient map Z (p) → Z (p) /Z;i no t h e r words, BSR(a)={α + Z | α ∈ BSR(a)}. Note that, by Lemma 4.14, BSR(a)i s closed under multiplication by p.

Let α ∈ Z (p) be a Bernstein-Sato root. Since BSR(a) is a finite set, there exist some n<msuch that p n α ≡ p m α mod Z; that is, there exists some c ∈ Z such that p n α = p m α + c. It follows that α = c/(p n (p m-n -1)) and thus α is rational (note that, a posteriori, we know that p n must divide c).

Lemma 4.17. Let R be an F -finite F -split ring. Let a be an r-generated ideal of R.L e tα ∈ Z (p) be a Bernstein-Sato root of a.

(i) If α>0, then there exists an increasing sequence {n j }⊂Z ≥0 such that α + n j is a Bernstein-Sato root of a for each j. (ii) If α<-r, then there exists an increasing sequence {n j }⊂Z ≥0 such that α -n j is a Bernstein-Sato root of a for each j.

Proof. For the first part, it suffices to show that there is some positive integer n such that α + n is a Bernstein-Sato root. We can write α = a 1 -p e + b with a, b ∈ Z ≥0 such that 0 ≤ a<p e -1a n db>0. By Lemma 4.14, there is some i ∈{0,...,r(p e -1)} such that p e α+i is a root. We have p e α+i = α+(p e -1)b-a+i.

Since a<p e -1 < (p e -1)b, the claim follows.

Likewise, for the second part, it suffices to show that there is some negative integer n such that α + n is a Bernstein-Sato root. We can write α = Proof. Pick a large prime p and let R = F p [x 1 ,...,x n ] be a polynomial ring over F p , m ⊆ R denote the maximal ideal m =(x 1 ,...,x n )andā ⊆ R denote the mod-p reduction of a. We may pick p large enough so that λ is a Bernstein-Sato root of ā [QG21a, Theorem 3.1]. Moreover, since the construction of the fiber cone of a with respect to m is compatible with mod-p reduction, we can further enlarge p to assumefoot_4 that (aR m )= (ā R m). Since a is homogeneous, we conclude that λ is a Bernstein-Sato root of ā R m (Proposition 4.9), and we conclude that -(ā R m) ≤ λ from Corollary 4.19.

In characteristic zero, whenever a ⊆ C[x 1 ,...,x n ] is a nonzero ideal, all the roots of the Bernstein-Sato polynomial of a are strictly negative. Since we have only shown that the Bernstein-Sato roots are nonpositive, the question of whether zero can be a Bernstein-Sato root arises. We can answer it for principal ideals as follows.

Proposition 4.21. Let R be an F -finite ring. The following are equivalent:

(a) The ring R is simple as a D R -module.

(b) For all nonzero f ∈ R we have 0 / ∈ BSR(f ). Moreover, if these hold then the nilradical √ 0 of R is a prime ideal. In particular, if R is reduced then it must be a domain.

Proof. Suppose that R is simple as a D R -module. Given a nonzero f ∈ R,w eh a v e D R •f = R and therefore there is some e large enough so that

) and, by Proposition 4.3, we conclude that 0 / ∈ BSR(f ). Conversely, suppose that 0 ∈ BSR(f ) for some nonzero f ∈ R.B yP r o p o s i t i o n 4.3, for all e ≥ 0w eha v e0∈B • f (p e ) and thus

and hence R is not simple as a D R -module.

For the last statement, suppose that f, g ∈ R are such that fg ∈ √ 0andf/ ∈ √ 0. Then the collection H 0 g (R)ofg-torsion elements of R is a D R -submodule of R which contains a power of f , and is therefore nonzero. If R is D R -simple, we conclude that 1 ∈ H 0 g (R) and thus some power of g is zero, i.e. g ∈ √ 0.

Question 4.22. Let R be an F -finite ring that is simple as a D R -module. Do we have 0 / ∈ BSR(a) for all nonzero ideals a ⊆ R?

In this section we introduce the other key numerical invariant of this paper: differential thresholds. These are related to F -jumping numbers, F -thresholds, Cartier thresholds, and Bernstein-Sato roots. 5.1. Definition and basic properties. Definition 5.1. Let R be an F -finite ring, and a ⊆ R an ideal. We say that λ ∈ R ≥0 is a differential threshold if there exists a sequence of elements ν e ∈B • a (p e ) such that λ = lim e→∞ ν e p e , where the limit is taken in the usual Euclidean topology. When R is F -split, it turns out that every differential threshold can be realized as a limit in a nice way.

Proposition 5.2. Let R be an F -finite F -split ring and λ ∈ R >0 .L e ta be an r generated ideal. The following are equivalent:

(a) λ is a differential threshold of a.

(b) For al l e ≥ 0, there is a differential jump of level e for a in the interval [p e λ -r, p e λ]. (c) There is an infinite set {e j }⊆Z >0 and differential jumps ν j ∈B • a (p e j ) such that (ν j /p e j ) converges to λ.

Proof. We start by showing that (a) implies (b) by contraposition. Suppose that [p e λ -r, p e λ] ∩B • a (p e )=∅ for some e ≥ 0. Since every differential jump is an integer, we get that [ p e λ -r, p e λ ] ∩B • a (p e )=∅. By Proposition 3.16(ii) we conclude that, for all integers a ≥ 0, p a p e λ -p a -r +1,p a p e λ + p a ∩B The implication (b) implies (c) is clear.

To show that (c) implies (a), let (e j ) be an infinite increasing sequence of integers; we need to show that we can extend (ν e j /p e j ) to a convergent sequence (ν i /p i ), i ∈ Z ≥0 . By Proposition 3.16, for ν j ∈B • a (p e j )a n df o ra>e j there is some ν a,j ∈ B • a (p a ) ∩ [ν j p a-e j , (ν j + r)p a-e j ]. For i = e j ,t a k eν i = ν j ,a n df o re j <i<e j+1 , take ν i = ν i,j .I fa, b ≥ e j ,a n du, v are such that e u ≤ a<e u+1 and e v ≤ b<e v+1 , then ν a p a -

Since (ν j /p e j ) is a Cauchy sequence, the right-hand side in the previous equation tends to zero as j →∞, and hence (ν i /p i ) is Cauchy, as required.

Remark 5.3. It follows from definition and from Remark 3.6 that if a = R,t h e na has no differential jumps. Conversely, if a R is a proper ideal with r generators, then by Remark 3.6 and Proposition 3.13 there is a differential jump of level e in the interval [0,rp e ] for every e, so by compactness of [0,r] there is a differential jump for a.

Proposition 5.4. Let R be a noetherian F -split ring of characteristic p, a ⊆ R be an ideal generated by r elements and k, l ≥ 0 be integers with l

for some e ≥ 0, then there are no differential thresholds of a in the interval (k + r -1)/p e ,l / p e .

Proof. If [k, l) ∩B • a (p e )=∅ then, by Proposition 3.16, we have kp a +(r -1)(p a -1),lp a ∩B • a (p e+a )=∅ for all a ≥ e, and thus

The result follows.

Remark 5.5. If R is F -split, and a is an r-generated ideal, we have that 0 is a differential threshold if and only if 0 is differential jump of level e for every e.I f 0 / ∈B • a (p e ), then

• a p a = R for all a, so there are no thresholds in the interval [0, 1/p e ). The other implication follows from the definition of differential threshold.

If R is F -split, we can also find differential thresholds that are close to differential jumps.

Lemma 5.6. Let R be an F -finite F -split ring, and a be an ideal with at most r generators. .

We now provide several properties of differential thresholds. We first show that the set formed by them remains the same after taking integral closure. Then, we show a version of Skoda's Theorem and a p-fractal property.

Proposition 5.8. Let R be an F -finite ring, and a, b ⊆ R be ideals with the same integral closure. Then, a and b have the same differential thresholds.

Proof. Since a and b have the same integral closure, there exists an integer a such that a n ⊆ b n+a ⊆ a n+2a for every n ∈ Z ≥0 . Then, = λ, we conclude that λ is a differential threshold.

Corollary 5.10. Let (R, m, K) be a local F -finite ring, a ⊆ m an ideal, and be its analytic spread. If λ> is a differential threshold of a,t h e nλ -1 is a differential threshold of a.

Proof. By Lemma 3.3, we may assume that K is infinite. Then, there exists an ideal b generated by elements with the same integral closure of a. Then, the result follows from Propositions 5.8 and 5.9.

5.2. Differential thresholds and numerical F -invariants. We now start comparing differential thresholds with other numerical invariants in prime characteristic. Proposition 5.12. Let R be an F -finite ring, and a ⊆ R be a proper ideal. Then: ) ⊇C e R • a ν e C e R • a ν e +1 ⊇ τ (a λ ). We conclude that λ is an F -jumping number.

Proposition 5.13. Let (R, m,K) be an F -finite F -split ring, and a ⊆ R an ideal. Then, fpt(a) is the smallest differential threshold of a.

Proof. We note that the first jump of level e is p e is the smallest differential threshold. 5.3. Discreteness and rationality results. We now show that the set of differential thresholds is closed under multiplication by p. This is known for F -thresholds, but not for F -jumping numbers outside Gorenstein rings. Then, this result shows one of the advantages of the unified approach provided by differential thresholds.

Lemma 5.14. Let R be an F -finite F -split ring, and a ⊆ R an ideal. If λ is a differential threshold, then pλ is also a differential threshold. In the following results we focus on Bernstein-Sato admissible ideals. In this case, we show discreteness and rationality. In Subsection 6.1 we use these results to provide new cases where the F -thresholds are rational numbers.

Theorem 5.15. Let R be an F -finite ring and a an ideal. If a ⊆ R is a Bernstein-Sato admissible ideal, then the set of differential thresholds for a is discrete. If R is F -split, then the converse holds.

Proof. Let r be the number of generators of a. We first show that if a is Bernstein-Sato admissible, then the set of differential thresholds is discrete. By Proposition 5.9, it suffices to show that the set of differential thresholds in (0,r)i sfi n i t e . S i n c e a is a Bernstein-Sato admissible ideal, there exists b ∈ Z ≥0 such that # B • a (p e ) ∩ [0,rp e ) ≤ b for every e, and we claim that there are at most b differential thresholds in (0,r).

Suppose, for a contradiction, that λ 1 ,...,λ b+1 ∈ (0,r) are distinct differential thresholds of a. Pick disjoint open intervals U 1 ,...,U b+1 ⊆ (0,r) with λ i ∈ U i . Then there is some e large enough and ν 1 ,...,ν b+1 ∈B • a (p e ) with ν i /p e ∈ U i for every i. It follows that ν 1 ,...,ν b+1 are distinct differential jumps of level e in the interval (0,rp e ), which gives a contradiction. Now suppose that R is F -split, and assume that the set of differential thresholds is discrete. In particular, there are finitely many differential thresholds in the interval [0,r]; let 0 ≤ λ 1 < ••• <λ c ≤ r be these differential thresholds. To obtain a contradiction, suppose that a is not Bernstein-Sato admissible. Then we can choose some e ∈ N such that the number of differential jumps of level e is greater than (r +1 ) c. By Lemma 5.6, every differential jump of level e lies in c i=1 [p e λ i -r, p e λ i ]. Since there are at most (r +1)c integers in this set, we obtain the desired contradiction.

Theorem 5.16. Let R be an F -finite F -split ring, and a ⊆ R be a Bernstein-Sato admissible ideal. Then, every differential threshold of a is a rational number.

Proof. Let λ be a differential threshold for a.W efi xe 0 such that p e 0 λ>r .F o r e ≥ e 0 ,w et a k eλ e = p e λ -p e λ + r -1. By construction λ e is a differential threshold for every e by Proposition 5.9 and Lemma 5.14. By Theorem 5.15, there exists e 1 <e 2 such that

Since e 2 >e 1 , we conclude that λ = p e 2 λ -p e 1 λ p e 2 -p e 1 ∈ Q.

Comparison between Bernstein-Sato roots and differential thresholds. We end this subsection with a comparison between differential thresholds and Bernstein-Sato roots. We note that we do not assume Bernstein-Sato admissibility in this result.

Theorem 5.17. Let R be F -split. Let a be an ideal with r generators.

(ii) Conversely, if λ ∈ (Z (p) ) ≥0 is a differential threshold for a, then there is some Bernstein-Sato root α for a such that

Thus, there is an equality of cosets in Z (p) /Z:

Proof. We start with (i). By Proposition 4.3, for every a,t h e r ei ss o m es ∈ {0,...,r -1} such that α <a + sp a ∈B • a (p a ). Thus, there is an s ∈{ 0,...,r -1} such that α <ea + sp ae ∈B • a (p ae ) for infinitely many a. If α ∈ Z <0 , then by Lemma 2.1, we have α <ea + sp ae = p ae + α + sp ae ∈B • a (p ae ). It then follows from Proposition 5.2 that lim a→∞ (s +1)p ae + α p ae = s+1 is a differential threshold.

If α/ ∈ Z <0 , then by Lemma 2.1, we have α <ea =(1-p ae )(α-α )+ α for a 0.

It then follows from Proposition 5.2 that lim

For (ii), let λ ∈ Z (p) be a differential threshold. For λ = 0, it follows from Remark 5.5 that 0 is a Bernstein-Sato root.

N e x t ,w ed e a lw i t ht h ec a s eλ ∈ Z >0 . By Proposition 5.2, for every a,t h e r ei s some ν a ∈B • a (p a ) such that p a λ -r ≤ ν a ≤ p a λ.W r i t i n gν a = p a λ -s a ,w eh a v e that s a ∈{ 0,...,r} for all a and p a λ -s a ∈B • a (p a ). There is some s ∈{ 0,...,r} such that s a = s and p a λ-s ∈B • a (p a ) for infinitely many values of a. It follows from Proposition 4.3 that -s, which is the p-adic limit of p a λ -s, is a Bernstein-Sato root of a.

Finally, suppose that λ/ ∈ Z ≥0 , and let e ∈ Z ≥0 be such that (p e -1)λ ∈ Z ≥0 . Write λ = λ -1+μ,s oμ ∈ (0, 1). We then have p ae λ -1=p ae λ ae = (p ae -1)μ + p ae ( λ -1) for all a ∈ Z ≥0 . By Proposition 5.2, for every a,t h e r e is some ν ae ∈B • a (p ae ) such that p ae λ -r ≤ ν ae ≤ p ae λ = p ae λ -1. Writing s ae = p ae λ -1 -ν ae , one has s ae ∈{0, 1,...,r-1} and (p ae -1)μ + p ae ( λ -1)s ae ∈B • a (p ae ). Therefore, there is some s ∈{ 0,...,r -1} such that (p ae -1)μ + p ae ( λ -1) -s ∈B • a (p ae ) for infinitely many a. It follows from Proposition 4.3 that lim a (p ae -1)μ + p ae λ -s = -μ -s is a Bernstein-Sato root of a.

Corollary 5.18. Let R be F -finite and F -split. If a =( f ) is principal and Bernstein-Sato admissible, then the set of Bernstein-Sato roots of (f ) is exactly the set of negatives of differential thresholds of (f ) in the interval [0, 1] ∩ Z (p) .

Corollary 5.19. If R is strongly F -regular and a is an ideal, then there is a containment in Z (p) /Z:

6. Classes of Bernstein-Sato admissible rings 6.1. Rings with finite F -representation type. In this section we prove that every ideal in a graded K-algebra with finite F -representation type is Bernstein-Sato admissible; therefore all ideals have a finite number of Bernstein-Sato roots. We closely mimic the strategy employed by Takagi and Takahashi [TT08] in their proof of discreteness of F -jumping numbers.

). Let R = n∈Z ≥0 R n be finitely generated Z ≥0 -graded K-algebra over a field R 0 = K. We say that an R has finite F -representation type if there exist a finite set of finitely generated graded R-modules, M 1 ,...,M such that for every e ∈ Z ≥0 there exist α e,i ∈ Z ≥0 and θ

where the grading on F e * R is as in Definition 2.2(ii). We say that M 1 ,...,M are the finite F -representation type factors of R.

n∈Z ≥0 R n be finitely generated graded Kalgebra with R 0 = K,a n da ⊆ R.L e tM be a finitely generated R-module. For e ∈ Z ≥0 ,w es e t I e (a,M)=Hom R (F e * R, M ) • F e * a. Lemma 6.3. Let R = n∈Z ≥0 R n be finitely generated graded K-algebra with R 0 = K, a ⊆ R be an ideal and M be a graded R-module. Suppose that R has finite Frepresentation type, and that a is generated in degree less or equal to N .T h e n , Licensed to Mathematical Sciences Research Institute. Prepared on Sat May 11 18:06:43 EDT 2024 for download from IP 173.239.64.5.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use there exists an integer C ∈ Z ≥0 such that I e (a,M) is generated in degree less or equal to C + N p e for every integer e ≥ 0.

Proof. Pick C>0 large enough so that, for all i =1,..., , the module Hom R (M i ,M) is generated in degrees ≤ C. This implies that for all i =1 ,..., and

and, since all θ (e) i,j are negative, we conclude that the module

Hom R (M i (θ

Hom F e * R, M

and so

As a consequence of Theorem 6.4, we obtain a new case where the differential thresholds satisfy rationality and discreteness. In particular, these recover and extend previous results known for Stanley such that δ = δ| R .

We note that large classes of invariant rings can be realized as level-differentially extensible direct summands of polynomial rings (see [BJNB19, Section 6] for more details). Theorem 6.7. Let R ⊆ S be a split extension of F -finite rings and a ⊆ R be an ideal. Then:

(i) For all integers e ≥ 0, B • a (p e ) ⊆B • aS (p e ). (ii) If aS ⊆ S is a Bernstein-Sato admissible ideal, then so is a ⊆ R. (iii) Every Bernstein-Sato root of a is a Bernstein-Sato root of aS. (iv) Every differential threshold of a is a differential threshold of aS. Assume furthermore that the extension R ⊆ S is level-differentially extensible. Then:

(v) For all integers e ≥ 0, B • a (p e )=B • aS (p e ). (vi) The Bernstein-Sato roots of a and the Bernstein-Sato roots of aS coincide. (vii) The differential thresholds of a and the differential thresholds of aS coincide.

Proof. Let us start with (i). Suppose that n/ ∈B • aS (p e ); that is, D

S • a n+1 , and we claim that

R • a n+1 . We observe it suffices to prove that a n ⊆ D (e) R • a n+1 . To prove this, suppose that f ∈ a n . We know that there exist differential operators ξ i ∈ D (e) S and elements g i ∈ a n+1 such that f = i ξ i • g i . Applying a splitting β : S → R to this equation, we obtain

R • a n+1 . This proves the claim, and thus (i) is proven.

Statement (ii) follows from (i): every bound for # B • aS (p e ) ∩ [0,rp e ) is a bound for # B • a (p e ) ∩ [0,rp e ) . Statement (iii) follows from (i) together with (ii). Statement (iv) follows from (i).

We now assume that the extension R ⊆ S is level-differentially extensible, and prove (v). Suppose that n/ ∈B • a (p e ); that is, D

S • a n+1 and, once again, we observe that it suffices to show that a n ⊆ D (e) S • a n+1 .W et h u sl e tf ∈ a n . We know that there exist ξ i ∈ D (e) R and g i ∈ a n+1 such that f = i ξ i • g i . For every i,l e t ξ i ∈ D (e) S be a lift of ξ i .W e conclude that f = i ξ i • g i and therefore f ∈ D (e) S • a n+1 . This proves the claim, and thus (v) is proven.

Statements (vi) and (vii) follow from (v).

where δ p e is just a formal symbol. In particular, H e a is the quotient of R[t]b ya D We let H a be the direct limit

-→ H 2 a → •••), which acquires the structure of a D R[t] -module. We note that the maps φ e are injective, and therefore each H e a is isomorphic to its image in H a .F r o mt h i sp o i n t onwards, we identify each module H e a with its image in H a ,a n dw et h i n ko fe v e r y element of H e a as an element of H a . For example, for all e ≥ 0w eh a v e δ 1 =(f -t) (p e -1)1 δ p e .

Lemma 7.1. There is an isomorphism

Proof. We let L denote the local cohomology module on the right hand side, which we construct via the Čech complex on the given generators. We have an R-module

where δ a denotes the class of (f -t) -a . We let ψ e : H e a → L be the unique R-linear map with ψ e ((f -t) a δ p e )=δ p e 1-a , which gives an isomorphism of H e onto the submodule a∈{1,...,p e } r Rδ a of L.O n e

In characteristic p>0, we view the algebra Cont( Z (p) , F p ) as a subalgebra of (D R[t] ) 0 by using the map Δ from Subsection 2.5, and this subalgebra plays the role of C[s]. Given a p-adic integer α ∈ Z (p) , we define (N a ) α := N a /m α N a , and our result is as follows.

Theorem 7.3. Let R be an F -finite ring, a ⊆ R be an ideal and α ∈ Z (p) be a p-adic integer. Let N a be the module defined above by using a choice of generators for a.T h e nα is a Bernstein-Sato root of a if and only if (N a ) α is nonzero. We begin working towards the proof of Theorem 7.3. The idea is to write (N a ) α as a direct limit (N a ) α = lim →e (N e a ) α , to understand the nonvanishing of the (N e a ) α for a fixed e, and to then analyze the effect of taking the direct limit.

We note that N a = lim →e N e a ,w h e r e N e a := (D α N e a . In particular, we have (N a ) α = lim →e (N e a ) α . Lemma 7.5. Let x =(x 1 ,...,x r ) and y =(y 1 ,...,y r ) be two sets of variables and e ≥ 0 be an integer. In the ring F p [x,y] we have

where the sum takes place over all multi-exponents b ∈ Z r ≥0 with 0 ≤ b i <p e . Proof. In the case where r = 1, this follows by observing that (x -y)(y p e -1 + y p e -2 x + •••+ yx p e -2 + x p e -1 )=x p e -y p e =(x -y) p e , together with the fact that F p [x, y] is a domain.

For the general case, first note that the multi-index binomial theorem states that the claim in the lemma is equivalent to the statement that (p e -1)1 b (-1) r(p e -1)-|b| ≡ 1m o d p, which in turn is equivalent to the claim that (p e -1)1 b ≡ (-1) |b| mod p. This latter statement follows directly from the r =1case.

Given an integer e ≥ 0a n dam u l t i -e x p o n e n ta ∈{ 0,...,p e -1} r we define Q e a to be the following element of H e a : Q e a := t (p e -1)1-a δ p e .

H e a = a∈{0,...,p e -1} r RQ e a .

Recall that we identified D F p [t] with subrings of D F p [t] ) 0 (see Lemma 2.12).

Lemma 7.6. For every integer e ≥ 0 we have the following equality of submodules of H e a :

(D (e)

Proof. We begin by noting that, by Lemma 7.5, we have

Given multi-exponents b,c ∈ Z r with 0 ≤ b i <p e and c i <p e we denote by σ

F p [t] such that for all k ∈{0, 1,...,p e -1} r we have

These operators form an F p -basis for D (e)

F p [t] , and since σ

F p [t] ) 0 , which proves the claim. Proposition 7.7. Let R be an F -finite ring, a ⊆ R be an ideal and e ≥ 0 be an integer.

(i) We have a direct sum decomposition

where Q e a denotes the image of Q e a in the quotient. F p [t] ) 0 (see Lemma 2.12) implies that the submodule (D

a is given by

Similarly, using Lemma 7.6 we observe that (D (e)

F p [t] ) 0 • δ 1 = 0≤a i <p e a |a|+1 Q e a , and by once again applying the fact that (D

and part (i) follows.

For part (ii), recall that the action of Cont e ( Z (p) , F p )o nN e a comes via the map Δ defined in Subsection 2.5. An easy computation yields that a function ϕ ∈ Cont e ( Z (p) , F p )a c t so nQ e a by the scalar ϕ(|a|), and therefore

0i f|a|≡α mod p e , F p Q e a otherwise. R • a n+1 for some n with 0 ≤ n ≤ r(p e -1) and n ≡ α mod p e . We conclude that (a) implies (c). To show that (c) implies (a), suppose that we are given a differential jump n as in part (c). By Proposition 3.13 we can subtract p e enough times to assume that 0 ≤ n ≤ r(p e -1), and the result follows by applying Corollary 7.8 once again.

Corollary 7.10. Suppose that (N e a ) α =0for some e ≥ 0.T h e n(N i a ) α =0for all i ≥ e.

Proof of Theorem 7.3. Suppose that α ∈ Z (p) is a Bernstein-Sato root of a;t h a ti s , there is a sequence (ν e ) ⊆ Z ≥0 such that ν e ∈B • a (p e )a n ds ot h a tα is the p-adic limit of ν e . By passing to a subsequence (see Remark 4.2) we may assume that ν e ≡ α mod p e . By Proposition 7.9, the image of δ 1 in (N e a ) α is nonzero for every integer e ≥ 0. We conclude that the image of δ 1 in (N a ) α is nonzero, and thus (N a ) α =0.

For the other direction, suppose that (N a ) α is nonzero. By Corollary 7.10, we must have that (N e a ) α is nonzero for every e ≥ 0. By Proposition 7.9 we conclude that for every e ≥ 0 there is a differential jump ν e ∈B • a (p e ) with ν e ≡ α mod p e . Since α is the p-adic limit of the sequence (ν e ), α is a Bernstein-Sato root of a.

7.2. The operators s p i and the algebra Cont( Z (p) , F p ). We give a few remarks about why the result on the previous subsection establishes that the definition of Bernstein-Sato root given in Section 4 is a natural extension of the notion in previous work on this subject [QG21b].

Given a (D R[t] ) 0 -module M and a p-adic integer α, we define

where the operators s p i are as given in Subsection 2.4 (note that this module was previously denoted as M α [QG21b]). Recall we also have the definition

where m α ⊆ Cont( Z (p) , F p ) is the ideal of functions that vanish at α. The module N a is a (D R[t] ) 0 -module. Our approach is to think of N a as a Cont( Z (p) , F p )-module by restriction of scalars, and α ∈ Z (p) is a Bernstein-Sato root of a precisely when (N a ) α is nonzero (Theorem 7.3). We recall that in earlier work α is defined to be a Bernstein-Sato root whenever (N a ) (α) is nonzero [QG21b]. A priori these two constructions are different, but Proposition 7.11 tells us that they agree whenever the module N a splits nicely; this is the case whenever a is a Bernstein-Sato admissible ideal, and therefore the two definitions agree when the ring R is regular (which is the only case considered in the third author's work [QG21b]).

Proposition 7.11. The following are equivalent for a left If these hold then for all α ∈ Z (p) , we have ψ(M (α) )=M α .I np a r t i c u l a r ,

Proof. With the notation from Subsection 2.4, we have that the ideal m α is generated by m α =(σ p i -α i | i ∈ Z ≥0 ) and, since Δ(σ p i )=s p i ,w es e et h a tM (α) =A n n M (m α ) for any (D R[t] ) 0 -module M and any α ∈ Z (p) .

Suppose that M splits as in part (a). Then for all β ∈ Z (p) we have

and the composition M → M β ∼ = M (β) is the projection map, which proves (b).

Suppose now that M splits as in part (b). The natural map M → α M α is Cont( Z (p) , F p )-linear, and thus for all β ∈ Z (p) we have

and the composition M β ∼ = M (β) → M is the inclusion map induced by the direct sum decomposition, which proves (a).

The last statement follows from the proof.

7.3. Alternative characterization of N f . For every e ≥ 0l e tH e f := R f [t]/(ft) p e δ p e ,w h e r eδ p e is a formal symbol. We have maps H e f → H e+1 f given by multiplication by (f -t) p e (p-1) =(f p e -t p e ) p-1 , and the limit

We get an action of the algebra Cont e ( Z (p) , F p )o nH e f by restriction of scalars through Δ e , and an action of the algebra Cont( Z (p) , F p )o nH f by restriction of scalars through Δ. Proposition 7.12. For every e ≥ 0 there is a unique additive isomorphism

that identifies 1 with δ 1 and that is linear over R f and over Cont e ( Z (p) , F p ).T h e s e isomorphisms glue to give an isomorphism

Proof. Fix an integer e ≥ 0. Given a ∈{0,...,p e -1} denote by χ We now observe that, given an e ≥ 0, we have the following equalities in Cont e ( Z (p) ,R f )a n dH e f respectively:

1=

and H e f = p e -1 a=0 R f T e a such a map Φ e exists and is indeed an isomorphism. Only the claim regarding the gluing remains, which follows from the following identities in Cont e ( Z (p) ,R f )a n dH e f :

a+cp e , T e a = f a t p e -1-a (f -t) p e (p-1) δ p e+1

With this notation, it follows from Proposition 7.12 that we have isomorphisms

Our next goal is to describe the Cont( Z (p) ,D R )-module structure of Cont( Z (p) ,R f )f s more explicitly. Note that R , the operators f -α <a δf α <a are equal for all a ≥ e. This construction defines a map

Note that Υ α,f respects the level filtration, and it therefore induces maps Υ e α,f : D

Proof. We retain the notation from the proof of Proposition 7.12. Let us also temporarily denote by ξ φ∈ Cont( Z (p) ,R f ) the function α → Υ α,f (ξ(α)) • φ(α)as before. We need to show that in H f we have the equality Δ(ξ)Φ(φ)=Φ(ξ φ) and, since the operation is bilinear, it suffices to prove it for ξ = δχ Given a p-adic integer α ∈ Z (p) , evaluation at α defines a surjective R f -module homomorphism Cont(

along which we can transfer the D R -module structure of the left hand side to R f . The module R f , equipped with this exotic D R -module structure, is denoted R f f α , where once again f α is a formal symbol.

We describ e the D R -module structure more explicitly: given δ ∈ D R and g ∈ R f we have

where a ∈ Z is an integer that p-adically approximates α; more precisely, if δ has level e, then we require that p e divides α -a.

If α ∈ Z (p) , then there exist b>0 such that α(p b -1) ∈ Z. Then, α(p eb -1) ∈ Z for all integers e>0.

for all g ∈ R f . This shows that R f f α agrees with the D R -module M -α as introduced by Blickle, Mustat ¸ȃ, and Smith [BMS09] and further studied by the second author and Pérez [NBP16].

Lemma 7.14 justifies the notation R f f α .

Lemma 7.14. Let R be an F -finite ring, f ∈ R.T h e n :

(iv) Suppose R is a domain and that for some h ∈ Frac(R),s o m em ∈ Z and some k ∈ Z pZ we have an equality

Proof. Fix an operator δ ∈ D R of level e,a ne l e m e n tg ∈ R f , α ∈ Z (p) and an integer a ∈ Z such that p e divides α -a. Part (i) follows because the given morphism sends

Similarly, in part (ii) we have

For part (iii) we note that p e divides nα -na, and we compute:

In order to prove part (iv), we may replace m and k by nm and nk respectively, and we may therefore assume that k = p b -1. We take our approximation to m/k to be a = -m k (p eb -1); since (p eb -1)/k is an integer, so is a and, moreover, 1) . Note then that the morphism sends

Proposition 7.15. Let R be an F -finite ring, f ∈ R be a nonzerodivisor and α ∈ Z (p) be a p-adic integer.

(i) We have a D R -module isomorphism

which gives (i). Statement (ii) follows from (i) together with Theorem 7.3.

We note that the analogue of statement (ii) in Proposition 7.15 does not hold in characteristic zero [Sai21].

Using Proposition 7.15, we show that positive Bernstein-Sato roots abound in rings with certain bad singularities. We recall that a domain R is seminormal whenever, for all a ∈ Frac(R) such that a 2 ,a 3 ∈ R,w eh a v ea ∈ R [Swa80].

Proposition 7.16. Let R be an F -finite domain. If R is not seminormal then, for every n ∈ Z ≥2 pZ,t h e r ei ss o m ef ∈ R such that 1/n is a Bernstein-Sato root of f . Proof. Since R is not seminormal, we may pick some a ∈ Frac(R) \ R such that a 2 ,a 3 ∈ R, and therefore a k ∈ R for all k ≥ 2. Let f = a n ;w et h e nh a v ef, fa ∈ R and a/ ∈ R. By Lemma 7.14(iv), the R f -module homomorphism R f f 1/n → Frac(R) that sends f 1/n → a is a D R -linear embedding. Since D R • fa ⊆ R,w eh a v e a/ ∈ D R • fa,s of 1/n / ∈ D R • f f 1/n . By Proposition 7.15, we have that 1/n is a Bernstein-Sato root of f . Corollary 7.17. Let R be an F -finite domain and suppose that all Bernstein-Sato roots of all elements of R are nonpositive. Then R is seminormal. Now let us assume (e) and prove (a). Once again we fix a ∈ Z >0 such that α(p a -1) ∈ Z. Pick some e large enough so that the interval (-α, -α + 1+α

p ea ] contains no differential thresholds of f . Note that, for all integers k ≥ 0a n da l l integers n with n ∈ -α(p (e+k)a -1) + 1, -α(p ea -1)p ka + p ka -1 , we have n p (e+k)a , n +1 p (e+k)a ⊆ -α, -α + 1+α p ea .

From Lemma 5.6 we conclude that, for all k ≥ 0,

f (p (e+k)a ) ∩ -α(p (e+k)a -1) + 1, -α(p ea -1)p ka + p ka = ∅, and hence there is some differential operator ξ k ∈ D ((e+k)a) R such that ξ k f -α(p ea -1)p ka +p ka = f -α(p (e+k)a -1)+1 .

We conclude that ξ k • f α(p ka -1)+p ka f α = f α(p (e+k)a -1) ξ k f -α(p (e+k)a -1) f α(p ka -1)+p ka f α = f α(p (e+k)a -1) ξ k f -α(p ea -1)p ka +p ka f α = f f α .

We conclude that f f α ∈ D R • f α(p ka -1)+p ka f α for all k ≥ 0 and, since α(p ka -1) + p ka = p ka (α +1)-α and α +1> 0, we have f f α ∈ D R • f t f α for all t ≥ 1.

Remark 8.5. For a non-F -split ring R and an arbitrary p-adic integer α the equivalence (b) ⇐⇒ (d) remains true. More generally, for all integers t we have

if and only if BSR(f ) ∩{α + t, α + t +1,...} = ∅ (see Lemma 7.14 and Proposition 7.15). Corollary 8.6. Let R be an F -finite F -split ring and f ∈ R be a Bernstein-Sato admissible nonzerodivisor. For all α ∈ Z (p) we have

Corollary 8.7. Let R be an F -finite F -split ring, f ∈ R be a nonzerodivisor and α ∈ Z (p) Z. The following are equivalent:

(a) The chain Definition 8.17. Suppose that R is a Bernstein algebra. Let M be a finitely generated nonzero D R -module. We say that M is holonomic if it admits a filtration of dimension dim(R) and finite multiplicity. We now show that G i is a B • R -filtration. We have that G i is a finite dimensional K-vector space and Hence,

We conclude that B i R G i ⊆ G i+j . Then, R f f α is a holonomic D R -module. We conclude that R f f α has finite length as D R -module by Theorem 8.18. 2 ). It suffices to show that xyz ∈ τ R (a λ )f o rλ< 3 2 .F o re>0, we have (xyz) 2p e -2 =( x 2 yz) (p e -1)/2 (xy 2 z) (p e -1)/2 (xyz 2 ) (p e -1)/2 ∈ a (3p e -3)/2 ,s o xyz ∈C e R • a (3p e -3)/2 .S i n c e (3p e -3)/2 p e → 3 2 , the claim follows. This example shows that the conclusion of Theorem 5.17 cannot be strengthened to say that the negative of a Bernstein-Sato root is a differential threshold. We have that 9 4 =1-α is an F -jumping number, and hence a differential threshold [QG21a, Example 3.5].

Example 9.3. Let R = F p [x 2 ,xy,y 2 ] ⊆ S = F p [x, y], with p odd. This inclusion is Cartier extensible, level differentially extensible, and is split as R-modules. Let m =( x 2 ,xy,y 2 ) be the homogeneous maximal ideal of R,a n dn =( x, y)b et h e homogeneous maximal ideal of S.W eh a v e τ R (m λ )=τ S ((mS) λ ) ∩ R = τ S (n 2λ ) ∩ R for all λ [ ÀMHJ + 22, Proposition 5.9]. We have τ S (n γ )=n γ-1 for γ ≥ 1, and τ S (n γ )=S for 0 ≤ γ<1. We observe that n γ ∩ R = m γ/2 .T h u s , t h e Fjumping numbers of mS are {1, 3 2 , 2, 5 2 , 3, 7 2 ,...},a n dt h eF -jumping numbers of m are {1, 2, 3,...}.

By Proposition 5.12(iv), the differential thresholds of mS are {1, 3 2 , 2, 5 2 , 3, 7 2 ,...}. Then, by Theorem 6.7(vii), the differential thresholds of m are {1, 3 2 , 2, 5 2 , 3, 7 2 ,...}. To compute the Bernstein-Sato roots of m, we may equivalently compute the Bernstein-Sato roots of mS = n 2 by Theorem 6.7(vi). We have that Passing to p-adic limits, we conclude that the set of Bernstein-Sato roots of m is {-1, -3 2 }. This shows that an ideal may have Bernstein-Sato roots that are not congruent modulo Z to F -jumping numbers. This also shows that an ideal may have differential thresholds that are not F -jumping numbers, even in a strongly F -regular hypersurface.