On a theory of the b-function in positive characteristic

We present a theory of the b-function (or Bernstein–Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic p>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>0.$$\end{document} Its b-function bf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document} is defined to be an ideal of the algebra of continuous k-valued functions on Zp.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p.$$\end{document} The zero-locus of the b-function is thus naturally interpreted as a subset of Zp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p,$$\end{document} which we call the set of roots of bf.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f.$$\end{document} We prove that bf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document} has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţă and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of bf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_f$$\end{document} and relate it to the test ideals of f.

The consideration of the solutions to the functional equation above was originally motivated by the problem of analytically continuing the zeta function attached to f, see [3,Theorem 1]. We would like to highlight two other types of consequences of that result.
On one hand the existence of the Bernstein-Sato polynomial of a general nonconstant polynomial is of theoretical interest for modules over rings of differential operators or D-modules. Namely it was the latter's first application and has been crucial in singling out the importance of holonomicity [3,Corollary (1.4.b)]. Moreover it is used to prove the stability of holonomicity under operations, e.g. direct image for an open embedding [4, p. 25], nearby and vanishing cycles [2, 2.1. Key Lemma].
On the other hand, the Bernstein-Sato polynomial of f is a rich invariant of the singularity { f = 0}, as was discovered by Malgrange [19]. In particular the roots of b f (s) are related to very many invariants of the singularities of f, see [15,22] and [17] for surveys. Of motivational interest for us is the following result of Ein-Lazarsfeld-Smith-Varolin [8,Theorem B], relating the jumping coefficients of the multiplier ideals of f to the roots of b f (s), see [8] for the definitions: Theorem B Let λ be a jumping coefficient of f which lies in (0, 1]. Then − λ is a root of the Bernstein-Sato polynomial of f. In this paper, we put forth a natural definition of Bernstein-Sato polynomial (or b-function) in positive characteristic and show that it satisfies statements analogous to Theorems A and B.

Positive characteristic
Let k be a perfect field of positive characteristic p and let g be a non-constant kpolynomial in n variables. A positive characteristic analogue of the multiplier ideals is provided by the (generalised) test ideals of Hara and Yoshida [12]. Where the Fjumping exponents of the test ideals of g are the pendant to the jumping coefficients of the multiplier ideals. These are rational numbers [6,Theorem 3.1].
In accordance with the problematic set forth by Mustaţȃ in [20] and in analogy with Theorem B, we would like the sought for b-function of g to be related to its F-jumping exponents. It does thus seem preferable for the b-function to be an object living beyond the positive characteristic.
The theory we propose here is p-adic, in the sense that the roots of our b-function are naturally p-adic integers. To describe it, let us first consider the usual abstraction of Bernstein's functional equation, see e.g. [13, §1].
Let D be the ring of complex differential operators with polynomial coefficients in n variables, and let O be the ring of complex polynomials in n variables. In the positive characteristic theory, the rôle of C[s] is played by the k-algebra A k generated by the binomial coefficients functions modulo p. It is not principal but there is nevertheless a natural notion of roots of an ideal. Indeed A k is isomorphic to k[Y e ; e ∈ N]/(Y p e − Y e ; e ∈ N) and the generators are canonically ordered. Thus for each homomorphism α from A k to a field, α(Y e ) is in the prime field, for all e. We associate to α the p-adic integer e α e p e , where α e is the unique lift of α(Y e ) to {0, . . . , p − 1}. In particular to each maximal ideal of A k , we may attach a padic integer for α the canonical quotient homomorphism. In fact, we show that A k is isomorphic to the k-algebra k of continuous k-valued functions on the p-adic integers. Thus to each ideal I of A k corresponds a set of p-adic integers. Namely those associated by the above to the maximal ideals containing I. We call them the roots of I.
Our approach to the b-function of a non-constant polynomial g in positive characteristic is to consider a left D ⊗ k A k -module N γ g for D the ring of Grothendieck differential operators, which is thought of as the positive characteristic analogue of D[s] f s /D[s] f s+1 (for g instead of f ), see Definition 1.2.4. We then define the bfunction b g of g to be the annihilator of the action of A k on N γ g . The roots of b g satisfy strong finiteness properties. Indeed, here are those corresponding to Theorems A and B: Theorem C The roots of the b-function of g form a finite set of strictly negative rational numbers. Furthermore they are not smaller than − 1.
In fact, we have the following precise relation to the F-jumping exponents of g, see Theorem 2.4.1.
Theorem D Let λ be an F-jumping exponent of g in (0, 1] ∩ Z ( p) . Then − λ is a root of the b-function of g. Moreover this exhausts the roots of b g .
In order to prove our main results, we study in some depth the D-module structure of N γ g . We show that it is a finitely generated unit F-module (Theorem 2.3.4), which serves as a replacement for holonomicity. In fact we exhibit an explicit generator of the unit F-module N γ g expressed in terms of the test ideals of g. This is what ultimately allows us to relate the roots of the b-function to the F-jumping exponents and prove Theorem 2.4.1.

Relation to the work of Mustaţȃ and motivation
In [20], which was our starting point, Mustaţȃ uses the action of Euler operators on bounded level analogues of N g to construct a whole sequence of Bernstein-Sato-type polynomials and his main result is that the information provided by these invariants corresponds to that provided by the F-jumping exponents of g. However, this correspondence leaves open the question of what is the natural analogue of the Bernstein-Sato polynomial. This is the question addressed here. We do so by focusing on D-modules (as opposed to D (e) -modules with bounded divided powers) and hence on the whole algebra of higher Euler operators. This leads us to a new notion of b-function (or Bernstein-Sato polynomial), naturally p-adic.

Nearby cycles
We have generalised our theory to unit F-modules coefficients and argue that these may be called nearby cycles. This will appear elsewhere.

Frobenius structure
The action of Frobenius on N g is given here via an explicit root for the corresponding unit F-structure. It would be preferable to have a direct description. We have failed to obtain one so far.

Relation to the Bernstein-Sato polynomial
In view of the analogy between the Bernstein-Sato polynomial of a, say, complex polynomial and the theory of the b-function presented here, it is natural to ask about the comparison between the Bernstein-Sato polynomial of a polynomial f with say rational coefficients and the b-function of its reduction f p modulo a prime p, for p large.
The situation is subtle, as shown in Sect. 3. Strikingly, there are polynomials f such that there are arbitrary large primes p for which there is a root of the b-function of f p which is not a root of the Bernstein-Sato polynomial of f, see Example 3.0.8.

The algebra of higher Euler operators
Let k be a field of characteristic p > 0 and let R be a commutative k-algebra. In the sequel, we will use the notation D R for the ring of global sections of the sheaf of Grothendieck differential operators D Spec(R)/k . We refer to [11, 16.8] for a general treatment of the sheaf of rings of Grothendieck differential operators.
After [20], we set the higher Euler operators to be the following differential operators in D k[t] :  For all n ≥ 0, let s n denote the n-th binomial coefficient function modulo p, We will consider them as k-valued functions. Proof Let n be a natural number and let n = N e=0 n e p e be its base p expansion. It follows directly from Lucas' Theorem that This proves that the k-algebra of binomial coefficients is generated by the binomial coefficients { s p e ; e ∈ N}. Since the s p e are F p -valued functions, it is clear that they satisfy the relations ; e ∈ N} is contained in the kernel of π k . Let us prove that the kernel is generated by {Y p e − Y e ; e ∈ N}. Since π k = k ⊗ F p π F p and k is flat over F p , it is enough to show the assertion for π F p .
Let P be in the kernel. That is P is a F p -polynomial in N variables for a natural number N and P( s p 0 , . . . , s p N −1 ) = 0. Thus by Lucas' Theorem, P is in the kernel of the evaluation morphism: where Fun(F N p , F p ) is the algebra of F p -valued functions on F N p . One easily sees by counting dimensions that this implies that P lies in the ideal generated by {Y The k-algebra of higher Euler operators and the k-algebra of binomial coefficients are isomorphic. Let us fix an isomorphism.
The morphism γ is an isomorphism.
Proof Let us first note that the definition of the higher Euler operators provides a natural embedding of k into the algebra of binomial coefficients. Indeed, since D A 1 k acts faithfully on k[t], the higher Euler operators are characterised by their action on the monomials. It is given by ν e (t n ) = d dt [ p e ] t p e (t n ) = p e +n p e t n = (1 + n p e )t n , the last equality holding by Lucas' Theorem. We thus see that the assignment ν e → 1 + s p e defines an embedding of k into the k-algebra of binomial coefficients. It clearly is an isomorphism.
We are thus left with having to show that the assignment s . This is obvious from the presentation given in Corollary 1.1.5.
The isomorphism γ is the analogue of the isomorphism from the complex theory.
Next we explicitly relate the algebra of binomial coefficients to the p-adic integers.
Theorem 1.1.8 Let F p and k be endowed with the discrete topology.
(1) For all natural numbers e, the binomial coefficient function s p e modulo p extends to a continuous F p -valued function c e on Z p , such that c e (z) = z e mod p, where z = e≥0 z e p e is the p-adic expansion. This induces an embedding i of k-algebras: where C(Z p , k) is the k-algebra of continuous k-valued functions on Z p . The morphism i is an isomorphism. (2) Moreover, for each p-adic integer z, let ev z be the evaluation morphism: The map from the p-adic integers to the set of maximal ideals Max(C(Z p , k)) of C(Z p , k) given by is a bijection.
Proof (1) The projection to the e-th coefficient in the p-adic expansion is a continuous map from Z p to F p . It is thus continuous as a function from Z p to k. Moreover it coincides with s p e on N by Lucas' Theorem (Theorem 1.1.4). This provides the morphism i of the statement. It is injective since N is dense in Z p . As the s p e generate the algebra of binomial coefficients by Corollary 1.1.5, it is a direct consequence of Mahler's Theorem, see e.g. [16,III 1.2.4], that i is surjective.
(2) Since s p e is the projection to the e-th coordinate in the p-adic expansion, it is clear that the map Z p → Max(C(Z p , k)) is injective. Indeed if z = e≥0 z e p e is different from z = e≥0 z e p e , then there is a number l ≥ 0 such that z l is different from z l . Thus s p l − z l belongs to ker(ev z ) but not to ker(ev z ). Let us show that is surjective. Let z = e≥0 z e p e , the maximal ideal ker(ev z ) is generated by { s p e − z e ; e ≥ 0}. Indeed the set { s p e − z e ; e ≥ 0} is contained in ker(ev z ), and since the ideal generated by { s p e − z e ; e ≥ 0} is obviously maximal, it has to be equal to ker(ev z ). Let m be a maximal ideal of C(Z p , k) and let The following proposition shows that the naive notion of multiplicity is not so interesting for the algebra of binomial coefficients. Proof (1) Let A denote the k-algebra of binomial coefficients. Let us show that every finitely generated A-algebra that is a field is a finitely generated A-module. This is one of the characterisations of Jacobson rings, see e.g. [1, Ch. 5 Exercise 25]. Let K be such a field and let υ be the homomorphism of k-algebras Since s p e p = s p e for all e ≥ 0, its image by υ is in the prime field and accordingly υ(A) = k. It is well-known that fields are Jacobson rings. Therefore K is a finitely generated k-module. Hence it is a finitely generated A-module. We have thus proved that A is a Jacobson ring. ( Clearly f p n = f, for all natural numbers n. Suppose that f is in the radical of an ideal I, i.e. f N is in I for some N ≥ 0. Let n ≥ 0 be such that p n ≥ N . We have that f = f p n is in I. Thus the ideal I is radical. Since by (1), the algebra of binomial coefficients is radical, we have that every radical ideal is the intersection of maximal ideals. Hence every ideal . .] is the intersection of the maximal ideals containing it. It is thus characterised by its set of roots.

Definition of the b-function
Let k be a field of characteristic p > 0 and let R be a commutative k-algebra. Recall that we denote by D R the ring of Grothendieck differential operators of R over k. Definition 1.2.1 Let R be a smooth commutative k-algebra and let f be an element of R. Denote by R[t] the ring of polynomials in one variable t over R.
be the subring of D R[t] generated by D R and the higher Euler

Lemma 1.2.2 For all natural numbers e, we have the following identity in D
where the product over the empty set, i.e. for e = 0, is 1. In particular, the k-algebra of higher Euler operators k satisfies k t = t k in D k [t] .
Proof For the identity to hold in D F p [t] , it is enough that both sides agree when evaluated at monomials t N , for all natural numbers N ≥ 0.
One easily sees that, for the left-hand side: and for the right-hand side: Since n p−1 = 1 mod p if and only if n = 0 mod p, one has that Since j=e−1 j=0 ( p − 1) p j = p e − 1, it is thus clear using Lucas' Theorem again that both sides of the identity evaluated at t N vanish if N = p e − 1 mod p e and that they are equal to t N +1 , otherwise. This concludes the proof of the identity.
The inclusion k t ⊂ t k follows immediately from the identity. The reverse inclusion t k ⊂ k t follows from the identity and an easy induction. Namely, it suffices to show that for all natural numbers m, n 0 , . . . , n m , the monomial tν n m m . . . ν n 0 0 is in k t. Let us well-order the finite sequences (m, n m , . . . , n 0 ) by the lexicographic order and induct on it. It is clear that for the smallest element (0, 0), . . . ν n 0 0 , which is in k t by the induction hypothesis. This concludes the proof of the lemma.

Corollary 1.2.3 The left D R -submodule t M f of M f is stable under the action of the higher Euler operators. Hence the quotient N
We can now give the definition of the b-function. . This allows us to globalise the definition of the b-function to f a function on a smooth k-variety, for example. We leave the details to the reader.

Bounded level versions of N f
Here we start analysing N f by considering differential operators of bounded level. Let us recall some definitions. From here on, let us suppose that the base field k is perfect.
f is set to be the quotient,

Test ideals
In this subsection and the next, unless otherwise mentioned, we use the following notations: k is a perfect field of characteristic p > 0, R is a smooth k-algebra and f is an element of R.
Let us recall the definition of the test ideals of f and sum up the properties we will use. The following finiteness result is crucial to us.

Theorem 1.4.6 The set of F-jumping exponents of f is (1) a discrete subset of R (2) a subset of Q.
Proof This is [6, Theorem 3.1], for a principal ideal.
We will also use the following definition.

Lemma 1.4.8 Let λ be a positive real number. Denote by λ the supremum of the subset S λ of the real numbers containing 0 and the F-jumping exponents of f strictly smaller than λ. Since S λ is discrete by (1) of Theorem 1.4.6, λ is in S λ . The test ideal immediately preceding τ ( f λ ) is τ ( f λ ) if and only if λ is not a F-jumping exponent of f. If λ is a F-jumping exponent of f, then the test ideal immediately preceding
Proof If λ is not a F-jumping exponent, then there exists a real number > 0 such that τ ( f λ− ) = τ ( f λ ). Since the filtration by test ideals is decreasing, If λ is a F-jumping exponent, then for all real numbers r in Proof It follows directly from Lemma 1.4.8

N (l) f and test ideals
By explicit computations, Mustaţȃ observed the following.

(3) The natural inclusion M
is transported by the isomorphisms of level l and l + 1 of (1) to where the subscript of an element indicates the direct summand to which it belongs.
Proof By (1) of Proposition 1.5.1 and Lemma 1.5.2, for all natural numbers l, we have that and thus that , unique by the hypothesis on l.
Finally, since, for varying n, the intervals ( n p l+1 , n+1 p l+1 ] form a partition of (0, 1], all F-jumping coefficients of f in (0, 1] appear in this way. Thus one has that 0≤n< p l+1 where the direct sum on the right-hand side is over the F-jumping exponents of f in the interval (0, 1]. One applies Corollary 1.4.9 to conclude the proof of the proposition.
In the next section, we use the relation between N f and the F-jumping exponents of f to get information about the roots of its b-function.

Relation to test ideals and rationality of the roots 2.1 Preliminaries on p-adic and 1 p -adic expansions
We will express the p-adic expansions of the roots of the b-function of f in terms of the " 1 p -adic expansions" of its F-jumping exponents. Let us precise what we mean. p -adic period of r is the period ρ r of the sequence (r n ) n≥n and the word r n . . . r n +ρ r −1 is its 1 p -adic repetend. A 1 p -adically periodic number is said to be purely 1 p -adically periodic if its 1 p -adic anteperiod is 0.
The following lemma characterises the periodicity properties of 1 p -adic expansions.

Lemma 2.1.3 Let p be a prime number. (1) A positive real number is 1 p -adically periodic if and only if it is rational. (2) A positive real number is purely 1 p -adically periodic if and only if it is in Z ( p) .
Proof (1) To prove that 1 p -adic periodicity implies rationality, one notices that for all e ≥ 1, If a number r ∈ (0, 1] is purely 1 p -adically periodic, then its 1 p -adic expansion is of the form r 0 n>0 where e is a natural number > 1 and r 0 is smaller than p e . Thus r is in Z ( p) .
Suppose that r is in Z ( p) ∩ (0, 1] and let r = a b be its reduced rational expression. Then it is well-known that, since b is prime to p, there is a positive natural number e such that b divides p e − 1. Thus r = a p e −1 = a n>0 1 p en , with 0 < a < p e . Hence r is purely 1 p -adically periodic.
To each 1 p -adically periodic number correspond finitely many "conjugated" numbers.

Definition 2.1.4 The 1
p -adic conjugates of a 1 p -adically periodic number r are the numbers whose 1 p -adic expansion is obtained from that of r by a cyclic permutation of the letters of its 1 p -adic repetend.
Thus for l the anteperiod of r and d its period, the 1 p -adic conjugates of r = b 1 p + · · · + b l p l + a 1 p l+1 + a 2 p l+2 + · · · + a d−1 We now present a pure periodicity lemma which will be useful in our description of the roots of the b-function. We start with a definition. Proof Since is a finite set, there is a number L greater than N such that for each l greater than L , there is an element λ in beginning with s l and beginning with s l for infinitely many l greater than l. Let us prove that such a rational number λ has to be purely 1 p -adically periodic. Clearly, one may suppose that λ coincides with its 1 p -adic fractional part. Let λ = j≥1 λ j p j be its 1 p -adic expansion. By (1) of Lemma 2.1.3, λ is 1 p -adically periodic. Let n be its anteperiod and ρ > 0 be its period. We claim that for all numbers j ≥ 1, λ j+ρ = λ j . This indeed implies that λ is purely 1 p -adically periodic. We will show that the hypothesis on the beginning of λ allows one to embed the beginning of its 1 p -adic expansion into its repeating part, thus forcing pure periodicity.
More precisely, since λ j+ρ = λ j for all j > n by definition of the anteperiod n, it is enough to show that λ j+ρ = λ j for all 1 ≤ j ≤ n. Consider l greater than max{n + ρ, L} such that λ begins by s l . Thus for all 1 ≤ i ≤ n + ρ, λ i is equal to s(l − i + 1). Since there is also l greater than l + n such that λ begins with s l , we have that for all 1 ≤ i ≤ n, Moreover l − l is not smaller than the anteperiod n, thus l − l + i > n. Hence λ l −l+i is equal to λ l −l+i+ρ . Using again that λ begins with s l , we get that the latter λ l −l+i+ρ = s(l − (l − l + i + ρ) + 1) = s(l − (i + ρ) + 1). Recall that λ begins also with s l and thus that s(l − (i + ρ) + 1) = λ i+ρ . In conclusion, we have proved that for each i between 1 and n, This completes the proof of the lemma.

Unit F-modules
In the next subsection, we will show that N f seen as a left D R -module is of a very particular type, namely it is a unit F-module. Let us first recall their definition and basic properties.
From now on, unless otherwise mentioned, k is a perfect field of characteristic p > 0, R is a smooth k-algebra, f is an element of R and F is the Frobenius endomorphism of R.
is an isomorphism.
The following is well-known.
There is a convenient way to generate unit F-modules, due to Lyubeznik [18].

Proposition 2.2.6 Let e be a positive integer. Let G be an R-module and β be an
and denote it lim − → β. Then the R-module lim − → β is naturally isomorphic to its pull-back (F e ) * lim − → β. Hence it is canonically endowed with a structure of unit F e -module.
Proof This is a direct consequence of the commutation of pull-back with direct limits.

Definition 2.2.7
Let e be a positive integer and M a unit F e -module. The data of a finitely generated R-module G, an R-linear morphism G β → (F e ) * G and an isomorphism ι of Frobenius modules lim − → β → M is called a generator of M. We will often omit to mention the isomorphism ι.
In fact, all finitely generated unit F-modules appear this way, as noted by Emerton-Kisin.

Theorem 2.2.8 Every finitely generated unit F e -module has a generator.
Proof This is a special case of [9, Theorem 6.1.3].
Here is a fundamental finiteness property of unit F-modules by which they stand out among D R -modules. It is due to Lyubeznik.

A unit F-structure on N f
Here we continue the study of N f via its bounded level versions started in Sect. 1.5. In particular, we show that the left D R -module N f is a unit F-module and the higher Euler operators are compatible with its Frobenius endomorphism.
for the unique m such that m p l+1 is the truncated 1 p -adic expansion of λ. We claim that there are numbers N and L such that if λ / ∈ Z ( p) , then for all i > N , and l > L , the above map vanishes.
Let us argue by contradiction and suppose that there are l and i, large at will, such that the above map does not vanish. In particular the target of the map is not trivial. Hence there is a number n i , congruent to m modulo p l+1 , such that n i p l+i+1 is the truncated 1 p -adic expansion of an F-jumping exponent of f. Thus n i = m + b i p l+1 , for a certain natural number b i = j=i−1 j=0 (b i ) j+1 p j and where m 1 p + m 2 p 2 +· · ·+ m l+1 p l+1 is the truncated 1 p -adic expansion of λ. Since the map is an i-th iterate, for it not to vanish it is necessary that the corresponding j-th iterate does not vanish either, for all j ≤ i. Hence one may assume that as i varies, the numbers b i are compatible with each others. Namely for all i ≤ i such that b i and b i are defined, one has that b i is congruent For every l as above, we define the sequence s := (m l+1 , m l , . . . , m 1 , (b) 1 , (b) 2 , . . . ). Let be the set of F-jumping exponents of f in (0, 1]. It is clear that for all n large enough, there is an element of beginning with s n , using the notation of Lemma 2.1.6. Hence by Lemma 2.1.6, for n large enough, there is a purely 1 p -adically periodic element of beginning with s n . As l can be taken as large as one wishes, this implies that the sequence (m 1 , m 2 , . . . ) is purely periodic, that is that λ is purely 1 p -adically periodic. This is absurd since λ is in the complement of Z ( p) , by hypothesis. We have thus shown the required vanishing. This allows us to give an inductive system for the D R -module N f expressed only in terms of the F-jumping exponents of f in (0, 1] ∩ Z ( p) . Recall from Proposition 1.5.4 that for all l large enough to separate the F-jumping exponents of f, we have maps: and π is the natural projection gr λ∈(0,1] (τ ( f λ )) π → gr λ∈(0,1]∩Z ( p) (τ ( f λ )). We also have to consider the following intermediate system: with c l+1 := (F l+2 ) * (i) • (F l+2 ) * (π ) • a l+1 . We will denote the corresponding inductive systems by (a j ), (b j ) and (c j ), respectively.  In particular N f is isomorphic to the direct limit of (b j ), lim − → (F j ) * (gr λ∈(0,1]∩Z ( p) (τ ( f λ ))).
Proof It is a straightforward consequence of Proposition 2.3.2.
We can now prove the Theorem 2.3.4 Let k be a perfect field of characteristic p > 0, R a smooth k-algebra and let f be an element of R.
(1) Let e be the lcm of the lengths of the periods of the 1 p -adic expansions of the F-jumping exponents of f in (0, 1] ∩ Z ( p) . The left D R -module N f is a finitely generated unit F e -module. (2) For each l separating the F-jumping exponents of f, the unit F e -module N f splits as a direct sum where the λ are the F-jumping exponents of f in (0, 1] ∩ Z ( p) . Each summand is a direct limit where the limit is for the maps induced by the system (b j ) and it is non-trivial, (3) The higher Euler operators act as endomorphisms of the unit F e -module N f .
Proof (1) We use the inductive system (b j ) of Corollary 2.3.3, which by that Corollary has N f as direct limit. Pick an l that separates the F-jumping exponents of f and denote β e the morphism b l+e • · · · • b l+2 • b l+1 : We claim that β e is a generator. It is enough to check that for all r ≥ 0, the pull-back (F re ) * (β e ) coincides with the composition of the structure maps from (b j ), that is (F re ) * (β e ) = b (r +1)e+l •· · ·•b re+l+1 . We see by the description of the structure maps from Proposition 1.5.1.3 and Proposition 2.3.1 that for all purely 1 p -adically periodic F-jumping exponent λ, both β e,r := b (r +1)e+l • · · · • b re+l+1 and (F re ) * (β e ) send Indeed let m p l+1 be the truncated 1 p -adic expansion of λ. Since e is a multiple of the periods, they both have to factor through the sum of the factors indexed by F-jumping exponents μ with the same truncated 1 p -adic expansion m p l+1 . But l separates the Fjumping exponents hence μ has to be equal to λ.
Moreover if m p l+1 + p l+1 n p e+l+1 is the truncated 1 p -adic expansion of λ, we have by the description of the structure maps that β e,r is the multiplication by c f p re+l+1 n , for a non-zero element c of F p . We thus have that β e,r = (F re ) * (β e,0 ). Since β e,0 is equal to β e , this concludes the proof of the point. Note that by [14,Example 6.19], the Bernstein-Sato polynomial of f = x 2 + y 3 is b f = s + 7 6 (s + 1) s + 5 6 .