F-THRESHOLDS OF HYPERSURFACES MANUEL BLICKLE, MIRCEA MUSTAT¸Aˇ, AND KAREN E. SMITH 8 0 0 2 1. Introduction n a J In characteristic zero one can define invariants of singularities using all divisors 0 over theambient variety. Akey result that makes these invariants computable says that 1 they can be determined by the divisors on a resolution of singularities. For example, if ] a is a sheaf of ideals on a nonsingular variety, then to every nonnegative real number G λ one associates the multiplier ideal J(aλ). The jumping exponents of a are those λ A such that J(aλ) 6= J(aλ′) for every λ′ < λ. It is an easy consequence of the formula . h giving the multiplier ideals of f in terms of a log resolution of singularities, that the t a jumping exponents form a discrete set of rational numbers. See for example [Laz], Ch. m 9 for the basic facts about multiplier ideals and their jumping exponents. [ In positive characteristic Hara and Yoshida defined in [HY] an analogue of the 3 v multiplier ideals, the (generalized) test ideals. The definition works in a very general 0 setting, involving a notion of tight closure for pairs. In this paper, however, we assume 1 2 that we work in a regular ring R of characteristic p > 0 that is F-finite, i.e. such 1 that the Frobenius morphism F: R −→ R is finite. If a is an ideal in R and if λ is a . 5 nonnegative real number, then the corresponding test ideal is denoted by τ(aλ). In this 0 context we say that λ is an F-jumping exponent (or an F-threshold) if τ(aλ) 6= τ(aλ′) 7 0 for every λ′ < λ. The following is our main result about F-jumping exponents in : v positive characteristic. i X Theorem 1.1. If R is an F-finite regular ring, and if a = (f) is a principal ideal, then r the F-jumping exponents of a are rational, and they form a discrete set. a The discreteness and the rationality of F-jumping numbers has been proved in [BMS] for every ideal when the ring R is essentially of finite type over an F-finite field. We mention also that for R = k[[x,y]], with k a finite field, the above result has been proved in [Ha] using a completely different approach. We stress that the difficulty in attacking this result does not come from the fact that there is no available resolution of singularities in positive characteristic. Even Key words and phrases. F-thresholds, test ideals, F-modules, non-standard extension. 2000Mathematics Subject Classification. Primary 13A35; Secondary 14B05. ThesecondauthorwaspartiallysupportedbytheNSFundergrantsDMS0500127andDMS0111298, and by a PackardFellowship. 1 2 M. Blickle, M. Musta¸t˘a and K.E. Smith in cases when such a resolution is known to exist, the F-jumping exponents are not simplygivenintermsofthenumericalinformationoftheresolution.Wereferto[MTW] for a discussion of the known and conjectural connections between the invariants in characteristic zero and those in characteristic p. In order to prove Theorem 1.1 it is enough to show that the set of F-jumping exponents is discrete. The rationality statement follows as in [BMS]: it is enough to use the fact that if λ is an F-jumping exponent, then so are the fractional parts of peλ, for all e ≥ 1. Moreover, we will see that it is enough to prove the result in the case when R is local. Thecrucialstepintheproofofthetheoremreliesonshowingthatifαisarational number, then α is not an accumulation point of F-jumping exponents of f (irrational α’s are excluded by an inductive argument). The key point in this step is that (after preparing α) we may rephrase the statement that “α is not an accumulation point of F-jumping exponents” as the statement that “a certain element e of a certain α D –module M (which can be thought of as the D –module generated by 1 , see R α R fα the paragraph before Lemma 2.3) is a D -generator of M ” (Corollary 2.8). Here D R α R denotes the ring of all differential operators of R. This D -module reformulation is R an extension of an argument due to Alvarez-Montaner, Blickle and Lyubeznik from [AMBL] (one can interpret the main result in loc. cit. as the case α = 1, when M = α R ). Since we may assume that R is local, one then finishes the argument as in loc. cit. f by using the fact that M has finite length as a D -module (see [Lyu]) to conclude that α R e indeed generates M as a D -module (Theorem 2.11). This argument is carried out α α R in detail in Section 2 where also the necessary background and notation is recalled. The second half of the paper deals with limits of F-purethresholds. We apply our rationality result for formal power series to deduce that every such limit is a rational number. Recall that the F-pure threshold of a is the smallest (positive) F-jumping exponent of a. This invariant has been introduced by Takagi and Watanabe in [TW] who pointed out the analogy with the log canonical threshold in characteristic zero. In a fixed characteristic p, we consider the set T consisting of all F-pure thresholds n of principal ideals in regular F-finite rings of characteristic p and dimension ≤ n. We consider also the set T ◦ of F-pure thresholds at the origin for polynomials f ∈ n k[x ,...,x ], where k is an algebraically closed field of characteristic p (the definition 1 n does not depend on k). It is easy to see that every element in T can be computed as n the F-pure threshold of a formal power series f, and therefore it is the limit of the F-pure thresholds of the various truncations of f. Conversely, we show that every limit of F-pure thresholds in bounded dimension is the F-pure threshold of some formal power series. Theorem 1.2. For every prime p > 0 and every n ≥ 1, the set T is the closure of n T ◦ In particular, every limit of F-pure thresholds of principal ideals in F-finite regular n rings of bounded dimension is a rational number. F-thresholdsof hypersurfaces 3 The proof of Theorem 1.2 uses nonstandard methods to construct a power series whose F-pure threshold is the limit of a given sequence of F-pure thresholds. The necessary background for Theorem 1.2 and its proof are given in Section 3. In the final Section 4 we record some peculiar features of F-pure thresholds and test ideals, and state some open problems in analogy with some well-known conjectures in birational geometry. For an application of non-standard techniques to the study of log canonical thresholds, see [dFM]. In that case the non-standard argument is more involved due to the fact that the definition of the log canonical threshold is ”less elementary”. While the F-pure threshold is a more subtle invariant than the log canonical threshold, its definition is ”simpler”, and this pays off when using non-standard extensions. Webelievethatexploitingtheconnectionsandanalogiesbetweentheinvariantsin positive and zero characteristic can be very fruitful. For example, results on test ideals such as the Subadditivity and the Restriction Theorems are much easier to prove than for multiplier ideals, and they imply their characteristic zero counterpart by reduction mod p. Moreover, there are results on multiplier ideals that so far have been proved only by reduction to characteristic p (see the work of Takagi [Ta1] and [Ta2]). On the other hand, certain phenomena that are well-understood (or just conjectural) in characteristic zero can point to interesting phenomena in positive characteristic. Acknowledgements. The second author is indebted to Caucher Birkar for explaining him the usefulness of non-standard methods. He would also like to thank the Institute for Advanced Study, where part of this work has been carried out. 2. Discreteness and rationality We start by reviewing the definition and some basic properties of the generalized test ideals from [BMS]. Let R be a regular ring of characteristic p > 0. We assume that R is F-finite, that is the Frobenius morphism F: R −→ R, F(u) = up is finite. Note that F-finiteness is preserved by taking quotients, localization and completion (see Example 2.1 in [BMS]). Moreover, if R is F-finite then so are R[x] and R[[x]]. For an ideal J and for e ≥ 1, we put J[pe] = (upe | u ∈ J). If b is an arbitrary ideal in R, then we denote by b[1/pe] the (unique) minimal ideal J such that b ⊆ J[pe]. Suppose now that a is a fixed ideal in R and λ is a positive real number. For every e ≥ 1 we have a⌈λpe⌉ [1/pe] ⊆ a⌈λpe+1⌉ [1/pe+1], (cid:16) (cid:17) (cid:0) (cid:1) where ⌈u⌉ denotes the smallest integer ≥ u. This sequence of ideals stabilizes since R is Noetherian, and the test ideal is defined as τ(aλ) := a⌈λpe⌉ [1/pe] for e ≫ 0. (cid:0) (cid:1) 4 M. Blickle, M. Musta¸t˘a and K.E. Smith Note that if λ > µ, then τ(aλ) ⊆ τ(aµ). It is shown in [BMS] that for every λ there is ε > 0 such that τ(aλ) = τ(aλ′) for every λ′ ∈ [λ,λ+ε). A positive λ is called an F-jumping exponent of a if τ(aλ) 6= τ(aλ′) for every λ′ < λ. It is convenient to make the convention that 0 is an F-jumping exponent, too. If S is a multiplicative system in R, then τ((S−1a)λ) = S−1τ(aλ). Similarly, if R is local and R its completion, then τ((aR)λ) = τ(aλ)R. In particular, if λ is an F-jumping exponent for S−1a or for aR, then it has to be an F-jumping exponent also b b b for a. b Using the identification of F-jumping exponents as F-thresholds one shows in [BMS] that if λ is an F-jumping exponent, then so is pλ. Alternatively, this follows from a strengthening of the Subadditivity Theorem in this context (see Proposition 4.1 below and the remark following it). From now on we specialize to the case of a principal ideal a = (f). In this case it is shown in [BMS] that for every λ ≥ 1 we have τ(fλ) = f ·τ(fλ−1). This implies that if λ ≥ 1, then λ is an F-jumping exponent of f if and only if λ−1 is such an exponent. Combining theabovetwo properties,itfollowsthatifλisanF-jumpingexponent for f, then the fractional parts {peλ} are also F-jumping exponents for all e ≥ 1. Hence if we know that the F-jumping exponents of f are discrete, λ has to be rational. Lemma 2.1. If λ = m for some positive integer m, then τ(fλ) = (fm)[1/pe]. pe ′ Proof. By definition, we have τ(fλ) = fmpe′−e [1/pe ] for some e′ ≥ e. Therefore it is (cid:16) (cid:17) enough to show that for every g ∈ R and every ℓ ≥ 1 we have (gp)[1/pℓ+1] = g[1/pℓ]. This in turn follows from the flatness of the Frobenius morphism: for an ideal J, we have g ∈ J[pℓ] if and only if gp ∈ J[pℓ+1]. (cid:3) WerecallnowsomebasicfactsaboutR[Fe]-modulesandD -modules. Fordetails R we refer to [Lyu] or [Bli]. Since R is an F-finite regular ring, the ring of differential operators DR ⊆ EndFp(R) admits the following description. For every e ≥ 0, let DRe = EndRpe(R), hence DR0 = R. We have DRe ⊆ DRe+1 and D = De. R R e[∈N Bydefinition,R hasacanonicalstructureofleftD -module.Notealso thatifS is R a multiplicative system inR, thenwe have a canonical isomorphism S−1(De) ≃ De . R S−1R The following lemma is a concrete special case of so called Frobenius descent (see F-thresholdsof hypersurfaces 5 [AMBL] for a fast introduction) which states that the Frobenius functor induces an equivalence of the category of R–modules and De–modules. In this explicit case it R shows the relevance of De-modules in our setting. R Lemma 2.2. The De-submodules of R are the ideals of the form J[pe] for some ideal J. R In particular, for every b, the ideal b[1/pe] [pe] is equal to the De-submodule generated R by b. (cid:0) (cid:1) Proof. By definition, if P ∈ De and a,b ∈ R, then P(apeb) = apeP(b). This implies R that every ideal of the form J[pe] is a De-submodule of R. R Conversely, suppose that I is such a submodule, and let J = {a ∈ R | ape ∈ I}. We clearly have J[pe] ⊆ I, and we show that equality holds. If q is a prime ideal in R, then J = {b ∈ R | bpe ∈ I } and (J[pe]) = (J )[pe]. Since I is a De -submodule of q q q q q q Rq R , it follows that it is enough to prove that I = J[pe] when R is local. Hence we may q assume that R is free (and finitely generated) over Rpe. If u ,...,u give a basis of R over Rpe, then we get morphisms P : R −→ R 1 N i that are Rpe-linear by mapping u = N apeu to ape. It follows that if u ∈ I, then i=1 i i i P (u) = ape ∈ I for every i, hence a ∈PJ and we have u ∈ J[pe]. (cid:3) i i i We denote by Re the R-R-bimodule on R, with the left structure being the usual one, and the right one being induced by the eth composition of the Frobenius morphism Fe: R −→ R. We use the scheme-theoretic notation for extension of scalars via Fe: if M is an R-module, then we denote by Fe∗M the R-module Re⊗ M. We have a canonical R isomorphism Re ⊗ Re′ ≃ Re+e′ that takes a⊗b to abpe. R The ring R[F] is the noncommutative ring extension of R generated by a variable F such that Fa = apF for every a ∈ R. For every e ≥ 1 we consider also the subring R[Fe] ⊆ R[F].AnR[Fe]-moduleishence nothingbut anR-moduleM togetherwithan “action of the eth composition of the Frobenius on M”, that is a group homomorphism Fe = Fe : M −→ M suchthatFe(au) = apeu(ormoreconcisely: Fe isanR-linearmap M M M −→ FeM). Due to the adjointness of Fe∗ and Fe this can be rephrased as follows: ∗ ∗ M is an R-module together with a morphism of left R-modules ϑe : Re ⊗ M −→ M. M R The adjointness is expressed through the equation ϑe (a⊗u) = aFe(u). M Aunit R[Fe]-module isanR[Fe]-moduleM suchthatϑe isanisomorphism.Note M that for every s ≥ 1, the inclusion R[Fse] ⊆ R[Fe] makes any (unit) R[Fe]-module into a (unit) R[Fse]-module. Moreover, ϑse can be described recursively as M 1⊗ϑ(s−1)e ϑe Rse ⊗ M ≃ Re ⊗ (R(s−1)e ⊗ M) −−−−M−−−→ Re ⊗ M −−−M→ M. R R R R Every unit R[Fe]-module M has a canonical structure of D -module. This is R describedasfollows:sinceD = Dse,itisenoughtodescribetheactionofP ∈ Dse R s≥1 R R S 6 M. Blickle, M. Musta¸t˘a and K.E. Smith onM.Usingtheisomorphismϑse: Rse⊗ M −→ M,weletP actbyP(a⊗u) = P(a)⊗u. M R A fundamental result of Lyubeznik [Lyu] says that if R is an algebra of finite type over a regular local F-finite ring, then every finitely generated unit R[Fe]-module has finite length in the category of D -modules. R It is a general fact that for every R-module P and every e ≥ 1, the pull-back Fe∗(M) has a natural structure of D -module. Moreover, if P is a unit R[Fe]-module, R then ϑe : Fe∗(P) −→ P is an isomorphism of D -modules. For a discussion of this and P R related facts we refer to [AMBL], §2. For simplicity, from now on we assume that R is a domain. A basic example of an R[F]-module is given by R , where f ∈ R is nonzero. The action of F on R is given f f by the Frobenius morphism of R . It is easy to see that R is a unit R[F]-module. In f f fact, we will check this for the following generalization. Suppose that α is a positive rational number such that p does not divide the denominator of α. Therefore we can find positive integers e and r such that α = r . pe−1 We define the R[Fe]-module M as being the R -free module with generator e . α f α We think of e formally as 1 . Since peα = r+α, this suggests the following action of α fα Fe on M : α b bpe Fe ·e = ·e . (cid:18)fm α(cid:19) fmpe+r α It is clear that this makes M an R[Fe]-module. α Lemma 2.3. For every α as above, M is a unit R[Fe]-module. α Proof. It follows from definition that the morphism ϑe : Re⊗ M −→ M is given by Mα R α α b abpe ϑe a⊗ e = e . Mα(cid:18) fm α(cid:19) fmpe+r α It is straightforward to check that the map c e −→ cfs(pe−1)+r ⊗ 1 e is well-defined fs α fs α and that it is an inverse of ϑe . (cid:3) Mα Remark 2.4. If e′ = es for some positive integer s, then we may write r r′ (1) α = = , pe −1 pe′ −1 ′ with r′ = r · pe −1. Since pe−1 1 1 (Fe)s(e ) = e = e , α fr(1+pe+···+p(s−1)e) α fr′ α we see that the action of Fe′ on e is the same for both ways of writing α in (1). In α particular, the D -module structure on M depends only on α. R α F-thresholdsof hypersurfaces 7 The following lemma relates the module M to some test ideals of f. If α = r α pe−1 as above and m ∈ N, we put α := pme−1 ·α. Hence the α form a strictly increasing m pme m sequence converging to α. Lemma 2.5. With the above notation, the following are equivalent: (i) τ(fαm) = τ(fαm+1). (ii) There is a differential operator P ∈ D(m+1)e such that P ·e = Fe(e ). R α α Proof. It follows from Lemma 2.1 that we have τ(fαm) = frpe(ppem−e1−1) [1/p(m+1)e], andτ(fαm+1) = frp(mp+e−1)1e−1 [1/p(m+1)e]. (cid:18) (cid:19) (cid:16) (cid:17) Therefore Lemma 2.2 implies τ(fαm)[p(m+1)e] = D(m+1)e ·frpe(ppem−e1−1) andτ(fαm+1)[p(m+1)e] = D(m+1)e ·frp(mp+e−1)1e−1. R R We always have τ(fαm+1) ⊆ τ(fαm) for every m. It follows from the above for- (m+1)e mulas that these ideals are equal if and only if there is P ∈ D such that R (2) frpe(ppem−e1−1) = P ·frp(mp+e−1)1e−1. We claim that this is the case if and only if P ·e = 1 e in M . Note first that since α fr α α (m+1)e P ∈ D , it follows from the description of the action of D on M that R R α P ·e = ϑ(m+1)e(P ⊗1)(ϑ(m+1)e)−1(e ). α Mα Mα α The formula for (ϑe )−1 in the proof of Lemma 2.3 implies that Mα (ϑM(mα+1)e)−1(eα) = frp(mp+e−1)1e−1 ⊗eα and therefore (ϑM(mα+1)e)−1(P ·eα) = P (cid:18)frp(mp+e−1)1e−1(cid:19)⊗eα = frp(m+1)e ·P (cid:18)frp(mp+e−1)1e−1(cid:19)⊗ f1reα. On the other hand, (ϑM(mα+1)e)−1(cid:18)f1reα(cid:19) = frpe(p(mpe+−11)e−1) ⊗ f1reα, hence P ·e = 1 e if and only if α fr α frp(m+1)eP frp(mp+e−1)1e−1 = frpe(p(mpe+−11)e−1), (cid:18) (cid:19) (cid:3) which is equivalent to (2). This completes the proof of the lemma. 8 M. Blickle, M. Musta¸t˘a and K.E. Smith Remark 2.6. Since we have D(m+1)e ⊆ D(m+2)e, it follows from the lemma that if R R τ(fαm) = τ(fαm+1), then τ(fαi) = τ(fαm) for every i ≥ m. In other words, there is no F-jumping exponent in (α ,α). See also Proposition 4.3 below for an alternative m proof of this statement. Remark 2.7. Using the language of roots and generators of finitely generated unit R[Fe]-modules as in [Lyu] one can easily show that M is naturally isomorphic to the α unit R[Fe]-module generated by β : R −−a7→−−fr−·a−⊗−1→ Re ⊗R ∼= Fe∗R . The unit module generated by β is by definition the inductive limit M of the direct α system one obtains by composition of Frobenius powers of the map β. As an R-module, f M is isomorphic to R but the action of the Frobenius is not the usual one (except α f in the case r = pe −1). One easily checks (by sending the image of 1 ∈ R in the limit f M to e ∈ M ) that M and M are isomorphic as R[Fe]-modules. By construction it α α α α α follows that Re ⊆ M is a root of M . Therefore [AMBL, Corollary 4.4] implies that α α α f f e generates M as a D -module. In Theorem 2.11 below we will give a direct proof α α R of this fact. Corollary 2.8. If α = r , then e generates M as a D -module if and only if α is pe−1 α α R not an accumulation point of F-jumping exponents of f. Proof. The α form a strictly increasing sequence converging to α, hence α is not m an accumulation point of F-jumping exponents if and only if the sequence of ideals {τ(fαm)} stabilizes. By Remark 2.6, this is the case if and only if τ(fαm) = τ(fαm+1) m for some m. Since D = ∪ Dme, it is clear from Lemma 2.5 that if M = D ·e , R m≥1 R α R α then α is not an accumulation point of F-jumping exponents. Conversely, if this is the case, then 1 e ∈ D ·e . By Remark 2.4, we see that in fact we have infinitely many fr α R α positive integers r such that 1 e lies in D ·e . Since these elements generate M m frm α R α α as an R-module, we see that M = D ·e . (cid:3) α R α Corollary 2.9. If α = r , then α is not an accumulation point of F-jumping expo- pe−1 nents of f ∈ R if and only if for every q ∈ Spec(R), α is not an accumulation point of F-jumping exponents of f ∈ R . 1 q Proof. We have seen that α is not an accumulation point of F-jumping exponents of f if and only if τ(fαm) = τ(fαm+1) for some m. Since taking test ideals commutes with localization, it is clear that if this property holds in R, then it holds in every R . For q the converse, note that if (3) τ((fR )αm) = τ((fR )αm+1), q q then the same holds for all primes q′ is a neighborhood of q. If U is the open subset m consisting of those q for which (3) holds, and if Spec(R) = ∪ U , then Spec(R) = U m m m0 F-thresholdsof hypersurfaces 9 for some m (we use the fact that Spec(R) is quasicompact and that U ⊆ U by 0 m m+1 Remark 2.6). This implies that τ(fαm0) = τ(fαm0+1), hence α is not an accumulation point of F-exponents of f. (cid:3) Remark 2.10. ItiseasytoseethatM isgeneratedbye asanR[Fe]-module.Indeed, α α M is generated as an R-module by the (Fe)m(e ) = 1 ·e , with m ≥ 1. α α frppmee−−11 α Theorem 2.11. Let R be an F-finite regular domain. If f is a nonzero element in R and α = r for some positive integers r and e, then M is generated over D by e . pe−1 α R α Proof. Note first that by Corollary 2.9, we may assume that R is local. Then the argument follows verbatim the argument for Theorem 4.1 in [AMBL]. Let N denote the D -submodule of M generated by e . Note that we have R α α ϑe Fe∗(N) ⊆ Fe∗(M ) −−−M−α→ M , α α hence we may consider Fe∗(N) as a submodule of M . We claim that N ⊆ Fe∗(N). α Since Fe∗(N) is a D -submodule of M , it is enough to show that e ∈ Fe∗(N). This R α α follows from e = fr ·ϑe (1⊗e ). α Mα α Theorem 4.3 in [AMBL] shows that this makes N a unit R[Fe]-module, i.e. we have in fact N = Fe∗(N). The idea is the following: if N 6= (F∗)(N), then the faithfull flatness of the Frobenius implies that we have a sequence of strict inclusions N ( Fe∗(N) ( (F2e)∗(N) ( ··· of D -submodules of M . This contradicts Lyubeznik’s Theorem [Lyu] saying that as R α a unit R[Fe]-module, M has finite length in the category of D -modules (we may α R apply the theorem, since we assume that R is local and F-finite). Therefore we have N = (Fme)∗(N) forevery m.Onthe other hand,every element in M lies in some (Fme)∗(N). This follows from α 1 e = ϑme(1⊗e ) ∈ (Fme)∗(N). frp(m−1)e α Mα α Therefore N = M . (cid:3) α Remark 2.12. Byputting togethertheaboveresults, weseethatunder thehypothesis of Theorem 2.11, every rational number α whose denominator is not divisible by p is not an accumulation point of F-jumping exponents of a given f. The above proofs extend to this setting the main result in [AMBL] which deals with the case α = 1. In addition, we have dropped the extra assumption that was imposed in loc. cit. in order to apply Lyubeznik’s Theorem. Before we proceed to the proof of Theorem 1.1 we show the following lemma which allows us to do induction. Note that this lemma itself does not require the ideal to be principal. 10 M. Blickle, M. Musta¸t˘a and K.E. Smith Lemma 2.13. Let R be an F-finite regular ring, and a an ideal in R. (i) If λ is an F-jumping exponent of a, then there is a prime ideal q in R such that λ is an F-jumping exponent also of aR . q (ii) If λ is an accumulation point of jumping numbers of a, then we can find a non-maximal prime ideal q such that λ is an F-jumping exponent of aR . q Proof. We may assume that λ > 0, and let us fix a strictly increasing sequence of posi- tive numbers {λm}m, with limm−→∞λm = λ. For every m we have τ(aλ) ⊆ τ(aλm+1) ⊆ τ(aλm). Let I be the ideal m (τ(aλ): τ(aλm)) = {h ∈ R | h·τ(aλm) ⊆ τ(aλ)}. Therefore I ⊆ I for every m, and since R is Noetherian, there is an ideal I such m m+1 that I = I for all m ≫ 0. m Note that λ is an F-jumping exponent of a if and only if for every m we have τ(aλ) 6= τ(aλm), or equivalently, I 6= R. Moreover, λ is an accumulation point of m F-jumping exponents if and only if τ(aλm) 6= τ(aλm+1) for every m. If λ is an F-jumping exponent of a, let q be a minimal prime containing I. Since (τ((aR )λ): τ((aR )λm)) = (τ(aλ)R : τ(aλm)R ) = I 6= R , q q q q q q it follows that λ is an F-jumping exponent of aR . This gives i). q We show now that if λ is an accumulation point of F-jumping exponents, then we can find q as above that is not a maximal ideal. Equivalently, we need to show that dim(R/I) ≥ 1, i.e. R/I is not Artinian. By assumption, if m ≫ 0 then we have a strictly decreasing sequence τ(aλm)/τ(aλ) ) τ(aλm+1)/τ(aλ) ) ... of finitely generated R/I-modules. Therefore R/I cannot be Artinian, and we get (cid:3) ii). Proof of Theorems 1.1. Since R is a regular ring, we may write R = R × ...× R , 1 m where R are regular domains. If we write f = (f ,...,f ), then the set of F-jumping i 1 m exponents of f is the union of the sets of F-jumping exponents of each of the f . i Therefore in order to prove Theorem 1.1 for R, we may assume that R is a domain and that f 6= 0, the case f = 0 being trivial. We have seen that for every R and f, the discreteness of the set of F-jumping exponents implies the rationality of every such exponent. Conversely, if we know that all F-jumping exponents are rational, then they form a discrete set. Indeed, if α is an accumulation point of F-jumping exponents, then α is an F-jumping exponent, too, hence α ∈ Q. We can find a positive integer m such that the denominator of pmα is not divisible by p. For every F-jumping exponent β, pmβ is again an F-jumping exponent.