aa r X i v : . [ m a t h . A C ] F e b F -PURITY OF HYPERSURFACES DANIEL J. HERN ´ANDEZ
Abstract.
Motivated by connections with birational geometry over C , the theory of F -purity for rings of positive characteristic may be extended to a theory of F -purity for “pairs”[HW02]. Given an element f of an F -pure ring of positive characteristic, this extensionallows us to define the F -pure threshold of f , denoted fpt p f q . This invariant measuresthe singularities of f , and may be thought of as a positive characteristic analog of thelog canonical threshold, an invariant that typically appears in the study of singularities ofhypersurfaces over C . In this note, we study F -purity of pairs, and show (as is the casewith log canonicity over C ) that F -purity is preserved at the F -pure threshold. We alsocharacterize when F -purity is equivalent to sharp F -purity, an alternate notion of purityfor pairs introduced in [Sch08]. These results on purity at the threshold generalize resultsappearing in [Har06, Sch08], and were expected to hold by many experts in the field. Weconclude by extending results in [BMS09] on the set of all F -pure thresholds to the mostgeneral setting. Introduction
Let R be a domain of characteristic p ą
0. The e th -iterated Frobenius map R F e Ñ R (defined by r ÞÑ r p e ) is a ring homomorphism whose image is the subring R p e Ď R consistingof all p p e q th powers of elements of R . The Frobenius map has been an important tool incommutative algebra since Kunz characterized regular rings as those for which R is flatover R p e [Kun69]. In general, singular rings exhibit pathological behavior with respect toFrobenius, and by imposing conditions on the structure of R as an R p e -module, new classesof singularities can be defined. For example, we say that R is F -finite if R is a finitelygenerated R p e module for every (equivalently, for some) e ě
1. We call
R F -pure (or F -split ) if the inclusion R p e Ď R splits as a map of R p e -modules for all (equivalently, for some) e ě
1. [HR76]. The notion of F -purity is a critical ingredient in the proof of the well-knownHochster-Roberts Theorem on the Cohen-Macaulay property of rings of invariants [HR74].By modifying the condition that R p e Ď R splits over R p e , one may extend the notionof purity for rings to a more general setting. A pair , denoted p R, λ ‚ f q , consists of thecombined information of an ambient ring R , a non-zero, non-unit element f P R , and anon-negative real parameter λ . We say that the pair p R, λ ‚ f q is F -pure if the inclusion R p e ¨ f t p p e ´ q λ u Ď R splits as a map of R p e -modules for all e " R p e ¨ f N denotesthe R p e -submodule of R generated by f N . The purity condition for pairs encapsulates theclassical one, as the ring R is F -pure if and only if the pair p R, ‚ f q is F -pure.Though technical, this extension adds great flexibility to the theory, and allows one todefine F -pure thresholds . The F -pure threshold of f , denoted fpt p f q , is the supremum over The author was partially supported by the National Science Foundation RTG grant number 0502170 atthe University of Michigan. ll λ ě p R, λ ‚ f q is F -pure. This definition is analogous to that of the log canonical threshold in complex algebraic geometry, which we now briefly recall.Let S be a regular ring of finite type over C , and let g be any non-zero, non-unit element of S . Via (log) resolution of singularities, one may define the notion of log canonical singularities for pairs p S, λ ‚ g q [Laz04, BL04]. When the ambient space is a polynomial ring over C , wehave the following concrete description: p C r x s , λ ‚ g q is said to be log canonical if for every0 ď ε ă λ , the real-valued function | g | ε is locally integrable in a neighborhood of every point.We define the log canonical threshold of g , denoted lct p g q , to be the supremum over all λ ě p S, λ ‚ g q is log canonical. The following theorem, one of many relating singularitiesdefined over C with those in positive characteristic, illustrates the tight relationship between F -purity and log canonical singularities. Theorem. [HW02]
Let g P S be as above, and let g p P S p denote the reduction of g and S to prime characteristic p ą ; see [Smi97] for a concrete discussion of this process. If p S p , λ ‚ g p q is F -pure for infinitely many p , then p S, λ ‚ f q is log canonical. The converse is conjectured to hold; see [Her11c, Tak11] for recent positive results.In [Sch08], we are introduced to an alternate notion of purity for pairs called sharp F -purity . A pair p R, λ ‚ f q is said to be sharply F -pure if R p e ¨ f r p p e ´ q λ s Ď R splits for some e ě
1, and the above-mentioned theorem holds after replacing “ F -pure” with “sharply F -pure” [Sch08]. Given the close ties between (sharp) F -purity and log canonical singularities,one is motivated to ask whether certain properties of log canonicity also hold for (sharp) F -purity in the positive characteristic setting.One such property of log canonical singularities, which follows essentially from its defi-nition, is that p S, lct p g q ‚ g q must be log canonical. We show that the analogous property,though not an immediate consequence of the definitions involved, also holds for F -purity.The situation for sharp F -purity is slightly more complicated. Theorem 4.9. If R is an F -pure ring of characteristic p ą , then p R, fpt p f q ‚ f q is F -pure,and is sharply F -pure if and only if p p e ´ q ¨ fpt p f q P N for some e ě . The condition that p p e ´ q ¨ fpt p f q P N appearing above is equivalent to the condition that fpt p f q be a rational number whose denominator not divisible by p . While it is known that fpt p f q P Q in many cases (see [BMS08, BMS09, KLZ09, BSTZ09]), explicit computationsof F -pure thresholds appearing in the literature show that very often the denominator of fpt p f q is divisible by p [Her11a, Her11b]. Thus, there are many instances in which F -purityand sharp F -purity are not equivalent. However, from the point of view of computations of F -pure thresholds of hypersurfaces (especially those reduced from characteristic zero), thecondition that p p e ´ q ¨ fpt p f q P N has many desirable consequences. We emphasize thatTheorem 4.9 holds assuming only that the ambient ring is F -pure, which is the minimalassumption needed to study F -purity of pairs. We also note that Theorem 4.9 generalizesresults appearing in [Har06, Sch08], in which the ambient ring is assumed to be an F -finite,(complete) regular local ring.If g again denotes an element of a regular ring of finite type over C , it follows easily from thedefinition of log canonicity (in terms of resolution of singularities) that lct p g q P Q X r , s .Furthermore, every number in Q X r , s may be realized as a log canonical threshold: if rs P Q X r , s , then rs “ lct p x s ` ¨ ¨ ¨ ` x sr q [How01a, Example 3], [How01b, Example 9]. The ssue of which numbers may be realized as F -pure thresholds is more complicated, and wasfirst considered in [BMS09].As with log canonical thresholds, it is easy to verify (e.g., see Lemma 1.5) that the F -purethreshold of a hypersurface is contained in r , s . If one considers only ambient rings that are F -finite and regular of a fixed characteristic p ą
0, it was shown in [BMS09, Proposition 4.3]that there exist infinitely many non-empty open intervals contained in Q X r , s that cannotcontain a number of the form fpt p f q ; the arguments given therein rely on the behaviorof F -jumping exponents in F -finite regular rings. In Proposition 4.8, we show that theaforementioned statement still holds if one replaces the condition that the ambient spacesbe F -finite and regular with the minimal requirement that they be F -pure. Proposition 4.8.
Let
FPT p denote the set of all fpt p f q , where f P R ranges over everyelement of every F -pure ring of characteristic p ą ; see Definition 4.5. Then, for every e ě and every β P r , s X p e ¨ N , we have that FPT p X ˆ β, p e p e ´ ¨ β ˙ “ H . Example 0.1.
Proposition 4.8 states that, for every e ě
1, there exist p e ´ r , s that do not intersect FPT p . To better appreciate this condition,consider the intervals corresponding to e “ , FPT . e “ e “ e “
18 14 38 12 58 34 78 Figure:
Intervals corresponding to e “ , e “
2, and e “ FPT .As was first observed in [BMS09], for every e ě ÿ β Pr , sX pe ¨ N length ˆ β, p e p e ´ ¨ β ˙ “ . Thus, Proposition 4.8 states that for every e ě
1, there is a set of Lebesgue measure thatdoes not intersect FPT p . Furthermore, Proposition 4.8 may be used to show that FPT p is a set of Lebesgue measure zero. This is not surprising because, as noted earlier, F -purethresholds are very often rational numbers. However, we stress that the issue of whether FPT p Ď Q remains open in general.The main results appearing in this article are obtained as corollaries of Key Lemma 3.4,which essentially says that all of the relevant “splitting data” for an element f in an F -purering is encoded by the digits appearing in the unique (non-terminating) base p expansionof fpt p f q ; see Definition 2.1 for the definition of a non-terminating base p expansion. Theproof of Key Lemma 3.4 depends mostly on taking p th roots of elements and morphisms (seeDefinition 3.1) in a careful way, and in the case that the ambient space is F -finite and regular,Key Lemma 3.4 is simply a translation of [MTW05, Proposition 1.9] into the language ofnon-terminating base p expansions. cknowledgements. This note is part of the author’s Ph.D thesis, which was completedat the University of Michigan under the direction of Karen Smith. The author would liketo thank Karen Smith, Mel Hochster, Emily Witt, Luis Nu r nez Betancourt, Daisuke Hirose,and Michael Von Korff for many enlightening discussions.1. Preliminaries
Let R be a reduced ring of prime characteristic p ą
0, and for e ě
1, let R { p e denote theset of formal symbols t f { p e : f P R u . We define a ring structure on R { p e via f { p e ` g { p e : “p f ` g q { p e and f { p e ¨ g { p e : “ p f g q { p e . As R is reduced, we have an inclusion R Ď R { p e given by r ÞÑ p r p e q { p e . If R is a domain, then R { p e admits a more concrete description: Let L denote the algebraic closure of the fraction field of R , and for every f P R , let f { p e denote theunique root of the equation T p e ´ f P L r T s in L . We may then describe R { p e as the subringof L consisting of all p p e q th -roots of elements of R . For example, if R “ F p r x , ¨ ¨ ¨ , x m s ,then R { p e “ F p r x { p e , ¨ ¨ ¨ , x { p e m s . Via the inclusion (of rings) R { p d Ď R { p e ` d given by r { p d “ p r p e q { p e ` d , we may identify p R { p d q { p e and R { p e ` d as R { p d -algebras.Let R F e Ñ R denote the e th iterated Frobenius morphism defined by r ÞÑ r p e , and let F e ˚ R denote R when considered as an R -algebra via F e . If R p e “ r p e : r P R ( is thesubring consisting of p p e q th powers of R , then the R -algebra structure of F e ˚ R and the R p e -algebra structure of R are isomorphic. We also note that F e ˚ R – R { p e as R -modules viathe isomorphism r ÞÑ r { p e . This map and its inverse are often referred to as “taking p p e q th roots” and “raising to p p e q th powers.” We say that R is F -finite if R { p (equivalently, F ˚ R )is a finitely-generated R -module. One can show that R is F -finite if and only if R { p e (equivalently, F e ˚ R ) is a finitely-generated R -module for some (equivalently, for all) e ě S Ă T is an inclusion of rings, and S ¨ t is the S -submodule of T generated by t P T ,we say that the inclusion S ¨ t Ď T splits over S (or splits as a map of S -modules) if thereexists a map θ P Hom S p T, S q with θ p t q “
1. Recall that R is said to be F -pure (or F -split )if the inclusion R “ R ¨ Ď R { p splits over R . An F -pure ring is necessarily reduced, and R is F -pure if and only if the inclusion R Ď R { p e splits as a map of R -modules for some(equivalently, for all) e ě g P R , note that the inclusion R ¨ g { p e Ď R { p e splitsover R if and only if R p e ¨ g Ď R splits over R p e if and only if R ¨ g Ď F e ˚ R splits over R . Asa matter of taste, we use the language of p p e q th roots when discussing such splittings.1.1. F -purity for pairs and F -pure thresholds.Definition 1.1. A pair p R, λ ‚ f q consists of the combined information of an ambient ring R , a hypersurface f P R , and a non-negative real number λ . Universal Hypothesis 1.2.
For the remainder of this article, R will be assumed to be an F -pure ring of characteristic p ą , and f will be assumed to be a non-zero, non-unit in R . In what follows, r α s (respectively, t α u ) will denote the least integer greater than or equalto α (respectively, the greatest integer less than or equal to α .) Definition 1.3. [Tak04, TW04, Sch08] The pair p R, λ ‚ f q is said to be(1) F -pure if R ¨ f t p p e ´ q λ u { p e Ď R { p e splits over R for all e ě strongly F -pure if R ¨ f r p e λ s { p e Ď R { p e splits over R for some e ě
1, and(3) sharply F -pure if R ¨ f r p p e ´ q λ s { p e Ď R { p e splits over R for some e ě emma 1.4 shows that strong F -purity implies sharp F -purity. We also have that sharp F -purity implies F -purity [Sch08] (though the converse need not be true; see Example 4.10). Lemma 1.4.
If the inclusion R ¨ f N { p e Ď R { p e splits as a map of R -modules, so must theinclusion R ¨ f a { p e Ď R { p e for all ď a ď N .Proof. Choose θ P Hom R p R { p e , R q with θ ` f N { p e ˘ “
1, and let φ denote the R -linear en-domorphism of R { p e given by multiplication by f N ´ ape . Then θ ˝ φ P Hom R p R { p e , R q and p θ ˝ φ q ` f a { p e ˘ “ (cid:3) Lemma 1.5. If p R, λ ‚ f q is F -pure, then so is p R, ε ‚ f q for every ď ε ď λ . Furthermore, p R, λ ‚ f q is not F -pure if λ ą .Proof. The pair p R, ‚ f q is F -pure as R is F -pure, and applying Lemma 1.4 to the inequality t p p e ´ q ε u ď t p p e ´ q λ u shows that p R, ε ‚ f q is F -pure whenever p R, λ ‚ f q is F -pure. Forthe second assertion, suppose that p R, λ ‚ f q is F -pure with λ “ ` ε for some ε ą
0. Bydefinition, we have that(1.1.1) R ¨ f t p p e ´ qp ` ε q u { p e Ď R { p e splits over R for every e ě . For e " , p p e ´ qp ` ε q “ p e ´ ` p p e ´ q ¨ ε ą p e for e "
0. Applying Lemma 1.4to (1.1.1) then shows that R ¨ f p e { p e “ R ¨ f Ď R { p e splits over R for e "
0. We concludethat there exists a map θ P Hom R p R { p e , R q with 1 “ θ p f q “ f ¨ θ p q , contradicting theassumption that f is not a unit. (cid:3) Of course, one can replace F -purity with strong (respectively, sharp) F -purity in Lemma1.5 to obtain analogous statements. Lemma 1.5 shows that F -purity for a given a param-eter implies F -purity for all smaller parameters, and so one may ask: What is the largestparameter for which a pair is F -pure? This leads to the notion of F -pure thresholds, whichwere first defined in [TW04]. It was shown in [TW04, Proposition 2.2] (respectively, [Sch08,Proposition 5.3]) that the F -pure threshold agrees with the “strongly F -pure threshold”(respectively, “sharply F -pure threshold”). We gather these facts in Definition 1.6. Definition 1.6.
The following supremums all agree, and we call their common value the F -pure threshold of f , denoted fpt p R, f q : fpt p R, f q : “ sup λ t p R, λ ‚ f q is F -pure u “ sup λ t p R, λ ‚ f q is strongly F -pure u“ sup λ t p R, λ ‚ f q is sharply F -pure u . We often suppress the ambient ring and write fpt p f q instead of fpt p R, f q .As a corollary of Lemma 1.5, we see that fpt p R, f q P r , s . Remark 1.7.
Note that fpt p f q ą e ě R ¨ f { p e Ď R { p e splits over R . One recognizes that fpt p f q ą R is a strongly F -regular domain [HH90]. Remark 1.8.
The issue of whether one can replace “sup” with “ max” in Definition 1.6 isprecisely the content of Theorem 4.9. . Some remarks on base p expansions In this section, we will consider the base p expansions of real numbers contained in theunit interval. By Lemma 1.5, fpt p f q is such a number, and in the next section we will usethe language developed here to obtain properties of f via fpt p f q . Definition 2.1. If α P p , s , we call the (unique) expression α “ ř e ě a e p e with the propertythat the integers 0 ď a e ď p ´ non-terminating base p expansion of α , and we call a e the e th digit of α .The adjective “non-terminating” in Definition 2.1 is only necessary when α is a rationalnumber with denominator a power of p . For example, p ` ř e ě p ´ p e is the the non-terminatingbase p expansion of p . Definition 2.2.
Consider α P p , s with non-terminating base p expansion α “ ř d ě a d p d .(1) We define the e th truncation of α by x α y e : “ a p ` ¨ ¨ ¨ ` a e p e . By convention, x y e “ e th tail of α by v α w e : “ α ´ x α y e “ ř d ě e ` a d p d . By convention, v w e “ Universal Hypothesis 2.3.
All truncations and tails will be taken with respect to somefixed prime p (which will always be the characteristic of any ambient ring in context). Lemma 2.4.
Let α P p , s . Then the following hold: (1) x α y e P p e ¨ N . (2) x α y e ă α and v α w e ą for all e . (3) v α w e ď p e , with equality if and only if α P p e ¨ N .Proof. These all follow easily from the definitions, keeping in mind that we are dealing withobjects derived from non-terminating expansions. The details are left to the reader. (cid:3)
Lemma 2.5.
Let α P p , s . Then the following hold: (1) r p e α s “ p e x α y e ` . (2) p e x α y e ´ ď t p p e ´ q α u ď p e x α y e . (3) p e x α y e ď r p p e ´ q α s ď p e x α y e ` . (4) If β P r , s X p e ¨ N and α ą β , then x α y e ě β .Proof. By definition, we have that(2.0.2) p e α “ p e x α y e ` p e v α w e . By Lemma 2.4, the first summand in (2.0.2) is an integer, while the second is contained in p , s . The first point follows. Substituting the decomposition of p e α appearing in (2.0.2)into the expression p p e ´ q ¨ α shows that(2.0.3) p p e ´ q ¨ α “ p e x α y e ` p e v α w e ´ α. By Lemma 2.4, both α and p e v α w e are contained in p , s , so that | p e v α w e ´ α | ă
1. Thus,the second and third points follow from (2.0.3). We now prove the last point. By Lemma2.4, we have that p e ` x α y e ě α ą β , and multiplying by p e shows that(2.0.4) 1 ` p e x α y e ą p e β. By assumption, both sides of 2.0.4 are integers, and we conclude that p e x α y e ě p e β . (cid:3) emma 2.6. Let α P p , s . If p p d ´ q ¨ α P N , then x α y ed ` d “ x α y ed ` p ed ¨ x α y d for all e ě .Proof. One can show that if p p d ´ q ¨ α P N , then the digits of the non-terminating base p expansion of α must begin repeating after the d th digit. Once this has been established, it iseasy to see that the p ed ` d q th truncation of α is determined by the ed th and d th truncationsin precisely the prescribed way. The details are left to the reader. (cid:3) Proof of the Key Lemma
The aim of this section is to prove Key Lemma 3.4, which says that all of the relevantsplitting data for f is encoded by the truncations of fpt p f q . The process of taking roots ofmaps is key to the proof of Key Lemma 3.4, so we isolate it in Definition 3.1. Definition 3.1. An R -linear map θ : R { p e Ñ R gives rise, in a natural way, to an R { p d -linear map θ { p d : R { p e ` d Ñ R { p d defined by the rule θ { p d p r { p e ` d q : “ θ p r { p e q { p d . We call θ { p d the p p d q th -root of θ . Remark 3.2. As R is always assumed to be F -pure, R Ď R { p splits over R . By taking p p e q th roots of this splitting, we see that R { p e Ď R { p e ` splits over R { p e , and a slight modificationof this argument shows that R { p e Ď R { p d splits over R { p e for every d ą e . Lemma 3.3.
Let α P r , s X p d ¨ N . If the inclusion R ¨ f r p e α s { p e Ď R { p e splits over R forsome e ě , then so must the inclusion R ¨ f α Ď R { p d .Proof. Fix θ P Hom R p R { p e , R q with the property that θ p f r p e α s { p e q “
1. If e ě d , then α P p d ¨ N Ď p e ¨ N , and so f α P R { p d Ď R { p e . It is then clear that f r p e α s { p e “ f α maps to 1under the composition R { p d Ď R { p e θ ÝÑ R .Instead, suppose that d ą e , so that R { p e Ď R { p d . Note that p e α ď r p e α s , so α ď r p e α s { p e .By Lemma 1.4, it suffices to show that there exists an R -linear map R { p d Ñ R sending f r p e α s { p e to 1. By Remark 3.2, there exists an R { p e -linear map φ : R { p d Ñ R { p e with φ p q “
1. Note that the R { p e -linearity of φ implies that φ ` f r p e α s { p e ˘ “ f r p e α s { p e . Thus, if Θdenotes the composition R { p d φ Ñ R { p e θ ÝÑ R , then Θ ` f r p e α s { p e ˘ “ (cid:3) Key Lemma 3.4. p d x fpt p f qy d “ max ! a P N : R ¨ f a { p d Ď R { p d splits over R ) .Proof. Set λ : “ fpt p f q and ν f p p d q : “ max ! a P N : R ¨ f a { p d Ď R { p d splits over R ) . As R is F -pure and f is a non-unit of R , we have that 0 ď ν f p p d q ď p d ´
1. If λ “
0, then ν f p p d q “ λ P p , s . By Lemma 2.4, x λ y d ă λ ,so it follows from Definition 1.6 that p R, x λ y d ‚ f q is strongly F -pure, and hence there exists e ě R ¨ f r p e x λ y d s { p e Ď R { p e splits. By Lemma 3.3, we may assume that e “ d ,and it follows that p d x λ y d ď ν f p p d q .We now show that the inclusion R ¨ f x λ y d ` pd Ď R { p d never splits over R . By Lemma 2.5, p d x λ y d ` “ P p d λ T , so it suffices to show that θ ´ f r p d λ s { p d ¯ ‰ θ P Hom R p R { p d , R q .If λ R p d ¨ N , then P p d λ T “ P p d p λ ` ε q T for 0 ă ε !
1. By Definition 1.6, λ is also the strongly F -pure threshold,” and it follows that θ ´ f r p d λ s { p d ¯ “ θ ´ f r p d p λ ` ε q s { p d ¯ ‰ θ P Hom R p R { p d , R q .Now, suppose that λ P p d ¨ N , so that P p d λ T { p d “ λ . By way of contradiction, supposethat θ ` f λ ˘ “ θ P Hom R p R { p d , R q . Note that 0 ‰ λ ě p d , so by Lemma 1.4,there exists an R -linear map R { p d Ñ R sending f { p d to 1. Taking p p d q th roots of this mapproduces an R { p d -linear map φ : R { p d Ñ R { p d with the property that φ p f { p d q “
1. Underthe composition R { p d φ ÝÑ R { p d θ ÝÑ R , it follows from the R { p d -linearity of φ that f λ ` p d “ f λ ¨ f { p d ÞÑ f λ ¨ φ p f { p d q “ f λ ¨ ÞÑ θ p f λ q “ . We see that R ¨ f λ ` p d Ď R { p d splits, contradicting the definition of λ “ fpt p f q . (cid:3) Remark 3.5.
Key Lemma 3.4 allows us to give a definition of fpt p f q which makes noreference to F -purity of pairs, which we now briefly describe. Set a “ max t a P N : R ¨ f a { p Ď R { p splits u . As R is F -pure and f is a non-unit, 0 ď a ď p ´
1. Inductively, define a e ` “ max t a P N : R ¨ f a { p `¨¨¨` a e { p e ` a { p e ` Ď R { p e ` splits as a map of R -modules u . Again, one may verify that 0 ď a e ď p ´
1. Furthermore, by taking p th roots of maps as inthe proof of Key Lemma 3.4, one may show that the a e are not all eventually zero as longas one a e ‰ fpt p f q ‰ fpt p f q “ ř e ě a e p e is the non-terminating base p expansion of fpt p f q .3.1. Some consequences of Key Lemma 3.4.
In this subsection, we gather some corol-laries of Key Lemma 3.4 that we will utilize in later sections.
Lemma 3.6.
Let α P r , s with p p d ´ q ¨ α P N . If R ¨ f x α y d Ď R { p d splits over R , then sodoes R ¨ f x α y ed Ď R { p ed for every e ě .Proof. We induce on e , the base case being our hypothesis. Suppose R ¨ f x α y d Ď R { p d and R ¨ f x α y ed Ď R { p ed split as maps of R -modules, so that there exist(1) an R -linear map R { p d Ñ R with f x α y d ÞÑ
1, and(2) an R -linear map θ : R { p ed Ñ R with θ p f x α y ed q “ R ¨ f x α y ed ` d Ď R { p ed ` d splits as a map of R -modules. By taking p p ed q th -rootsof the map in (1), we obtain(3) an R { p ed -linear map φ : R { p ed ` d Ñ R { p ed with φ ˆ f x α y dped ˙ “ x α y ed ` d “ x α y ed ` x α y d p ed for all e ě
1, and it follows that(3.1.1) f x α y ed ` d “ f x α y dped ¨ f x α y ed . Under the composition R { p ed ` d φ ÝÑ R { p ed θ ÝÑ R , it follows from (3.1.1) and the R { p ed -linearity of φ , that f x α y ed ` d “ f x α y dped ¨ f x α y ed ÞÑ f x α y ed ¨ φ ˆ f x α y dped ˙ “ f x α y ed ¨ ÞÑ θ p f x α y ed q “ (cid:3) orollary 3.7. Let α P r , s with p p d ´ q¨ α P N . If R ¨ f x α y d Ď R { p d splits, then α ď fpt p f q .Proof. By Lemma 3.6, R ¨ f x α y ed Ď R { p ed splits for all e ě
1. By Key Lemma 3.4, we havethat x α y ed ď x fpt p f qy ed for every e ě
1, and taking the limit as e Ñ 8 gives the desiredinequality. (cid:3)
Corollary 3.8. fpt p f q “ if and only if R ¨ f p p ´ q{ p Ď R { p splits as a map of R -modules. Inparticular, if R is F -finite, regular and local, then R {p f q is F -pure if and only if fpt p f q “ .Proof. If fpt p f q “
1, then x fpt p f qy “ p ´ p , and so R ¨ f p p ´ q{ p Ď R { p splits over R by KeyLemma 3.4. On the other hand, if R ¨ f p p ´ q{ p Ď R { p splits over R , then it follows fromsetting α “ d “ fpt p f q ě
1, while fpt p f q ď m denote the maximal ideal of R . That R {p f q is F -pureif and only if f p ´ R m r p s is known as Fedder’s Criteria [Fed83, Proposition 2.1]. On theother hand, f p ´ R m r p s if and only if f p p ´ q{ p R m ¨ R { p . Since R is regular and F -finite, R { p is a finitely generated free R -module [Kun69], and so Nakayama’s lemma shows that f p p ´ q{ p R m ¨ R { p if and only if R ¨ f p p ´ q{ p Ď R { p splits over R . (cid:3) The main results
In this section, we prove our main results, Theorem 4.9 and Proposition 4.8.4.1.
More remarks on base p expansions. We begin with a few straightforward lemmasregarding non-terminating base p expansions that will be used later to impose conditions onthe set of all F -pure thresholds. Definition 4.1. If α P r , s , let x α y e “ ř d ě x α y e p ed “ x α y e ¨ ř d ě p ed “ x α y e ¨ p e p e ´ . Remark 4.2.
Note that x α y e is the rational number whose non-terminating base p expansionis obtained by “repeating” the first e digits of the non-terminating base p expansion of α . Lemma 4.3. If α P r , s , then the following hold: (1) A x α y e E e “ x α y e . (2) p e x α y e “ p p e ´ qx α y e .Proof. These assertions are intuitively clear given the description in Remark 4.2, and thetask of verifying the details is left to the reader. (cid:3)
Lemma 4.4.
Let α P r , s . For e ě , the following conditions are equivalent: (1) t p p e ´ q α u “ p e x α y e . (2) α ď p e v α w e . (3) α ě x α y e . Lemma 4.4 is closely related to Lemma 2.5, and its proof proceeds along the same lines.
Proof.
We will assume that α ą
0. By definition, α “ x α y e ` v α w e , and it follows that(4.1.1) p p e ´ q α “ p e x α y e ` p e v α w e ´ α. emma 2.4 shows that both α and p e v α w e are contained in p , s , so that | p e v α w e ´ α | ă p e x α y e “p p e ´ qx α y e , and substituting this into p . . q shows that(4.1.2) p p e ´ q ´ α ´ x α y e ¯ “ p e v α w e ´ α. From (4.1.2), we see that (2) holds if and only if (3) holds. (cid:3)
The set of all F -pure thresholds.Definition 4.5. Let
FPT p denote the set of all characteristic p ą F -pure thresholds: FPT p : “ t fpt p R, f q : f is a non-zero, non-unit in an F -pure ring R of characteristic p u .We stress that both the ambient ring R and the element f P R are allowed to vary inDefinition 4.5. By Lemma 1.5, we have that FPT p Ď r , s . Proposition 4.6.
For any λ P FPT p and e ě , we have that x λ y e ď λ .Proof. There exists an F -pure ring R and an element f P R such that λ “ fpt p R, f q . Set α : “ x λ y e . By Lemma 4.3, we have that(1) x α y e “ A x λ y e E e “ x λ y e , and(2) p p e ´ q ¨ α “ p e x λ y e P N . By Key Lemma 3.4 and p q , the inclusion R ¨ f x α y e “ R ¨ f x λ y e Ď R { p e splits as a map of R -modules. Corollary 3.7 and p q then imply that λ ě α . (cid:3) Corollary 4.7.
For any e ě and λ P FPT p , the following hold and are equivalent: (1) t p p e ´ q λ u “ p e x λ y e . (2) λ ď p e v λ w e . (3) λ ě x λ y e .Proof. The third point is Proposition 4.6, and all points are equivalent by Lemma 4.4. (cid:3)
Corollary 4.7 places severe restrictions on the set
FPT p , as we see in Proposition 4.8. Proposition 4.8.
For every e ě and β P r , s X p e ¨ N , we have that FPT p X ˆ β, p e p e ´ ¨ β ˙ “ H . Proposition 4.8 generalizes [BMS09, Proposition 4.3], in which R is assumed to be an F -finiteregular ring. Proof.
Let λ P FPT p . If λ ą β , then x λ y e ě β by Lemma 2.5. Combining this withProposition 4.6 and Definition 4.1, we conclude that λ ě x λ y e “ p e p e ´ ¨ x λ y e ě p e p e ´ ¨ β . (cid:3) .3. On purity at the threshold.
We conclude by addressing the limiting behavior of thevarious types of purity given in Definition 1.3.
Theorem 4.9.
The pair p R, fpt p f q ‚ f q is F -pure and is not strongly F -pure. Moreover, p R, fpt p f q ‚ f q is sharply F -pure if and only if p p e ´ q ¨ fpt p f q P N for some e ě . Theorem 4.9 generalizes [Har06, Proposition 2.6] and [Sch08, Corollary 5.4 ` Remark 5.5],in which R is assumed be an F -finite (complete) regular local ring. Proof.
By Corollary 4.7, we have that t p p e ´ q fpt p f q u “ p e x fpt p f qy e , and so Key Lemma3.4 implies that the inclusion R ¨ f t p p e ´ q fpt p f q u { p e “ R ¨ f x fpt p f qy e Ď R { p e splits over R . Thisshows that p R, fpt p f q ‚ f q is F -pure.By Lemma 2.5, r p e fpt p f q s “ p e x fpt p f qy e `
1, and applying Key Lemma 3.4 shows thatthe inclusion R ¨ f x fpt p f qy e ` { p e Ď R { p e never splits as a map of R -modules. This shows that p R, fpt p f q ‚ f q is not strongly F -pure.Finally, suppose that p R, fpt p f q ‚ f q is sharply F -pure. By Definition 1.3,(4.3.1) R ¨ f r p p e ´ q fpt p f q s { p e Ď R { p e splits as a map of R -modules for some e ě . By Key Lemma 3.4, we know that (4.3.1) holds if and only if(4.3.2) r p p e ´ q fpt p f q s ď p e x fpt p f qy e “ t p p e ´ q fpt p f q u , where the last equality in (4.3.2) comes from Corollary 4.7. Thus, we must have equalitythroughout in (4.3.2), which we observe holds if and only if p p e ´ q ¨ fpt p f q P N . (cid:3) Example 4.10.
There are many instances for which F -purity and sharp F -purity are dis-tinct. The simplest example such that p p e ´ q ¨ fpt p f q R N for any e ě fpt p F p rr x ss , x p q “ p . For more examples where fpt p f q is a rational number whose denomi-nator is divisible by p , see [Her11a, Her11b]. References [BL04] Manuel Blickle and Robert Lazarsfeld. An informal introduction to multiplier ideals. In
Trends incommutative algebra , volume 51 of
Math. Sci. Res. Inst. Publ. , pages 87–114. Cambridge Univ.Press, Cambridge, 2004. 2[BMS08] Manuel Blickle, Mircea Mustat¸ˇa, and Karen E. Smith. Discreteness and rationality of F -thresholds. Michigan Math. J. , 57:43–61, 2008. Special volume in honor of Melvin Hochster. 2[BMS09] Manuel Blickle, Mircea Mustat¸˘a, and Karen E. Smith. F -thresholds of hypersurfaces. Trans. Amer.Math. Soc. , 361(12):6549–6565, 2009. 1, 2, 3, 10[BSTZ09] Manuel Blickle, Karl Schwede, Shunsuke Takagi, and Wenliang Zhang. Discreteness and rationalityof F -jumping numbers on singular varieties. Mathematische Annalen, Volume 347, Number 4, 917-949, 2010 , 06 2009. 2[Fed83] Richard Fedder. F -purity and rational singularity. Trans. Amer. Math. Soc. , 278(2):461–480, 1983.9[Har06] Nobuo Hara. F-pure thresholds and F-jumping exponents in dimension two.
Math. Res. Lett. , 13(5-6):747–760, 2006. With an appendix by Paul Monsky. 1, 2, 11[Her11a] Daniel J. Hern´andez. F -invariants of diagonal hypersurfaces. arXiv:1112.2425v1 [math.AC] , 2011.2, 11[Her11b] Daniel J. Hern´andez. F -pure thresholds of binomial hypersurfaces. arXiv:1112.2427v1 [math.AC] ,2011. 2, 11[Her11c] Daniel J. Hern´andez. F -purity versus log canonicity for polynomials. arXiv:1112.2423v1 [math.AC] ,2011. 2 HH90] Melvin Hochster and Craig Huneke. Tight closure, invariant theory, and the Brian¸con-Skoda theo-rem.
J. Amer. Math. Soc. , 3(1):31–116, 1990. 5[How01a] J. A. Howald. Multiplier ideals of monomial ideals.
Trans. Amer. Math. Soc. , 353(7):2665–2671(electronic), 2001. 2[How01b] Jason Howald. Multiplier ideals of sufficiently general polynomials.
Preprint, math.AG/0303203 ,2001. 2[HR74] Melvin Hochster and Joel L. Roberts. Rings of invariants of reductive groups acting on regular ringsare Cohen-Macaulay.
Advances in Math. , 13:115–175, 1974. 1[HR76] Melvin Hochster and Joel L. Roberts. The purity of the Frobenius and local cohomology.
Advancesin Math. , 21(2):117–172, 1976. 1, 4[HW02] Nobuo Hara and Kei-Ichi Watanabe. F-regular and F-pure rings vs. log terminal and log canonicalsingularities.
J. Algebraic Geom. , 11(2):363–392, 2002. 1, 2[KLZ09] Mordechai Katzman, Gennady Lyubeznik, and Wenliang Zhang. On the discreteness and rationalityof F -jumping coefficients. J. Algebra , 322(9):3238–3247, 2009. 2[Kun69] Ernst Kunz. Characterizations of regular local rings for characteristic p . Amer. J. Math. , 91:772–784, 1969. 1, 9[Laz04] Robert Lazarsfeld.
Positivity in algebraic geometry. II , volume 49 of
Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematicsand Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] . Springer-Verlag, Berlin,2004. Positivity for vector bundles, and multiplier ideals. 2[MTW05] Mircea Mustat¸ˇa, Shunsuke Takagi, and Kei-ichi Watanabe. F-thresholds and Bernstein-Sato poly-nomials. In
European Congress of Mathematics , pages 341–364. Eur. Math. Soc., Z¨urich, 2005. 3[Sch08] Karl Schwede. Generalized test ideals, sharp F -purity, and sharp test elements. Math. Res. Lett. ,15(6):1251–1261, 2008. 1, 2, 4, 5, 11[Smi97] Karen E. Smith. Vanishing, singularities and effective bounds via prime characteristic local algebra.In
Algebraic geometry—Santa Cruz 1995 , volume 62 of
Proc. Sympos. Pure Math. , pages 289–325.Amer. Math. Soc., Providence, RI, 1997. 2[Tak04] Shunsuke Takagi. F-singularities of pairs and inversion of adjunction of arbitrary codimension.
In-vent. Math. , 157(1):123–146, 2004. 4[Tak11] S. Takagi. ”Adjoint ideals and a correspondence between log canonicity and F-purity”,. arXiv:1105.0072v2 [math.AG] , 2011. 2[TW04] Shunsuke Takagi and Kei-ichi Watanabe. On F-pure thresholds.
J. Algebra , 282(1):278–297, 2004.4, 5, 282(1):278–297, 2004.4, 5