Frobenius splitting of valuation rings and F-singularities of centers
aa r X i v : . [ m a t h . A C ] N ov FROBENIUS SPLITTING OF VALUATION RINGS AND F -SINGULARITIES OF CENTERS RANKEYA DATTA
Abstract.
Using a local monomialization result of Knaf and Kuhlmann, we prove that thevaluation ring of an Abhyankar valuation of a function field over a perfect ground field ofprime characteristic is Frobenius split. We show that a Frobenius splitting of a sufficientlywell-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigateproperties of valuations centered on arbitrary Noetherian domains of prime characteristic. Incontrast to [DS16, DS17], this paper emphasizes the role of centers in controlling Frobeniusproperties of valuation rings in prime characteristic. Introduction
One of the goals of this paper is to prove the following result.
Theorem A.
Let K be a finitely generated field extension of a perfect field k of prime char-acteristic. Then the valuation ring of any Abhyankar valuation of K/k is Frobenius split.
For a valuation ν of K/k , if Γ ν is the value group and ( R ν , m ν , κ ν ) the valuation ring, then ν is Abhyankar if dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /k = tr . deg K/k.
Abhyankar valuations, often also called quasimonomial valuations in characteristic , extendthe class of valuations associated to prime divisors on normal models, since dim Q ( Q ⊗ Z Γ ν ) =1 and tr . deg κ ν /k = tr . deg K/k − for divisorial valuations. Nevertheless, non-divisorialAbhyankar valuations arise naturally in geometry (see [Spi90, ELS03, FJ04, FJ05, JM12,Mus12, Tem13, RS14, Tei14, Pay14, Blu16] for some applications), and they possess many ofthe good properties of their Noetherian counterparts. For example, the value group of anyAbhyankar valuation is a free abelian group of finite rank, and its residue field is a finitelygenerated extension of the ground field.Divisorial valuation rings over perfect fields of prime characteristic are Frobenius split.Indeed, when a Noetherian, local ring R is F -finite , that is when the Frobenius ( p -th power)endomorphism F : R → R is a finite ring map, a famous result of Kunz shows that regularityof R is characterized by R being free over its p -th power subring R p [Kun69, Theorem 2.1].Therefore F -finite regular rings, and consequently divisorial valuation rings over perfect fields,are Frobenius split (the same conclusion can be drawn using the Direct Summand Conjecture).Thus Theorem A extends a well-known fact about divisorial valuation rings to a class ofvaluation rings that behaves the most like divisorial ones.A key ingredient in our proof of Theorem A is the following result of Knaf and Kuhlmannwhich says that one can locally monomialize finite subsets of Abhyankar valuation rings in any characteristic. Theorem 1. [KK05, Theorem 1]
Let K be a finitely generated field extension of a field k ofany characteristic, and let ν be an Abhyankar valuation of K/k such that the residue field ν is separable over k . Then for any finite set Z ⊆ R ν , R ν is centered on a regular localring ( A, m A , κ A ) essentially of finite type over k with fraction field K satisfying the followingproperties:(1) The Krull dimension of A equals d := dim Q ( Q ⊗ Z Γ ν ) .(2) Z ⊆ A , and there exists a regular system of parameters { x , . . . , x d } of A such thatevery z ∈ Z admits a factorization z = ux a . . . x a d d , for some u ∈ A × , and a i ∈ N ∪ { } . When the ground field k is perfect, any Abhyankar valuation ν admits a local monomializationsince the residue field of the valuation is automatically separable over k . Theorem 1 thenallows us to choose a center of ν on a regular local k -algebra A such that A has a regularsystem of parameters { x , . . . , x d } whose valuations freely generate Γ ν , and the residue fieldof A coincides with κ ν (Lemma 3.0.1). Our strategy then is to identify a suitable Frobeniussplitting of A that lifts to a Frobenius splitting of R ν .A proof of Theorem A was announced in [DS16, Theorem 5.1], where Frobenius splittingwas deduced as a consequence of the incorrect assertion that Abhyankar valuation ringsare F-finite. On the contrary, [DS17, Theorem 0.1] shows that finiteness of Frobenius forvaluation rings of function fields (a function field is a finitely generated extension of a basefield) is equivalent to the associated valuation being divisorial, and so non F-finite Abhyankarvaluations are abundant. The current paper rectifies the error in [DS16].On the other hand, the proof of [DS16, Theorem 5.1] does establish that a valuation ν ofan F -finite function field K/k is Abhyankar if and only if [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . (1.0.0.1)Said differently, ν is Abhyankar precisely when the extension of valuations ν/ν p is defectless ,where ν p denotes the restriction of ν to the subfield K p . We refer the reader to [FV11, Page281] for the definition of defect of an extension of valuations.The second goal of this paper is to generalize (1.0.0.1) to valuations, not necessarily offunction fields, that admit a Noetherian local center. When a valuation ν of an arbitrary field K is centered on a Noetherian, local domain ( R, m R , κ R ) such that Frac( R ) = K , one has thefollowing beautiful inequality established by Abhyankar [Abh56, Theorem 1]: dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /κ R ≤ dim R. (1.0.0.2)When equality holds in (1.0.0.2), ν behaves a lot like an Abhyankar valuation of a functionfield. For example, the value group Γ ν is then again a free abelian group of finite rank, andthe residue field κ ν is finitely generated over κ R . However, whether a valuation of a functionfield is Abhyankar is intrinsic to the valuation, while equality in (1.0.0.2) with respect to acenter depends, unsurprisingly, on the center as well (see Example 4.0.1 for an illustration).Bearing this difference in mind, we call a Noetherian center R an Abhyankar center of ν , if ν satisfies equality in (1.0.0.2) with respect to R .In practice one is often interested in centers satisfying additional restrictions. For example,in the local uniformization problem for valuations of function fields, one seeks centers thatare regular. Similarly, in geometric applications centers are usually local rings of varieties,and consequently essentially of finite type over a ground field. Although satisfying equality n (1.0.0.2) is not intrinsic to a valuation, the property of possessing Abhyankar centers froma more restrictive class of local rings may become independent of the center. For example,when K/k is a function field and C is the collection of local rings that are essentially of finitetype over k with fraction field K , then a valuation ν admits an Abhyankar center from thecollection C precisely when ν is an Abhyankar valuation of K/k , and then all centers of ν from C are Abhyankar centers of ν (see Section 4). In other words, the property of possessingAbhyankar centers that are locally of finite type over k is intrinsic to valuations of functionfields over k .Our investigation reveals that even in a non function field setting, one can find a reasonablybroad class of Noetherian local rings such that the property of admitting an Abhyankar centerfrom this class is independent of the choice of the center. More precisely, we show the following: Theorem 4.0.3.
Let ( R, m R , κ R ) be a Noetherian, F -finite, local domain of characteristic p > and fraction field K . Suppose ν is a non-trivial valuation of K centered on R withvalue group Γ ν and valuation ring ( V, m ν , κ ν ) . Then R is an Abhyankar center of ν if andonly if [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . Since the identity [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] does not depend on the center, Theorem4.0.3 implies that possessing F -finite Abhyankar centers is intrinsic to a valuation. Althoughimposing finiteness of Frobenius on Noetherian centers appears to be a strong restriction,the class of such rings include any center one is likely to encounter is geometric applications.For example, any generically F -finite excellent domain of prime characteristic is also F -finite.Thus Theorem 4.0.3 implies that the property of a valuation possessing generically F -finite,excellent Abhyankar centers is independent of the choice of such centers.We can draw interesting conclusions from Theorem 4.0.3 even for valuations of functionfields. First note that it generalizes (1.0.0.1) (see Remark 4.0.8(i)). Also, if K is a functionfield over a perfect field k of prime characteristic, then Theorem 4.0.3 and (1.0.0.1) imply thatany valuation of K/k that possesses an excellent Abhyankar center is an Abhyankar valuationof
K/k (Corollary 4.0.5). Admitting excellent Abhyankar centers is a priori much weaker thanadmitting Abhyankar centers that are essentially of finite type over k , and so Corollary 4.0.5is not at all obvious. In fact, the corresponding statement is false when the ground field k hascharacteristic – one can easily construct a non-Abhyankar valuation of K/ C which has anAbhyankar center that is an excellent local domain (see Remark 4.0.8(vi) or [ELS03, Example1(iv)]).Theorem 4.0.3 does not claim that if a valuation ν of an F -finite field K satisfies [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] , then ν is centered on an excellent, local domain; the latter assertionis false even when K is not perfect. Indeed, using the theory of F -singularities of valuationsdeveloped in [DS16, DS17], one can show that if the valuation ring of ν is F -finite, thenthe identity [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] is always satisfied (Section 5, (3)(ii)). However,we prove in this paper that a non-Noetherian, F -finite valuation ring cannot be centered onany excellent local domain of its fraction field (Section 5, (8)), and explicitly construct sucha valuation ring of K = L ( X ) , where L is a perfect field admitting a non-trivial valuation(Example 5.0.5). The valuation ring of Example 5.0.5 does not contain the ground field L ,and so its existence does not contradict local uniformization in positive characteristic. Onthe contrary, this paper and the author’s prior work with Karen Smith [DS17, Theorem 0.1]shows it is impossible to construct non-Noetherian, F -finite valuation rings of function fields hen the valuation rings contain the ground field (see also Remark 5.0.6). Thus, pathologiessuch as Example 5.0.5 cease to exist for valuations trivial on the ground field of a functionfield (see also (Section 5, (8))).In the final section of this paper (Section 5), we summarize all known results on F-singularities of valuation rings for the convenience of the reader, mainly drawing from thepaper [DS16] and the accompanying corrigendum [DS17]. Results are grouped according tothe type of F -singularity they characterize, which we hope will make it easier for the readerto navigate [DS16, DS17]. The summary is not just limited to a recollection of old results;many new results also appear in it with complete proofs. For example, in (Section 5, 14) weshow that when ν/ν p is totally unramified or has maximal defect, that is, when [Γ ν : p Γ ν ][ κ ν : κ pν ] = 1 , then the valuation ring of ν cannot be Frobenius split. To put this in context, Theorem Aand (1.0.0.1) imply that, in contrast, at least for valuations ν of function fields over perfectground fields, if ν/ν p is defectless then the valuation ring of ν is Frobenius split. At the sametime, Frobenius splitting is not at all well understood when the defect of ν/ν p is not one oftwo possible extremes. Acknowledgments:
The question of Frobenius splitting of valuation rings arose in con-versations of Zsolt Patakfalvi, Karl Schwede and Karen Smith. While Frobenius splitting ofAbhyankar valuations was suspected in my prior work with Karen, the idea of using localuniformization took shape while I was visiting University of Utah. I thank Karl Schwede,Raymond Heitmann, Linquan Ma and Anurag Singh for many fruitful conversations duringmy stay in Utah, and Karl for the invitation. I am also grateful to Karen, Linquan andRaymond Heitmann for helpful comments on a draft of this paper. Further thanks go toFranz-Viktor Kuhlmann, whose question on the relationship between defect and Abhyankarvaluations inspired the material of Section 4, and to Steven Dale Cutkosky for helpful con-versations. My work was supported by department and summer fellowships at University ofMichigan and NSF Grant DMS
Background and conventions
Valuations.
All valuations are written additively, and local rings are not necessarilyNoetherian. We say a valuation ν with valuation ring V is centered on a local ring A , or A isa center of ν (or V ) if A ⊆ V , and the maximal ideal of V contracts to the maximal ideal of A . It is always assumed that the fraction field of a center coincides with the fraction field ofthe valuation ring.Let K be a finitely generated field extension of a field k , that is K is a function field over k . A valuation ν of K/k (this means ν is trivial on k ) with valuation ring ( R ν , m ν , κ ν ) andvalue group Γ ν satisfies the fundamental inequality dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /k ≤ tr . deg K/k. (2.1.0.1)If equality holds in the above inequality, then ν is called an Abhyankar valuation or a quasi-monomial valuation of K/k , and the associated valuation ring of ν is called an Abhyankar valuation ring of K/k . or Abhyankar valuations it is well-known [Bou89, VI, § . , Corollary 1] that Γ ν is a freeabelian group of finite rank equal to d := dim Q (Γ ν ⊗ Z Q ) , and the residue field κ ν is a finitely generated field extension of k of transcendence degree n − d , where n := tr . deg K/k.
If the ground field k has prime characteristic p and [ k : k p ] < ∞ , this implies [ K : K p ] = [ k : k p ] p n and [ κ ν : κ pν ] = [ k : k p ] p n − d . An important fact, used implicitly throughout the paper, is that if x, y ∈ K × are non-zeroelements such that x + y = 0 and ν ( x ) = ν ( y ) , then ν ( x + y ) = inf { ν ( x ) , ν ( y ) } . A discrete valuation ring (abbreviated DVR) is a Noetherian valuation ring which is not afield. Equivalently, a DVR is a dimension regular local ring.2.2. Frobenius.
Let R be a ring of prime characteristic. We have the (absolute) Frobeniusmap F : R → R, which maps an element r ∈ R to its p -th power. The target copy of R is usually consideredas an R -module by restriction of scalars via F , and is then denoted F ∗ R . In other words,if r ∈ R and x ∈ F ∗ R , then r · x = r p x . For an ideal I of R , by I [ p e ] we mean the idealgenerated by the p e -th powers of elements of I . Thus I [ p e ] is the expansion of I under F e , the e -th iterate of Frobenius.Quite remarkably, the Frobenius map can detect when a Noetherian ring is regular, andthe foundational result in the theory of F -singularities is the following: Theorem 2.2.1 ([Kun69], Theorem 2.1) . Let R be a Noetherian ring of prime characteristic.Then R is regular if and only if F : R → F ∗ R is a flat ring homomorphism. If Frobenius is a finite map and R is regular local, the above theorem implies that F ∗ R is afree R -module. The freeness of F ∗ R will be important when proving Theorem A.In geometry, finiteness of Frobenius is a mild restriction. For instance, Frobenius is afinite map for the localization of any finitely generated algebra over a perfect field of primecharacteristic, and also for any complete local ring whose residue field k satisfies [ k : k p ] < ∞ .A ring for which Frobenius is finite is called F -finite .Kunz’s theorem shows that when R is regular local and F -finite, F ∗ R has many free R -summands. For an arbitrary ring R , if F ∗ R has at least one free R -summand, we say R is Frobenius split . More formally, R is Frobenius split if F : R → F ∗ R has a left inverse, calleda Frobenius splitting , in the category of R -modules. When R is reduced, Frobenius is anisomorphism onto its image R p , and the existence of a Frobenius splitting is equivalent to theexistence of an R p -linear map R → R p that maps .Weakening the notion of Frobenius splitting leads to F -purity . We say that R is F -pure when Frobenius is a pure map of R -modules. This means for any R -module M , the inducedmap F ⊗ id M : M → F ∗ R ⊗ R M is injective. Regular rings are F -pure because Frobeniusis flat hence faithfully flat for such rings, and faithful flatness implies purity. Also Frobenius plitting clearly implies F -purity, but the converse is false. For example, any non-excellentDVR of a function field over a perfect ground field is F-pure but not Frobenius split. See (1)and (10) in Section 5 for further discussion, and Example 4.0.1 for a non-excellent DVR of afunction field. 3. Proof of Theorem A
Unless otherwise specified, throughout this section we assume that K is a finitely generatedfield extension of an F -finite ground field k , and ν is an Abhyankar valuation of K/k whoseresidue field κ ν is separable over k . We will prove more generally under these assumptionsthat the valuation ring R ν is Frobenius split. Also, we let d := dim Q ( Q ⊗ Z Γ ν ) and n := tr . deg K/k .The goal is to choose a regular local center of ν satisfying some nice properties, and thenextend a Frobenius splitting of this center to a Frobenius splitting of R ν . The center we seekis given by the following lemma. Lemma 3.0.1.
Let ν be an Abhyankar valuation as in Theorem 1. Then there exists a regularlocal ring ( A, m A , κ A ) which is essentially of finite type over k with fraction field K satisfyingthe following properties:(1) R ν is centered on A , and κ A ֒ → κ ν is an isomorphism.(2) A has Krull dimension d , and there exist a regular system of parameters { x , . . . , x d } of A such that { ν ( x ) , . . . , ν ( x d ) } freely generates the value group Γ ν .Proof. Choose r , . . . , r d , s , . . . , s t ∈ R ν such that { ν ( r ) , . . . , ν ( r d ) } freely generates the valuegroup Γ ν , and the images of s , . . . , s t in κ ν generate the latter over k . Taking Z := { r , . . . , r d , s , . . . , s t } , by local monomialization (Theorem 1) there exists a regular local ring ( A, m A , κ A ) essentiallyof finite type over k with fraction field K such that R ν dominates A , A has dimension d , Z ⊆ A , and there exists a regular system of parameters { x , . . . , x d } of A such that every z ∈ Z admits a factorization z = ux a . . . x a d d , for some u ∈ A × , and a i ∈ N ∪ { } . In particular, each ν ( r i ) is Z -linear combination of ν ( x ) , . . . , ν ( x d ) , which implies that { ν ( x ) , . . . , ν ( x d ) } also freely generates Γ ν . Moreover,our choice of s , . . . , s t implies that κ A = κ ν . (cid:3) Remark 3.0.2.
For a valuation ν of K/k , the existence of a center which is an essentially offinite type k -algebra of Krull dimension equal to dim Q ( Q ⊗ Z Γ ν ) implies that ν is Abhyankar.Thus, only Abhyankar valuations admit a center as in Lemma 3.0.1.From now on A will denote a choice of a regular local center of ν that satisfies Lemma3.0.1, and { x , . . . , x d } a regular system of parameters of A whose valuations freely generate Γ ν . Observe that A is F -finite since it is essentially of finite type over an F -finite field.Then Theorem 2.2.1 implies that A is free over its p -th power subring A p of rank equal to [ K : K p ] = [ k : k p ] p n . For f := [ κ ν : κ pν ] = [ k : k p ] p n − d , f we choose y , y , . . . , y f ∈ A, such that the images of y i in κ A = κ ν form a basis of κ ν over κ pν , then B := { y j x β . . . x β d d : 1 ≤ j ≤ f, ≤ β i ≤ p − } , is a free basis of A over A p . Note the elements y j are units in A .With respect to the basis B , A has a natural Frobenius splitting η B : A → A p , given by mapping y x . . . x d , and all the other basis elements to . Extending η B uniquely to a K p -linear map f η B : K → K p of the fraction fields, we will show that the restriction of f η B to R ν yields a Frobenius splittingof R ν , or in other words, f η B | R ν maps into R pν . Claim 3.0.3.
For any a ∈ A , either η B ( a ) = 0 or ν ( η B ( a )) ≥ ν ( a ) . Theorem A follows easily from the claim using the following general observation.
Lemma 3.0.4.
Let ν be a valuation of a field K of characteristic p > with valuation ring R ν , and A a subring of R ν such that Frac( A ) = K . Suppose ϕ : A → A p e is an A p e -linearmap, for some e ≥ . Consider the following:(i) For all a ∈ A , ϕ ( a ) = 0 or ν ( ϕ ( a )) ≥ ν ( a ) .(ii) For all a, b ∈ A such that ν ( a ) ≥ ν ( b ) , if ϕ ( ab p e − ) = 0 , then ν ( ϕ ( ab p e − )) ≥ ν ( b p e ) .(iii) ϕ extends to an R p e ν -linear map R ν → R p e ν .(iv) ϕ extends uniquely to an R p e ν -linear map R ν → R p e ν .Then (ii), (iii) and (iv) are equivalent, and (i) ⇒ (ii). Moreover, if ϕ is a Frobenius splittingof A satisfying (i) or (ii), then ϕ extends to a Frobenius splitting of R ν .Proof. For the final assertion on Frobenius splitting, note that the extension of a Frobeniussplitting remains a Frobenius splitting since in the extension.(i) ⇒ (ii): If ϕ ( ab p e − ) = 0 , we have ν ( ϕ ( ab p e − )) ≥ ν ( ab p e − ) ≥ ν ( b p e ) , where the first inequality follows from (i), and the second inequality follows from ν ( a ) ≥ ν ( b ) .(ii) ⇒ (iii): Extending ϕ to a K p e -linear map e ϕ : K → K p , it suffices to show that e ϕ | R ν maps into R p e ν . Let r ∈ R ν be a non-zero element. Since K is the fraction field of A and R ν ,one can express r as a fraction a/b , for non-zero a, b ∈ A . Note ν ( a ) ≥ ν ( b ) . Then e ϕ ( r ) = e ϕ (cid:18) ab (cid:19) = 1 b p e ϕ ( ab p e − ) . (3.0.4.1)If ϕ ( ab p e − ) = 0 , then e ϕ ( r ) = 0 , and r maps into R p e ν . Otherwise by assumption, ν ( ϕ ( ab p e − )) ≥ ν ( b p e ) , nd so, ν ( e ϕ ( r )) = ν ( ϕ ( ab p e − )) − ν ( b p e ) ≥ , that is e ϕ ( r ) is an element of K p e ∩ R ν = R p e ν .(iii) ⇒ (iv): Since A and R ν have the same fraction field, any extension of ϕ to R ν isobtained as a restriction to R ν of the unique extension of ϕ to a K p e -linear map e ϕ : K → K p e ,and so is also unique. See (3.0.4.1) above for a concrete description of how ϕ extends to R ν .To finish the proof of the lemma, it suffices to show (iv) ⇒ (ii). But this also follows easilyfrom (3.0.4.1). (cid:3) Proof of Claim 3.0.3 . Recall that B = { y j x β . . . x β d d : 1 ≤ j ≤ f, ≤ β i ≤ p − } is a basis of A over A p , where the x i and y j are chosen such that { ν ( x ) , . . . , ν ( x d ) } freelygenerates the value group Γ ν , and the images of y , y , . . . , y f in κ ν form a basis of κ ν over κ pν . The A p -linear Frobenius splitting η B is given by η B (cid:18) f X j =1 X ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d (cid:19) = c p , , ,..., . Thus, we need to show that either c p , , ,..., = 0 or ν ( c p , , ,..., ) ≥ ν (cid:18) f X j =1 X ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d (cid:19) . Assuming without loss of generality that P fj =1 P ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d = 0 , we willprove the stronger fact that ν (cid:18) f X j =1 X ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d (cid:19) = inf { ν ( c pj,β ,...,β d y j x β . . . x β d d ) : c pj,β ,...,β d = 0 } . (3.0.4.2)For two non-zero terms c pj,α ,...,α d y j x α . . . x α d d and c pk,β ,...,β d y k x β . . . x β d d in the above sum, ν ( c pj,α ,...,α d y j x α . . . x α d d ) = ν ( c pk,β ,...,β d y k x β . . . x β d d ) (3.0.4.3)if and only if pν ( c j,α ,...,α d ) + α ν ( x ) + · · · + α d ν ( x d ) = pν ( c k,β ,...,β d ) + β ν ( x ) + · · · + β d ν ( x d ) . (3.0.4.4)By Z -linear independence of ν ( x ) , . . . , ν ( x d ) , for all i = 1 , . . . , d , we get p | ( α i − β i ) . Since ≤ α i , β i ≤ p − , this means that α i = β i for all i , and moreover, then ν ( c pj,α ,...,α d ) = ν ( c pk,β ,...,β d ) . Thus, (3.0.4.3) holds precisely when ν ( c pj,α ,...,α d ) = ν ( c pk,β ,...,β d ) and α i = β i ,for all i = 1 , . . . , d .For ease of notation, let us use α as a shorthand for α , . . . , α d , and x α for x α . . . x α d d .Then for a fixed non-zero term c pj ,α y j x α , consider the set { c pj ,α y j x α , c pj ,α y j x α , . . . , c pj i ,α y j i x α } f all non-zero terms of P fj =1 P ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d having the same valuation as c pj ,α y j x α . In particular, by the above reasoning we also have ν ( c pj ,α ) = ν ( c pj ,α ) = · · · = ν ( c pj i ,α ) . Adding these terms of equal valuation, in the valuation ring R ν one can write c pj ,α y j x α + c pj ,α y j x α + · · · + c pj i ,α y j i x α = (cid:18) y j + (cid:18) c j ,α c j ,α (cid:19) p y j + · · · + (cid:18) c j i ,α c j ,α (cid:19) p y j i (cid:19) c pj ,α x α , where y j + (cid:18) c j ,α c j ,α (cid:19) p y j + · · · + (cid:18) c j i ,α c j ,α (cid:19) p y j i is a unit in R ν by the κ pν -linear independence of the images of y j , . . . , y j i in κ ν and the factthat ( c j ,α /c j ,α ) p , . . . , ( c j i ,α /c j ,α ) p are units in R pν . Thus, the valuation of the sum c pj ,α y j x α + · · · + c pj i ,α y j i x α equals the valuation of any of its terms. Now rewriting P fj =1 P ≤ β i ≤ p − c pj,β ,...,β d y j x β . . . x β d d by collecting non-zero terms having the same valuation, (3.0.4.2), hence also the claim, follows. (cid:3) Examples 3.0.5. (a) A valuation ring of a function field of a curve over an F -finite ground field is alwaysFrobenius split. Indeed, such a valuation ring is always centered on some normal affine modelof dimension of the function field, and so is an F -finite DVR.(b) For a positive integer n , consider Z ⊕ n with the lexicographical order. That is, if { e , . . . , e n } denotes the standard basis of Z ⊕ n , then e > e > · · · > e n . There exists a unique valuation ν lex on F p ( x , . . . , x n ) / F p such that for all i ∈ { , . . . , n } , ν lex ( x i ) = e i . The valuation ν lex is clearly Abhyakar since dim Q ( Q ⊗ Z Z ⊕ n ) = n , which coincides withthe transcendence degree of F p ( x , . . . , x n ) / F p . One can also show that the valuation ring R ν lex has Krull dimension n and residue field F p . The valuation is centered on the regularlocal ring F p [ x , . . . , x n ] ( x ,...,x n ) such that the valuations of the obvious regular system ofparameters freely generate Z ⊕ n and the residue field coincides with the residue field of ν lex .Then a Frobenius splitting of R ν lex → R pν lex is obtained by extending the canonical splittingon F p [ x , . . . , x n ] ( x ,...,x n ) with respect to the basis { x β . . . x βn : 0 ≤ β i ≤ p − } . This splitting of F p [ x , . . . , x n ] ( x ,...,x n ) maps x α . . . x α n n ( x α . . . x α n n if p | α i for all i , otherwise.(c) Let Γ = Z ⊕ Z π ⊂ R . There exists a valuation ν on F p ( x, y, z ) / F p given by ν ( x ) = ν ( y ) = 1 , ν ( z ) = π. hen dim Q ( Q ⊗ Z Γ) = 2 , and the transcendence degree of the residue field κ ν / F p is at least since the image of y/x in the residue field is transcendental over F p . Therefore the fundamentalinequality (2.1.0.1) implies that ν is Abhyankar. Although the valuation ν is centered on theregular local ring F p [ x, y, z ] ( x,y,z ) , no regular system of parameters can freely generate thevalue group because the center has dimension , whereas the value group is free of rank .However, blowing up the origin in A F p , we see that ν is now centered on the regular local ring F p [ x, y/x, z/x ] ( x,z/x ) , and the valuations of the regular system of paramaters { x, z/x } freelygenerate Γ . Furthermore, the residue field of F p [ x, y/x, z/x ] ( x,z/x ) can be checked to coincidewith the residue field of the valuation ring. Relabelling y/x and z/x as u, w respectively, aFrobenius splitting on R ν is obtained by extending the Frobenius splitting of F p [ x, u, w ] ( x,w ) given by the same rule as in (a) with respect to the algebraically independent elements x, u, w over F p . 4. Valuations centered on Noetherian, local domains
The proof of [DS16, Theorem 5.1] shows that a valuation ν of an F -finite function field K/k of characteristic p is Abhyankar precisely when [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . (4.0.0.1)If ν p denotes the restriction of ν to the subfield K p of K , then the value group of ν p is easilyverified to be p Γ ν , and the residue field κ ν p can be identified with κ pν . Thus, [Γ ν : p Γ ν ] is the ramification index , and [ κ ν : κ pν ] the residue degree of the extension of valuations ν/ν p [DS16,Remark 4.3.3]. Note ν is the unqiue extension of ν p to K since K is a purely inseparableextension of K p .In terms of the theory of extensions of valuations, (4.0.0.1) can be reinterpreted as sayingthat a necessary and sufficient condition for ν to be Abhyankar is for the unique extension ofvaluations ν/ν p to be defectless (see [FV11, Page 281] for the definition of defect). There isa natural generalization of the notion of an Abhyankar valuation for valuations of arbitraryfields. The goal of this section is to introduce this more general notion, and investigate towhat extent such valuations can be characterized in terms of the defect of ν/ν p .We fix some notation. Let K denote a field of characteristic p > (not necessarily a functionfield), and ν a valuation of K with valuation ring ( V, m ν , κ ν ) centered on a Noetherian localring ( R, m R , κ R ) such that dim( R ) < ∞ . Recall that centers, by convention, always have the same fraction field as the valuation ring.Let Γ ν be the value group of ν .Abhyankar greatly generalized (2.1.0.1) in [Abh56, Theorem 1], establishing that in theabove setup, dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /κ R ≤ dim( R ) . (4.0.0.2)Moreover, he showed that if equality holds in the above inequality, then Γ ν is a free abeliangroup and κ ν is a finitely generated extension of κ R .When K/k is a function field, there is a close relationship between Abhyankar valuationsof
K/k , and those valuations of
K/k that admit a Noetherian center with respect to whichequality holds in (4.0.0.2). Indeed, if equality holds in (4.0.0.2) for an arbitrary valuation of K/k with respect to a center R which is essentially of finite type over k , then ν is anAbhyankar valuation of K/k . To see this, let n = tr . deg K/k . Then tr . deg κ R /k = n − dim( R ) , because R is essentially of finite type over k with fraction field K , and so tr . deg κ ν /k = tr . deg κ ν /κ R + n − dim( R ) . This implies dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /k = (dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /κ R ) + n − dim( R )= dim( R ) + n − dim( R ) = tr . deg K/k.
Conversely, a similar reasoning shows that if ν is an Abhyankar valuation of K/k , then ν satisfies equality in (4.0.0.2) with respect to any center which is essentially of finite type over k . However, despite the similarity between (4.0.0.2) and (2.1.0.1), whether a valuation satisfiesequality in (4.0.0.2) is not an intrinsic property of the valuation, but also depends on thecenter R . In contrast, the property of being an Abhyankar valuation is intrinsic to valuationsof function fields. To better illustrate this difference, we construct a valuation of F p ( X, Y ) with two different Noetherian centers such that equality in (4.0.0.2) is satisfied with respectto one center, but not the other. In our example we work over a base field of characteristic p > , but the construction goes through when the ground field has characteristic . Example 4.0.1. (see also [DS16, Example 4.0.5]) Consider the laurent series field F p (( t )) with the canonical t -adic valuation, ν t , whose corresponding valuation ring is the DVR F p [[ t ]] .Choose an embedding of fields i : F p ( X, Y ) ֒ → F p (( t )) by mapping X t and Y p ( t ) , where p ( t ) ∈ F p [[ t ]] such that { t, p ( t ) } are algebraically independent over F p . Such a power-series exists because F p (( t )) is uncountable, but F p ( t ) is countable. Moreover, multiplying p ( t ) by t , we may evenassume that t | p ( t ) . Then we get a new valuation ν on F p ( X, Y ) , given by the composition ν := F p ( X, Y ) × i −→ F p (( t )) × ν t −→ Z . The corresponding valuation ring V is a DVR with maximal ideal generated by X . Since ν ( X ) = ν t ( t ) , ν ( Y ) = ν t ( p ( t )) ≥ , ( p ( t ) was scaled so that t | p ( t ) ), we see that ν is centered on F p [ X, Y ] ( X,Y ) . Furthermore, ν isalso trivially centered on its own valuation ring. As F p [[ t ]] dominates V and has residue field F p , κ ν = F p . Clearly ν satisfies equality in (4.0.0.2) with respect to its valuation ring as a center (this is truemore generally for any discrete valuation), but not with respect to the center F p [ X, Y ] ( X,Y ) .Note that ν is not an Abhyankar valuation of F p ( X, Y ) / F p , since dim Q ( Q ⊗ Z Z ) + tr . deg κ ν / F p = 1 = tr . deg F p ( X, Y ) / F p . oreover, the valuation ring of ν is not an F -finite DVR. This follows from results stated inthe next section, but we include the justification here. Indeed, since the maximal ideal m of V is principal, by [Section 5, (3)(i)] dim κ pν ( V / m [ p ] ) = p [ κ ν : κ pν ] = p = [ F p ( X, Y ) : F p ( X, Y ) p ] , and so V is not F -finite by [Section 5, (2)]. It turns out that V is also not excellent [Section5, (10)].Given the example above, we make the following definition. Definition 4.0.2.
Let ν be a valuation centered on a Noetherian local domain R . We say R is an Abhyankar center of ν if dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /κ R = dim( R ) . To summarize our observations, the property of being an Abhyankar valuation of a functionfield is intrinsic to a valuation, while whether ν admits Abhyankar centers depends on thecenters. However, if additional restrictions are imposed on the class of centers (for instance,if we require centers to be essentially of finite type over k ), then the property of possessingthese more restrictive Abhyankar centers becomes intrinsic to ν .The interplay between Abhyankar valuations and valuations admitting Abhyankar centersraises the natural question: does (4.0.0.1) have an analogue for valuations of fields that arenot necessarily function fields? Moreover, can the feature of possessing Abhyankar centersbecome intrinsic to a valuation if we restrict the class of admissible centers? The next resultprovides an affirmative answer for a broad class of Noetherian centers. Theorem 4.0.3.
Let ( R, m R , κ R ) be a Noetherian, F -finite local domain of characteristic p > and fraction field K . Suppose ν is a non-trivial valuation of K centered on R withvalue group Γ ν and valuation ring ( V, m ν , κ ν ) . Then R is an Abhyankar center of ν if andonly if [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . Theorem 4.0.3 has some interesting consequences that we illustrate first.
Corollary 4.0.4.
Let ν be a valuation of a field K of characteristic p > . If ν admits aNoetherian, F -finite center which is Abhyankar, then any other Noetherian, F -finite centerof ν is also an Abhyankar center of ν .In other words, the property of possessing Noetherian, F -finite, Abhyankar centers is in-trinsic to a valuation. Proof of Corollary 4.0.4 . The proof follows easily from Theorem 4.0.3 using the obser-vation that the identity [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] is independent of the choice of acenter. (cid:3) Corollary 4.0.5.
Let ν be a valuation of a function field K/k over a perfect field k of char-acteristic p > (it suffices for k to be F -finite). The following are equivalent:(1) ν is an Abyankar valuation of K/k .(2) ν admits an Abhyankar center which is an excellent local ring. roof of Corollary 4.0.5 . For (1) ⇒ (2), any center of the Abhyankar valuation ν whichis essentially of finite type over k , hence also excellent, is an Abhyankar center of ν . Forthe converse, let R be an excellent, Abhyankar center of ν . As K is F -finite, R is also F -finite. This follows from the fact that since R p is excellent (it is isomorphic to R ), its integralclosure S in K is module finite over R p , because K is a finite extension of K p . But R is an R p -submodule of S , and submodules of finitely generated modules over Noetherian rings arefinitely generated. So R is also module finite over R p , that is, R is F -finite. Thus ν satisfiesthe identity [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] by Theorem 4.0.3, and since K/k is a functionfield, (4.0.0.1) implies that ν is an Abhyankar valuation of K/k . (cid:3) We will prove Theorem 4.0.3 by first developing a connection between the inequality dim Q ( Q ⊗ Z Γ ν ) + tr . deg κ ν /κ R ≤ dim( R ) and the quantities [Γ ν : p Γ ν ] and [ κ ν : κ pν ] . This will also shed light on precisely where F -finiteness is used in the proof of Theorem 4.0.3.In order to achieve the above goal, we need the following general facts about torsion-freeabelian groups and F -finite fields. Lemma 4.0.6.
Let p be a prime number, K an F -finite field of characteristic p , and Γ atorsion-free abelian group such that dim Q ( Q ⊗ Z Γ) is finite. We have the following:(1) If L is an algebraic extension of K , then [ L : L p ] ≤ [ K : K p ] , with equality if K ⊆ L is a finite extension. In particular, L is then also F -finite.(2) If L is field extension of K of transcendence degree t , then [ L : L p ] ≤ p t [ K : K p ] , with equality if L is finitely generated over K .(3) If s = dim Q ( Q ⊗ Z Γ) , then [Γ : p Γ] ≤ p s , with equality if Γ is finitely generated.Indication of proof of Lemma 4.0.6. (2) clearly follows from (1), and (3) follows from [DS16,Lemma 5.5], with equality obviously holding when Γ is finitely generated, since Γ is then free.We prove (1) here, which is a minor generalization of [DS16, Lemma 5.8].To show [ L : L p ] = [ K : K p ] when K ⊆ L is finite is easy (see [DS16, Section 4.6]). Sosuppose K ⊆ L is algebraic, and [ K : K p ] < ∞ . It suffices to show that if a , . . . , a n ∈ L arelinearly independent over L p , then n ≤ [ K : K p ] . Let e L := K ( a , . . . , a n ) . Since L is algebraic over K , e L is a finite extension K , and so by what we already established, [ e L : e L p ] = [ K : K p ] . n the other hand, since a , . . . , a n are linearly independent over L p , and e L p ⊆ L p , it followsthat a , . . . , a n are also linearly independent over e L p . Thus, n ≤ [ e L : e L p ] = [ K : K p ] , as desired. (cid:3) Using the previous lemma, we can now relate the ramification index (i.e. [Γ ν : p Γ ν ] ) andresidue degree (i.e. [ κ ν : κ pν ] ) of the extension of valuations ν/ν p to (4.0.0.2): Proposition 4.0.7.
Let ν be a valuation of a field K of characteristic p > with valuationring ( V, m ν , κ ν ) , centered on Noetherian local domain ( R, m R , κ R ) such that [ κ R : κ pR ] < ∞ . We have the following:(1) [Γ ν : p Γ ν ][ κ ν : κ pν ] ≤ p dim( R ) [ κ R : κ pR ] .(2) R is an Abhyankar center of ν if and only if [Γ ν : p Γ ν ][ κ ν : κ pν ] = p dim( R ) [ κ R : κ pR ] . Proof of Proposition 4.0.7 . Throughout the proof, let s := dim Q ( Q ⊗ Z Γ ν ) and t := tr . deg κ ν /κ R . (1) Abhyankar’s inequality (4.0.0.2) implies s + t ≤ dim( R ) . In particular, s and t are both finite. Using Lemma 4.0.6(3), we get [Γ ν : p Γ ν ] ≤ p s . On the other hand, since κ R is F -finite by hypothesis, and κ ν has transcendence degree t over κ R , Lemma 4.0.6(2) shows [ κ ν : κ pν ] ≤ p t [ κ R : κ pR ] . Thus, [Γ ν : p Γ ν ][ κ ν : κ pν ] ≤ p s + t [ κ R : κ pR ] ≤ p dim( R ) [ κ R : κ pR ] . (4.0.7.1)(2) Suppose R is an Abhyankar center of ν , that is, s + t = dim( R ) . By [Abh56, Theorem 1], Γ ν is a free abelian group of rank s , and κ ν is a finitely generatedfield extension of κ R of transcendence degree t . Again using Lemma 4.0.6, we get [Γ ν : p Γ ν ] = p s and [ κ ν : κ pν ] = p t [ κ R : κ pR ] , and so [Γ ν : p Γ ν ][ κ ν : κ pν ] = p s + t [ κ R : κ pR ] = p dim( R ) [ κ R : κ pR ] , proving the forward implication.Conversely, if [Γ ν : p Γ ν ][ κ ν : κ pν ] = p dim( R ) [ κ R : κ pR ] then p dim( R ) [ κ R : κ pR ] = [Γ ν : p Γ ν ][ κ ν : κ pν ] ≤ p s + t [ κ R : κ pR ] ≤ p dim( R ) [ κ R : κ pR ] , where the inequalities follow from (4.0.7.1). Thus, dim( R ) = s + t , which by definition meansthat R is an Abhyankar center of ν . (cid:3) heorem 4.0.3 now follows readily from Proposition 4.0.7. Proof of Theorem 4.0.3 . Suppose R is a Noetherian, F -finite, local domain with fractionfield K . Then [ κ R : κ pR ] and [ K : K p ] < ∞ . In particular, R satisfies the hypotheses of Proposition 4.0.7, and so Theorem 4.0.3 follows ifwe can show that [ K : K p ] = p dim( R ) [ κ R : κ pR ] . (4.0.7.2)This is a well-known result that is implicit in the proof of [Kun76, Proposition 2.1]. However,since (4.0.7.2) is crucial for our proof, we briefly indicate how it is established. In [Kun76,Proposition 2.1], Kunz uses the analogue of Noetherian normalization for complete rings toshow that when R is F -finite, then for any minimal prime ideal P of the m R -adic completion b R , [ K : K p ] = p dim( b R/ P ) [ κ R : κ pR ] . This shows that dim( b R/ P ) is independent of P , or in other words that b R is equidimensional.However, since P is minimal, we then have dim( b R/ P ) = dim( b R ) = dim( R ) , which confirms (4.0.7.2). (cid:3) Remarks 4.0.8. (i) A key point in the proof of Theorem 4.0.3 is that when ( R, m , κ ) is an F -finite,Noetherian, local domain of characteristic p > and fraction field K , then [ K : K p ] = p dim( R ) [ κ R : κ pR ] . (4.0.8.1)A careful analysis of the proof of [Kun76, Proposition 2.1] reveals that (4.0.8.1) holdsfor any Noetherian, local domain R such that Ω R/ Z is a finitely generated R -moduleand the completion b R is reduced, that is, if R is analytically unramified. We notethat when R is F -finite, it satisfies both these properties. Indeed, since R is a finitelygenerated R p -module, Ω R/ Z = Ω R/R p is then a finitely generated R -module, and b R is reduced by [Kun69, Lemma 2.4].Theorem 4.0.3, hence Corollary 4.0.4, clearly hold more generally for the class ofNoetherian centers of any F -finite field that satisfy (4.0.8.1). While such centers arequite common, it is not difficult to construct generically F -finite Noetherian local,domains that do not satisfy (4.0.8.1). For instance, (4.0.8.1) fails for the non F -finiteDVR of F p ( X, Y ) constructed in Example 4.0.1. In particular, since regular localrings are analytically unramified, our observations imply that the module of absoluteKähler differentials of the DVR of Example 4.0.1 must not be finitely generated.(ii) Theorem 4.0.3 generalizes (4.0.0.1). Indeed, if K/k is an F -finite function field, thenany valuation ν of K/k admits an F -finite, Noetherian center. For instance, by thevaluative criterion of properness, ν is centered on a proper k -variety with functionfield K , and the local ring of this variety at the center is F -finite. Since we observedthat ν is an Abhyankar valuation if and only if any center of ν which is essentiallyof finite type over k is an Abhyankar center, it follows by Theorem 4.0.3 that ν isAbhyankar precisely when [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . iii) One can reinterpret Theorem 4.0.3 as saying that an F -finite, Noetherian center R of ν is an Abhyankar center of ν if and only if the extension of valuations ν/ν p isdefectless.(iv) When [ K : K p ] < ∞ , the condition [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] does not imply thatthe valuation ν admits a Noetherian, F -finite center. See Example 5.0.5 for a counter-example, which shows that counter-examples can be constructed even for valuationsof fields that are not perfect.(v) For a Noetherian domain R with F -finite fraction field K , the following are all equiv-alent:(a) R is F -finite.(b) R is excellent.(c) R is a Japanese/N-2 ring.(d) The integral closure of R p in K is a finite R p -module.We already saw (b) ⇒ (c) ⇒ (d) ⇒ (a) in the proof of Corollary 4.0.5. The hard partis to show (a) ⇒ (b), which follows from [Kun76, Theorem 2.5]. As a consequence, inTheorem 4.0.3, the F -finiteness assumption on centers can be replaced by excellence,provided we assume the ambient field K is F -finite. Hence when the fraction fieldof a valuation is F -finite, the property of admitting excellent Abhyankar centers isintrinsic to the valuation. In particular, this is true for valuations of function fieldsover perfect ground fields of prime characteristic.(vi) The analogue of (v) is false for valuations of function fields over ground fields ofcharacteristic , that is, whether a valuation admits an excellent Abhyankar centerdepends on the excellent center. For instance, any DVR of characteristic 0 is auto-matically excellent [Sta17, Tag 07QW], and by imitating the construction of Example4.0.1 using the fields C ( X, Y ) and C (( t )) instead, one can show that there exists adiscrete valuation ν of C ( X, Y ) / C centered on C [ X, Y ] ( X,Y ) such that the latter is notan Abhyankar center of ν (see [ELS03, Example 1(iv)] for more details). However, ν is also trivially centered on its own valuation ring which is an excellent, Abhyankarcenter of ν because ν is discrete. The same example also shows that Corollary 4.0.5is false over ground fields of characteristic .This remark and (v) indicate that excellent rings in characteristic p > behave verydifferently from excellent rings in characteristic , and that the notion of excellencein prime characteristic is more restrictive than in characteristic .5. Summary of F -singularities for valuation rings We summarize all known results on F-singularities of valuation rings, grouping them ac-cording to the type of F-singularity they characterize. While most results are proved in [DS16]and the erratum [DS17], some new results also appear below (with complete proofs).For a valuation ring ( V, m , κ ) of a field K of characteristic p > with associated valuation ν , singularities of V defined using the Frobenius map are intimately related to properties ofthe extension of valuations ν/ν p , where ν p denotes the restriction of ν to the subfield K p of K . Recall that the valuation ring of ν p is V p , and the residue field of ν p can be identifiedwith κ p . Furthermore, ν is the unique extension of ν p to K (up to equivalence of valuations),and V is the integral closure of V p in K . n what follows Γ or Γ ν will always denote the value group of a valuation (ring), and κ or κ ν its residue field. Flatness of Frobenius and F-purity: (1) [DS16, Theorem 3.1] The Frobenius endomorphism on any valuation ring of primecharacteristic is always faithfully flat. Hence a valuation ring of prime characteristicis F -pure, and so close to being Frobenius split. Remark 5.0.1.
It is not difficult to construct valuation rings for which Frobeniusis pure but not split. For example, the non F -finite DVR of Example 4.0.1 is notFrobenius split, because any Frobenius split Noetherian domain with F -finite fractionfield has to F -finite [DS16, Theorem 4.2.1] (see also (10) below). F -finiteness in general: Let ( V, m , κ ) be a valuation ring of a field K , with associated valuation ν . A necessarycondition for V to be F -finite is that [ K : K p ] < ∞ , that is, K is F -finite. So we implicitlyassume in our discussion of F -finiteness of V that K is F -finite to begin with. Note F -finiteness of K also implies [ κ : κ p ] < ∞ , that is, the residue field is always F -finite. This follows by observing that [ κ : κ p ] is the residue degree of the extension of valuations ν/ν p and then using [Bou89, VI, § . , Lemma2]. (2) The following are equivalent:(a) V is F -finite.(b) V is a free V p -module of rank [ K : K p ] .(c) dim κ p ( V / m [ p ] ) = [ K : K p ] . Proof of (2) . The equivalence of (a) and (b) is shown in [DS16, Theorem 4.1.1].Although used in the proof of [DS17, Erratum, revised Corollary 4.3.2], the equivalenceof (c) to (a) and (b) is not explicitly stated in [DS16, DS17]. Thus, we include acomplete proof here.Let n := [ K : K p ] . We show (b) and (c) are equivalent. Suppose V is a free moduleof rank n over the subring V p , which is a valuation ring of K p . If η is the maximalideal of V p , we see that V / m [ p ] ∼ = V ⊗ V p V p /η is a free κ p = V p /η -module of rank n , which proves (b) ⇒ (c). For the converse,suppose dim κ p ( V / m [ p ] ) = [ K : K p ] = n . Choose x , . . . , x n ∈ V such that the imagesof x i in V / m [ p ] form a κ p -basis, and let L := V p x + · · · + V p x n . Note L is a finitely generated, torsion free V p -module, hence free over V p since finitelygenerated torsion-free modules over valuation rings are free. To prove (b), it sufficesto show that L = V. The rank of L equals dim κ p L/ηL , and it is easy to see that the images of x , . . . , x n in L/ηL form a κ p -basis of L/ηL . Thus, L is a free V p -module of rank n , and so { x , . . . , x n } is a V p -basis of L . bserve that { x , . . . , x n } is also linearly independent over K p , and since [ K : K p ] = n , this means that { x , . . . , x n } is a K p -basis of K . Let s ∈ V be a non-zeroelement, and r , . . . , r n ∈ K p such that s = r x + · · · + r n x n . Clearly V = L , if we can prove that all the r i are elements of V p . By renumberingthe x i , and using the fact that V p is a valuation ring of K p , we may assume withoutloss of generality that r = 0 and r i r − ∈ V p , for all i ≥ . If r ∈ V p , then all the r i are already in V p . If not, then r − is anelement of the maximal ideal, η , of V p . Thus, r − s = x + r r − x + · · · + r n r − x n , which contradicts κ p -linear independence of the images of x , . . . , x n in V /ηV = V / m [ p ] . Hence all the r i are elements of V p , and we are done. (cid:3) (3) (i) [DS17, Erratum, Lemma 2.2] For ( V, m , κ ) and K as above, dim κ p ( V / m [ p ] ) = ( [ κ : κ p ] if m is not finitely generated, p [ κ : κ p ] if m is finitely generated. Sketch of proof of (i) . The assertion follows from the short exact sequence of κ p -vector spaces → m / m [ p ] → V / m [ p ] → κ → , with the additional observations that when m is not finitely generated, m [ p ] = m [DS17, Lemma 2.1], and when m is finitely generated, m is principal, so that dim κ p ( m / m [ p ] ) = ( p − κ : κ p ] . (cid:3) (ii) For an F-finite , valuation ring V with value group Γ = 0 :(a) [ K : K p ] = [Γ : p Γ][ κ : κ p ] = dim κ p ( V / m [ p ] ) ([DS17, corrected Thm 4.3.1]and (2)).(b) The value group Γ satisfies [Γ : p Γ] = 1 or [Γ : p Γ] = p .(c) If the maximal ideal of V is not finitely generated, then Γ is p -divisible.(d) If Γ is finitely generated, then V is a DVR. Remark 5.0.2.
The error in [DS16] arose from the incorrect assertion that [Γ : p Γ][ κ : κ p ] = [ K : K p ] ⇒ V is F-finite (although the assertion is true when V is a DVR by (3)(iv) below).(iii) If [ K : K p ] = [ κ : κ p ] , then V is F-finite [DS17, revised Corollary 4.3.2].(iv) If V is a DVR, then V is F -finite if and only if [Γ : p Γ][ κ : κ p ] = [ K : K p ] . Proof of (iv) . If V is F -finite, then [Γ : p Γ][ κ : κ p ] = [ K : K p ] by (ii)(a) above.For the converse, we have [ K : K p ] = [Γ : p Γ][ κ : κ p ] = [ Z : p Z ][ κ : κ p ] = p [ κ : κ p ] = dim κ p ( V / m [ p ] ) , where we use (3)(i) for the final equality. So V is F -finite by (2). (cid:3) Remark 5.0.3.
The proof of (iv) shows more generally that if V is a valuationring with principal maximal ideal m and such that [Γ : p Γ] = p , then [Γ : p Γ][ κ : κ p ] = [ K : K p ] implies V is F -finite. -finiteness and finite field extensions: (4) [DS17, Section 3] Let K ⊆ L be a finite extension of F -finite fields. Let ν be avaluation on K and w an extension of ν to L . Then the valuation ring of ν is F -finiteif and only if the valuation ring of w is F -finite. F -finiteness in function fields: Let K be a finitely generated field extension of an F-finite field k .(5) [DS17, Theorem 1.1] A valuation ν of K/k is Abhyankar if and only if [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . See also Theorem 4.0.3 for a generalization of the above result.(6) [DS17, Erratum, Theorem 0.1] For a non-trivial valuation ν of K/k , the associatedvaluation ring V is F -finite if and only if ν is divisorial. Sketch of proof of (6) . The non-trivial implication is that when V is F -finite, itsvaluation ν is divisorial. However, (3)(ii)(a) and (5) imply that ν is Abhyankar, andso has a finitely generated value group. Using (3)(ii)(d) one then concludes V is aDVR. Now a classical result of Zariski shows that any Abhyankar DVR is divisorial[SZ60, VI, § , Theorem 31]. (cid:3) Remarks 5.0.4. (i) [DS16, Theorem 5.1] erroneously states that an Abhyankar valuation ring of
K/k is F-finite. The justification for why this is wrong is given in Remark 5.0.2.For a counter-example, the valuation ring V of the lexicographical valuation on F p ( X, Y ) / F p with value group Z ⊕ (ordered lexicographically) is Abhyankar (seeExample 3.0.5(a)), but not F -finite because dim κ p ( V / m [ p ] ) = p [ κ : κ p ] = p [ F p : F p ] = p = [ F p ( X, Y ) : F p ( X, Y ) p ] . Here the first equality holds by (3)(i) because the maximal ideal of V is principal,with Y being a generator. The second equality holds because the residue field of V is F p .(ii) When not in a function field, it is easy to construct non-Noetherian F -finitevaluation rings. For example, the perfection F p [[ t /p ∞ ]] := S e ∈ N F p [[ t /p e ]] of thepower series ring F p [[ t ]] is a non-Noetherian, F -finite valuation ring of its fractionfield F p (( t /p ∞ )) . More generally, a non-trivial valuation ring of any perfect fieldof prime characteristic is not Noetherian, but F -finite because Frobenius is anisomorphism for such a ring. Rings of prime characteristic for which Frobeniusis an isomorphism are called perfect rings . Such rings have been extensivelyinvestigated of late since finding applications in Scholze’s work on perfectoidspaces (see [Sch12], [BS16]). While perfect rings are trivially F -finite, there existnon-Noetherian, F -finite valuation rings that are not perfect.Suppose L is a perfect field of prime characteristic equipped with a non-trivialvaluation ν with value group Γ ν . For instance L can be a perfectoid field, or thealgebraic closure of a field which has non-trivial valuations. Then the residuefield κ ν of the associated valuation ring is also perfect. Now consider the group Γ ′ := Γ ν ⊕ Z rdered lexicographically, and the field L ( X ) , where X is an indeterminate. Thereexists a unique extension w of the valuation ν to L ( X ) with value group Γ ′ suchthat for any polynomial f = P ni =0 a i X i in L [ X ] , we have w ( f ) = inf { ( ν ( a i ) , i ) : i = 0 , . . . , n } . The residue field κ w of w equals the residue field κ ν [Bou89, VI, § . , Proposition1], hence is also perfect. Also, Γ ′ has a smallest element > in the lex order,namely (0 , . Thus, if ( R w , m w ) is the valuation ring of w , the maximal ideal m w is principal, and in fact generated by X . Using (3)(i), we see that dim κ pw ( R w / m [ p ] w ) = p [ κ w : κ pw ] = p = [ L ( X ) : L ( X ) p ] . (5.0.4.1)Then R w is F -finite by (2), not Noetherian because Γ ′ = Γ ν ⊕ Z has rational rankat least , and not perfect because the field L ( X ) is not perfect.(iii) Curiously, if instead of taking Γ ′ = Γ ν ⊕ Z ordered lexicographically we take Γ ′ = Z ⊕ Γ ν ordered lexicographically in the above construction, the resultingextension w of ν to L ( X ) (with obvious modifications to the definition of w ) doesnot have an F -finite valuation ring R w . Indeed, then the maximal ideal of R w isnot finitely generated, while the residue field κ w still coincides with κ ν , which isperfect. Thus dim κ pw ( R w / m [ p ] w ) = [ κ w : κ pw ] = 1 = [ L ( X ) : L ( X ) p ] . F -finiteness and valuations centered on Noetherian domains: Using Theorem 4.0.3, one can generalize (6) to a non function field setting as follows.(7) Let ν be a non-trivial valuation on K centered on an F -finite, Noetherian local domain R . Then the valuation ring V of ν is F -finite if and only if V is a DVR and R is anAbhyankar center of ν . Proof of (7) . For the forward implication, if V is F-finite, then [Γ ν : p Γ ν ][ κ ν : κ pν ] =[ K : K p ] holds automatically (see (3)(ii)), and then Theorem 4.0.3 implies that R is an Abhyankar center of ν . In particular, Γ ν is a non-trivial finitely generatedabelian group, and so (3)(ii)(d) ⇒ V is a DVR. This proves the forward implication.Conversely, if R is an Abhyankar center of ν , then Theorem 4.0.3 again implies [Γ ν : p Γ ν ][ κ ν : κ pν ] = [ K : K p ] . Since V is also a DVR by hypothesis, it is F -finite by (3)(iv). (cid:3) The above result has the following interesting consequence that we would like to highlightseparately.(8) Suppose ν is a valuation of an F -finite field K with valuation ring V that satisfieseither of the following conditions:(a) V is F -finite, but not Noetherian.(b) dim( V ) > s , where [ K : K p ] = p s .Then ν is not centered on any excellent local domain whose fraction field is K . Proof of (8) . Since K is F -finite, a Noetherian domain with fraction field K isexcellent if and only if it is F -finite (see Remark 4.0.8(v)). Thus, it suffices to showthat ν is not centered on any F -finite, Noetherian local domain if it satisfies (a) or(b). uppose ν satisfies (a). As V is not Noetherian, (7) implies that ν cannot becentered on any F -finite, Noetherian local domain with fraction field K .If R is an F -finite, Noetherian local ring with fraction field K , then recall that wehave the identity p dim( R ) [ κ R : κ pR ] = [ K : K p ] . In particular, this means dim( R ) ≤ s , where s is as above. Now if ν is centered on R ,then Abhyankar’s inequality (4.0.0.2) shows in particular that dim Q ( Q ⊗ Z Γ ν ) ≤ dim( R ) ≤ s. However, it is well-known that the Krull dimension of V is at most dim Q ( Q ⊗ Z Γ ν ) [Bou89, § dim( V ) ≤ s , which contradicts thehypothesis of (b). Hence ν cannot be centered on any F -finite, Noetherian domainwith fraction field K . (cid:3) Example 5.0.5.
Let w be the valuation of L ( X ) (where L is a perfect field) con-structed in Remark 5.0.4(ii). The valuation ring R w satisfies conditions (a) and (b) of(8). We have already observed that R w satisfies (a). To see that R w satisfies (b), notethat the value group of w has a proper, non-trivial isolated/convex subgroup. Thus R w has Krull dimension at least 2 [Bou89, § [ L ( X ) : L ( X ) p ] = p .Although R w is a valuation ring of a function field, it does not contain the groundfield L . So even though w/w p is defectless, this example does not contradict (5), orthe problem of local uniformization in prime characteristic. Remark 5.0.6. If K/k is an F -finite function field, and ν is a valuation of K/k withvaluation ring V , then (5) shows that V cannot satisfy (8)(a), while (2.1.0.1) showsthat V cannot satisfy (8)(b). Thus, the pathologies of (8) do not arise for valuationsof function fields that are trivial on the ground field. Frobenius splitting: (9) [DS16, Corollary 4.1.2] Any F -finite valuation ring is Frobenius split. This followsfrom (2) because F ∗ V is then a free V -module. Remark 5.0.7. (9) is a special case of a general phenomenon of valuation rings whichis independent of the characteristic of the ring. Recall, that a ring R is called a splinter if any module finite ring extension ϕ : R → S has an R -linear left inverse. For example, from recent work of André [And16] (see also[Bha16, HM17]) and earlier work of Hochster [Hoc73], it follows that any regular ringis a splinter. We want to show that valuation rings are also splinters. So let V be avaluation ring (of any characteristic), and ϕ : V → S a module finite ring extension. Choose a prime ideal P of S that lies over the zeroideal of V . Then the composition V ϕ −→ S ։ S/P is also a module finite ring extension, and it suffices to show this composite extensionsplits. Thus we may assume S is a domain, which makes S a finitely generated torsion-free V module. But finitely generated torsion free modules over valuation rings are ree. Nakayama’s lemma implies there exists a basis of S over V containing , andthen one can easily construct many splittings of ϕ with respect to such a basis.(10) [DS16, Corollary 4.2.2] The following are equivalent for a Noetherian valuation ring ( V, m , κ ) with F-finite fraction field K : • V is Frobenius split. • V is F -finite. • V is excellent. • dim κ p ( V / m p ) = [ K : K p ] . • [Γ : p Γ][ κ : κ p ] = [ K : K p ] .(11) Any DVR that admits a Noetherian, F -finite, Abhyankar center is Frobenius split. Proof of 11 . V is F -finite by (7), hence Frobenius split by (9). (cid:3) (12) Any complete DVR of prime characteristic is Frobenius split. Proof of 12 . Let V be a complete DVR of prime characteristic with residue field κ .Since V is equicharacteristic, by Cohen’s Structure Theorem for complete rings, V ∼ = κ [[ t ]] , as rings, and clearly κ p [[ t p ]] is a direct summand of κ [[ t ]] , even when [ κ : κ p ] is notfinite. (cid:3) (13) [Section 3] For an Abhyankar valuation ring V of an F -finite function field K/k , if theresidue field κ is separable over k , then V is Frobenius split. In particular Abhyankarvaluation rings over perfect ground fields of prime characteristic are always Frobeniussplit.(14) We have seen in (5) that Abhyankar valuations in F -finite function fields are charac-terized as those valuations ν such that ν/ν p is defectless, that is, [Γ ν : p Γ ν ][ κ ν : κ pν ] =[ K : K p ] . At the opposite extreme is a valuation ν such that the extension ν/ν p is totally unramified , which means that [Γ ν : p Γ ν ][ κ ν : κ pν ] = 1 . (5.0.7.1)We want to show that when ν/ν p is totally unramified and K is not perfect, thevaluation ring V of ν cannot be Frobenius split. Note that (5.0.7.1) implies Γ ν = p Γ ν and κ ν = κ pν , that is, the value group is p -divisible and the residue field is perfect.The p -divisibility of Γ ν shows that m = m [ p ] . Then any Frobenius splitting ϕ : V → V p maps the maximal ideal m of V into themaximal ideal of V p , thereby inducing a Frobenius splitting of residue fields e ϕ : κ ν → κ pν . However, κ ν is perfect, so that e ϕ is just the identity map. Since K is not perfect, ϕ hasa non-trivial kernel, that is, some non-zero x ∈ V gets mapped to . By p -divisibility,one can write x = uy p , for a unit u in V , and y = 0 . Then ϕ ( x ) = y p ϕ ( u ) , whichshows that ϕ ( u ) = 0 . But this contradicts injectivity of e ϕ , proving that no Frobeniussplitting of V exists. Remark 5.0.8.
The author thanks Ray Heitmann for showing him a proof of (14) inthe special case of Q -valuations of F p ( X, Y ) . Our argument above is a generalizationof Heitmann’s argument to all totally unramified extensions ν/ν p . -regularity: Two notions of F -regularity were introduced in [DS16], generalizing strong F -regularityto a non-Noetherian and non F -finite setting– split F -regularity [DS16, Definition 6.6.1] and F -pure regularity [DS16, Definiton 6.1.1]. Split F -regularity just drops the F -finite andNoetherian hypotheses from the definition of strong F -regularity, while F -pure regularityreplaces splitting of certain maps by purity and seems to be the better notion for rings thatare not F -finite. Split F -regularity ⇒ F -pure regularity, but the converse is false. Indeed,any non-excellent DVR with an F -finite fraction field will be F -pure regular but not split F -regular. In particular, the DVR of Example 4.0.1 is not excellent, so not split F -regular.(15) [DS16, Thm 6.5.1 and Cor. 6.5.4] For a valuation ring V of prime characteristic, V is F -pure regular if and only if it is Noetherian.(16) [DS16, Corollary 6.6.3] Let V be a valuation ring whose fraction field is F -finite. Thefollowing are equivalent (see also (10)): • V is split F -regular. • V is Noetherian and F -finite. • V is excellent. • V is Noetherian and Frobenius split. • If V is a valuation ring of a function field, then V is divisorial. Remark 5.0.9. (15) and (16) indicate that F -regularity is perhaps a useful notion of singu-larity only for Noetherian rings. Open Questions:
Just as is the case in geometry, our results indicate that Frobenius splittingis the most mysterious F-singularity for valuation rings with many basic open questions.The proof of Frobenius splitting of Abhyankar valuation rings of function fields over F-finite ground fields uses the local monomialization result of Knaf and Kuhlmann (Theorem1), hence also the hypothesis that the residue field of the valuation ring is separable over theground field. However, it is probably the case that any Abhyankar valuation ring of an F -finitefunction field is Frobenius split, and this will from our proof if one can remove the separabilityhypothesis from Theorem 1. Moreover, a natural question is if one can generalize our result onFrobenius splitting of Abhyankar valuations to valuations, not necessarily of function fields,that admit F -finite, Noetherian, Abhyankar centers satisfying ‘mild’ singularities such as F -regularity. For example, (7) shows that a discrete valuation admitting a Noetherian, F -finite,Abhyankar center is Frobenius split. F -singularities of a valuation ring V are intimately related to basic properties of the cor-responding extension of valuations ν/ν p . For example, (5) shows that at least for functionfields over perfect ground fields, when ν/ν p is defectless, ν is Abhyankar and its valuationring is Frobenius split. On the other hand, when ν/ν p has maximal defect, that is when ν/ν p is totally unramified, (14) shows that V cannot be Frobenius split unless it is a perfect ring.However, Frobenius splitting of V remains mysterious when the defect of ν/ν p is not one twopossible extremes.For instance, suppose K/k is an F -finite function field. Is there a non-Abhyankar valuationof K/k whose valuation ring is Frobenius split? We can use (16) to conclude that suchvaluations, if they exist, cannot be discrete. On the other hand, Example 5.0.5 shows thatif we relax the condition that the valuation is trivial on the ground field, then there exists avaluation w of K , not trivial on k , whose valuation ring is Frobenius split. Moreover, w has he feature that it is not centered on any excellent domain whose fraction field is K . However,even in this example, the extension w/w p is defectless.There are interesting open questions pertaining to Frobenius splitting even for Noetherianvaluation rings. As far as we know, it is not known if every excellent DVR of prime charac-teristic is Frobenius split. (10) (resp. (12)) provides an affirmative answer when the fractionfield of a DVR is F -finite (resp. when the DVR is complete). At the same time it is worthrecalling that Frobenius is always pure for any valuation ring of prime characteristic by (1),and purity seems to be a better notion than Frobenius splitting for non F-finite rings. References [Abh56] S. Abhyankar,
On the valuations centered in a local domain , American Journal of Mathematics (1956), no. 2, 321–348.[And16] Y. André, La conjecture du facteur direct , arXiv:1609.00345, August 2016.[Bha16] B. Bhatt,
On the direct summand conjecture and its derived variant , arXiv:1608.08882, August2016.[Blu16] H. Blum,
Existence of valuations with smallest normalized volume , arXiv:1606.08894, 2016.[Bou89] N. Bourbaki,
Commutative algebra, Chapters 1-7 , Springer-Verlag Berlin Heidelberg, 1989.[BS16] B. Bhatt and P. Scholze,
Projectivity of the Witt vector affine Grassmannian , Inventiones math-ematicae (2016), no. 2, 329–423.[DS16] R. Datta and K.E. Smith,
Frobenius and valuation rings , Algebra and Number Theory (2016), no. 5, 1057–1090.[DS17] , Correction to the article “Frobenius and valuation rings” , Algebra and Number Theory (2017), no. 4, 1003-1007.[ELS03] L. Ein and R. Lazarsfeld and K.E. Smith, Uniform approximation of Abhyankar valuation idealsin smooth function fields , American Journal of Mathematics (2003), no. 2, 409–440.[FJ04] C. Favre and M. Jonsson,
The Valuative Tree , Lecture notes in Mathematics, vol. 1853, Springer-Verlag Berlin Heidelberg, 2004.[FJ05] ,
Valuations and multiplier ideals , Journal of Amer. Math. Soc. (2005), no. 3, 655-684.[FV11] F-V. Kuhlmann, The defect , Commutative Algebra– Noetherian and Non-Noetherian Perspec-tives, Springer New York, pp. 277–318, 2011.[HM17] R. Heitmann and L. Ma,
Big Cohen-Macaulay algebras and the vanishing conjecture for mapsof Tor in mixed characteristic , arXiv:1703.08281, March 2017.[Hoc73] M. Hochster,
Contracted ideals from integral extensions of regular rings , Nagoya Math. J. (1973), 25–43.[JM12] M. Jonsson and M. Mustaţă, Valuations and asymptotic invariants for sequences of ideals ,Annales de l’institut Fourier (2012), no. 6, 2145–2209.[KK05] H. Knaf and F-V. Kuhlmann, Abhyankar places admit local uniformization in any characteristic ,Annales scientifiques de l’École Normale Supérieure, Série 4 (2005), no. 6, 833–846.[Kun69] E. Kunz, Characterizations of regular local rings of characteristic p , American Journal of Math-ematics (1969), no. 3, 772–784.[Kun76] , On Noetherian rings of characteristic p , American Journal of Mathematics (1976),no. 4, 999–1013.[Mus12] M. Mustaţă, IMPANGA lecture notes on log canonical thresholds , Contributions to algebraicgeometry: Impanga ecture notes, EMS (2012), 407–442.[Pay14] S. Payne, Topology of non-Archimedean analytic spaces and relations to complex algebraicgeometry, Bull. Amer. Math. Soc. (2014), no. 2, 223–247.[RS14] G. Rond and M. Spivakovsky, The analogue of Izumi’s theorem for Abhyankar valuations , Jour-nal of London Math. Soc. (2014), no. 3, 725–740. Sch12] P. Scholze,
Perfectoid Spaces , Publications mathématiques de l’IHÉS (2012), Issue 1, 245–313.[Spi90] M. Spivakovsky,
Valuations in function fields of surfaces , American Journal of Mathematics (1990), no. 1, 107–156.[Sta17] The Stacks Project Authors,
Stacks Project , http://stacks.math.columbia.edu , 2017.[SZ60] P. Samuel and O. Zariski, Commutative Algebra, Vol II , Springer-Verlag Berlin Heidelberg,1960.[Tei14] B. Teissier,
Overweight deformations of affine toric varieties and local uniformization , vol.Valuation Theory in Interaction, EMS Series of Congress Reports, pp. 474–565, EMS PublishingHouse, Zürich, 2014.[Tem13] M. Temkin,
Inseparable local uniformization , Journal of Algebra (2013), no. 1, 65–119.
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