arXiv:1707.01649v2 [] 28 Nov 2017

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 the
valuation ring of an Abhyankar valuation of a function ﬁeld over a perfect ground ﬁeld of
prime characteristic is Frobenius split. We show that a Frobenius splitting of a suﬃciently
well-behaved center lifts to a Frobenius splitting of the valuation ring. We also investigate
properties of valuations centered on arbitrary Noetherian domains of prime characteristic. In
contrast to [DS16, DS17], this paper emphasizes the role of centers in controlling Frobenius
properties of valuation rings in prime characteristic.

1. 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 characteristic. Then the valuation ring of any Abhyankar valuation of K/k is Frobenius split.
For a valuation ν of K/k, if Γν is the value group and (Rν , mν , κν ) the valuation ring, then ν
is Abhyankar if
dimQ (Q ⊗Z Γν ) + tr. deg κν /k = tr. deg K/k.
Abhyankar valuations, often also called quasimonomial valuations in characteristic 0, extend
the class of valuations associated to prime divisors on normal models, since dimQ (Q ⊗Z Γν ) =
1 and tr. deg κν /k = tr. deg K/k − 1 for divisorial valuations. Nevertheless, non-divisorial
Abhyankar valuations arise naturally in geometry (see [Spi90, ELS03, FJ04, FJ05, JM12,
Mus12, Tem13, RS14, Tei14, Pay14, Blu16] for some applications), and they possess many of
the good properties of their Noetherian counterparts. For example, the value group of any
Abhyankar valuation is a free abelian group of finite rank, and its residue field is a finitely
generated 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 regularity
of R is characterized by R being free over its p-th power subring Rp [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 of
valuation rings that behaves the most like divisorial ones.
A key ingredient in our proof of Theorem A is the following result of Knaf and Kuhlmann
which 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 of
any characteristic, and let ν be an Abhyankar valuation of K/k such that the residue field
1

κν is separable over k. Then for any finite set Z ⊆ Rν , Rν is centered on a regular local
ring (A, mA , κA ) essentially of finite type over k with fraction field K satisfying the following
properties:
(1) The Krull dimension of A equals d := dimQ (Q ⊗Z Γν ).
(2) Z ⊆ A, and there exists a regular system of parameters {x1 , . . . , xd } of A such that
every z ∈ Z admits a factorization
z = uxa11 . . . xadd ,
for some u ∈ A× , and ai ∈ N ∪ {0}.
When the ground field k is perfect, any Abhyankar valuation ν admits a local monomialization
since the residue field of the valuation is automatically separable over k. Theorem 1 then
allows us to choose a center of ν on a regular local k-algebra A such that A has a regular
system of parameters {x1 , . . . , xd } whose valuations freely generate Γν , and the residue field
of A coincides with κν (Lemma 3.0.1). Our strategy then is to identify a suitable Frobenius
splitting of A that lifts to a Frobenius splitting of Rν .
A proof of Theorem A was announced in [DS16, Theorem 5.1], where Frobenius splitting
was deduced as a consequence of the incorrect assertion that Abhyankar valuation rings
are F-finite. On the contrary, [DS17, Theorem 0.1] shows that finiteness of Frobenius for
valuation rings of function fields (a function field is a finitely generated extension of a base
field) is equivalent to the associated valuation being divisorial, and so non F-finite Abhyankar
valuations 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 of
an F -finite function field K/k is Abhyankar if and only if
[Γν : pΓν ][κν : κpν ] = [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, Page
281] 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 of
function fields, that admit a Noetherian local center. When a valuation ν of an arbitrary field
K is centered on a Noetherian, local domain (R, mR , κR ) such that Frac(R) = K, one has the
following beautiful inequality established by Abhyankar [Abh56, Theorem 1]:
dimQ (Q ⊗Z Γν ) + tr. deg κν /κR ≤ dim R.

(1.0.0.2)

When equality holds in (1.0.0.2), ν behaves a lot like an Abhyankar valuation of a function
field. For example, the value group Γν is then again a free abelian group of finite rank, and
the residue field κν is finitely generated over κR . However, whether a valuation of a function
field is Abhyankar is intrinsic to the valuation, while equality in (1.0.0.2) with respect to a
center 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 that
are 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
2

in (1.0.0.2) is not intrinsic to a valuation, the property of possessing Abhyankar centers from
a 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 finite
type over k with fraction field K, then a valuation ν admits an Abhyankar center from the
collection 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 possessing
Abhyankar centers that are locally of finite type over k is intrinsic to valuations of function
fields over k.
Our investigation reveals that even in a non function field setting, one can find a reasonably
broad class of Noetherian local rings such that the property of admitting an Abhyankar center
from this class is independent of the choice of the center. More precisely, we show the following:
Theorem 4.0.3. Let (R, mR , κR ) be a Noetherian, F -finite, local domain of characteristic
p > 0 and fraction field K. Suppose ν is a non-trivial valuation of K centered on R with
value group Γν and valuation ring (V, mν , κν ). Then R is an Abhyankar center of ν if and
only if
[Γν : pΓν ][κν : κpν ] = [K : K p ].
Since the identity [Γν : pΓν ][κν : κpν ] = [K : K p ] does not depend on the center, Theorem
4.0.3 implies that possessing F -finite Abhyankar centers is intrinsic to a valuation. Although
imposing 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 function
fields. First note that it generalizes (1.0.0.1) (see Remark 4.0.8(i)). Also, if K is a function
field over a perfect field k of prime characteristic, then Theorem 4.0.3 and (1.0.0.1) imply that
any valuation of K/k that possesses an excellent Abhyankar center is an Abhyankar valuation
of K/k (Corollary 4.0.5). Admitting excellent Abhyankar centers is a priori much weaker than
admitting Abhyankar centers that are essentially of finite type over k, and so Corollary 4.0.5
is not at all obvious. In fact, the corresponding statement is false when the ground field k has
characteristic 0 – one can easily construct a non-Abhyankar valuation of K/C which has an
Abhyankar center that is an excellent local domain (see Remark 4.0.8(vi) or [ELS03, Example
1(iv)]).
Theorem 4.0.3 does not claim that if a valuation ν of an F -finite field K satisfies [Γν :
pΓν ][κν : κpν ] = [K : K p ], then ν is centered on an excellent, local domain; the latter assertion
is false even when K is not perfect. Indeed, using the theory of F -singularities of valuations
developed in [DS16, DS17], one can show that if the valuation ring of ν is F -finite, then
the identity [Γν : pΓν ][κν : κpν ] = [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 on
any excellent local domain of its fraction field (Section 5, (8)), and explicitly construct such
a 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. On
the 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
3

when the valuation rings contain the ground field (see also Remark 5.0.6). Thus, pathologies
such as Example 5.0.5 cease to exist for valuations trivial on the ground field of a function
field (see also (Section 5, (8))).
In the final section of this paper (Section 5), we summarize all known results on Fsingularities of valuation rings for the convenience of the reader, mainly drawing from the
paper [DS16] and the accompanying corrigendum [DS17]. Results are grouped according to
the type of F -singularity they characterize, which we hope will make it easier for the reader
to 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) we
show that when ν/ν p is totally unramified or has maximal defect, that is, when
[Γν : pΓν ][κν : κpν ] = 1,
then the valuation ring of ν cannot be Frobenius split. To put this in context, Theorem A
and (1.0.0.1) imply that, in contrast, at least for valuations ν of function fields over perfect
ground fields, if ν/ν p is defectless then the valuation ring of ν is Frobenius split. At the same
time, Frobenius splitting is not at all well understood when the defect of ν/ν p is not one of
two possible extremes.
Acknowledgments: The question of Frobenius splitting of valuation rings arose in conversations of Zsolt Patakfalvi, Karl Schwede and Karen Smith. While Frobenius splitting of
Abhyankar valuations was suspected in my prior work with Karen, the idea of using local
uniformization took shape while I was visiting University of Utah. I thank Karl Schwede,
Raymond Heitmann, Linquan Ma and Anurag Singh for many fruitful conversations during
my stay in Utah, and Karl for the invitation. I am also grateful to Karen, Linquan and
Raymond Heitmann for helpful comments on a draft of this paper. Further thanks go to
Franz-Viktor Kuhlmann, whose question on the relationship between defect and Abhyankar
valuations inspired the material of Section 4, and to Steven Dale Cutkosky for helpful conversations. My work was supported by department and summer fellowships at University of
Michigan and NSF Grant DMS #1501625.

2. Background and conventions
2.1. Valuations. All valuations are written additively, and local rings are not necessarily
Noetherian. We say a valuation ν with valuation ring V is centered on a local ring A, or A is
a 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 of
the 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 (Rν , mν , κν ) and
value group Γν satisfies the fundamental inequality
dimQ (Q ⊗Z Γν ) + tr. deg κν /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.
4

For Abhyankar valuations it is well-known [Bou89, VI, §10.3, Corollary 1] that Γν is a free
abelian group of finite rank equal to
d := dimQ (Γν ⊗Z Q),
and the residue field κν 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 : kp ]pn and [κν : κpν ] = [k : kp ]pn−d .
An important fact, used implicitly throughout the paper, is that if x, y ∈ K × are non-zero
elements such that x + y 6= 0 and ν(x) 6= 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 a
field. Equivalently, a DVR is a dimension 1 regular local ring.
2.2. Frobenius. Let R be a ring of prime characteristic. We have the (absolute) Frobenius
map
F : R → R,
which maps an element r ∈ R to its p-th power. The target copy of R is usually considered
as an R-module by restriction of scalars via F , and is then denoted F∗ R. In other words,
e
if r ∈ R and x ∈ F∗ R, then r · x = r p x. For an ideal I of R, by I [p ] we mean the ideal
e]
e
[p
generated by the p -th powers of elements of I. Thus I
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, and
the 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 a
free 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 a
finite map for the localization of any finitely generated algebra over a perfect field of prime
characteristic, 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 Rsummands. 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, called
a Frobenius splitting, in the category of R-modules. When R is reduced, Frobenius is an
isomorphism onto its image Rp , and the existence of a Frobenius splitting is equivalent to the
existence of an Rp -linear map R → Rp that maps 1 7→ 1.
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 induced
map F ⊗ idM : M → F∗ R ⊗R M is injective. Regular rings are F -pure because Frobenius
is flat hence faithfully flat for such rings, and faithful flatness implies purity. Also Frobenius
5

splitting clearly implies F -purity, but the converse is false. For example, any non-excellent
DVR 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 a
function field.
3. Proof of Theorem A
Unless otherwise specified, throughout this section we assume that K is a finitely generated
field extension of an F -finite ground field k, and ν is an Abhyankar valuation of K/k whose
residue field κν is separable over k. We will prove more generally under these assumptions
that the valuation ring Rν is Frobenius split.
Also, we let
d := dimQ (Q ⊗Z Γν ) and n := tr. deg K/k.
The goal is to choose a regular local center of ν satisfying some nice properties, and then
extend a Frobenius splitting of this center to a Frobenius splitting of Rν . The center we seek
is given by the following lemma.
Lemma 3.0.1. Let ν be an Abhyankar valuation as in Theorem 1. Then there exists a regular
local ring (A, mA , κA ) which is essentially of finite type over k with fraction field K satisfying
the following properties:
(1) Rν is centered on A, and κA ֒→ κν is an isomorphism.
(2) A has Krull dimension d, and there exist a regular system of parameters {x1 , . . . , xd }
of A such that {ν(x1 ), . . . , ν(xd )} freely generates the value group Γν .
Proof. Choose r1 , . . . , rd , s1 , . . . , st ∈ Rν such that {ν(r1 ), . . . , ν(rd )} freely generates the value
group Γν , and the images of s1 , . . . , st in κν generate the latter over k. Taking
Z := {r1 , . . . , rd , s1 , . . . , st },
by local monomialization (Theorem 1) there exists a regular local ring (A, mA , κA ) essentially
of finite type over k with fraction field K such that Rν dominates A, A has dimension d,
Z ⊆ A, and there exists a regular system of parameters {x1 , . . . , xd } of A such that every
z ∈ Z admits a factorization
z = uxa11 . . . xadd ,
for some u ∈ A× , and ai ∈ N ∪ {0}. In particular, each ν(ri ) is Z-linear combination of
ν(x1 ), . . . , ν(xd ), which implies that {ν(x1 ), . . . , ν(xd )} also freely generates Γν . Moreover,
our choice of s1 , . . . , st implies that κA = κν .

Remark 3.0.2. For a valuation ν of K/k, the existence of a center which is an essentially of
finite type k-algebra of Krull dimension equal to dimQ (Q ⊗Z Γν ) 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 Lemma
3.0.1, and {x1 , . . . , xd } a regular system of parameters of A whose valuations freely generate
Γν . 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 Ap of rank equal to
[K : K p ] = [k : kp ]pn . For
f := [κν : κpν ] = [k : kp ]pn−d ,
6

if we choose
1 = y1 , y2 , . . . , yf ∈ A,
such that the images of yi in κA = κν form a basis of κν over κpν , then
B := {yj xβ1 1 . . . xβd d : 1 ≤ j ≤ f, 0 ≤ βi ≤ p − 1},
is a free basis of A over Ap . Note the elements yj are units in A.
With respect to the basis B, A has a natural Frobenius splitting
ηB : A → Ap ,
given by mapping 1 = y1 x01 . . . x0d 7→ 1, and all the other basis elements to 0. Extending ηB
uniquely to a K p -linear map
p
ηf
B :K →K
of the fraction fields, we will show that the restriction of ηf
B to Rν yields a Frobenius splitting
p
of Rν , or in other words, ηf
B |Rν maps into Rν .

Claim 3.0.3. For any a ∈ A, either ηB (a) = 0 or ν(ηB (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 > 0 with valuation ring
e
e
Rν , and A a subring of Rν such that Frac(A) = K. Suppose ϕ : A → Ap is an Ap -linear
map, for some e ≥ 1. Consider the following:
(i)
(ii)
(iii)
(iv)

For all a ∈ A, ϕ(a) = 0 or ν(ϕ(a)) ≥ ν(a).
e
e
e
For all a, b ∈ A such that ν(a) ≥ ν(b), if ϕ(abp −1 ) 6= 0, then ν(ϕ(abp −1 )) ≥ ν(bp ).
e
e
ϕ extends to an Rνp -linear map Rν → Rνp .
e
e
ϕ extends uniquely to an Rνp -linear map Rν → Rνp .

Then (ii), (iii) and (iv) are equivalent, and (i) ⇒ (ii). Moreover, if ϕ is a Frobenius splitting
of A satisfying (i) or (ii), then ϕ extends to a Frobenius splitting of Rν .
Proof. For the final assertion on Frobenius splitting, note that the extension of a Frobenius
splitting remains a Frobenius splitting since 1 7→ 1 in the extension.
(i) ⇒ (ii): If ϕ(abp

e −1

) 6= 0, we have
ν(ϕ(abp

e −1

)) ≥ ν(abp

e −1

e

) ≥ ν(bp ),

where the first inequality follows from (i), and the second inequality follows from ν(a) ≥ ν(b).
e

e : K → K p , it suffices to show that ϕ|
e Rν
(ii) ⇒ (iii): Extending ϕ to a K p -linear map ϕ
e
p
maps into Rν . Let r ∈ Rν be a non-zero element. Since K is the fraction field of A and Rν ,
one can express r as a fraction a/b, for non-zero a, b ∈ A. Note
ν(a) ≥ ν(b).
Then

If ϕ(abp

e −1

 
a
1
e
ϕ(r)
e
=ϕ
e
= pe ϕ(abp −1 ).
b
b
e

) = 0, then ϕ(r)
e
= 0, and r maps into Rνp . Otherwise by assumption,
ν(ϕ(abp

e −1

e

)) ≥ ν(bp ),

7

(3.0.4.1)

and so,
ν(ϕ(r))
e
= ν(ϕ(abp

e −1

e

)) − ν(bp ) ≥ 0,

e

that is ϕ(r)
e
is an element of K p ∩ Rν = Rνp .
e

(iii) ⇒ (iv): Since A and Rν have the same fraction field, any extension of ϕ to Rν is
e
e
e : K → Kp ,
obtained as a restriction to Rν of the unique extension of ϕ to a K p -linear map ϕ
and so is also unique. See (3.0.4.1) above for a concrete description of how ϕ extends to Rν .
To finish the proof of the lemma, it suffices to show (iv) ⇒ (ii). But this also follows easily
from (3.0.4.1).

Proof of Claim 3.0.3. Recall that
B = {yj xβ1 1 . . . xβd d : 1 ≤ j ≤ f, 0 ≤ βi ≤ p − 1}
is a basis of A over Ap , where the xi and yj are chosen such that {ν(x1 ), . . . , ν(xd )} freely
generates the value group Γν , and the images of 1 = y1 , y2 , . . . , yf in κν form a basis of κν
over κpν . The Ap -linear Frobenius splitting ηB is given by
X

f
X
ηB
cpj,β1,...,βd yj xβ1 1 . . . xβd d = cp1,0,0,...,0 .
j=1 0≤βi ≤p−1

Thus, we need to show that either cp1,0,0,...,0 = 0 or
X
f

X

cpj,α1 ,...,αd yj xα1 1

. . . xαd d

ν(cp1,0,0,...,0 )

≥ν

cpj,β1 ,...,βd yj xβ1 1

. . . xβd d

j=1 0≤βi ≤p−1



.

P
P
Assuming without loss of generality that fj=1 0≤βi ≤p−1 cpj,β1 ,...,βd yj xβ1 1 . . . xβd d 6= 0, we will
prove the stronger fact that

X
f
X
β1
βd
p
cj,β1 ,...,βd yj x1 . . . xd = inf{ν(cpj,β1 ,...,βd yj xβ1 1 . . . xβd d ) : cpj,β1,...,βd 6= 0}.
ν
j=1 0≤βi ≤p−1

(3.0.4.2)

For two non-zero terms

and

cpk,β1 ,...,βd yk xβ1 1

. . . xβd d

ν(cpj,α1 ,...,αd yj xα1 1 . . . xαd d ) = ν(cpk,β1 ,...,βd yk xβ1 1 . . . xβd d )

in the above sum,
(3.0.4.3)

if and only if
pν(cj,α1 ,...,αd ) + α1 ν(x1 ) + · · · + αd ν(xd ) = pν(ck,β1 ,...,βd ) + β1 ν(x1 ) + · · · + βd ν(xd ). (3.0.4.4)
By Z-linear independence of ν(x1 ), . . . , ν(xd ), for all i = 1, . . . , d, we get
p|(αi − βi ).
Since 0 ≤ αi , βi ≤ p − 1, this means that αi = βi for all i, and moreover, then ν(cpj,α1 ,...,αd ) =
ν(cpk,β1 ,...,βd ). Thus, (3.0.4.3) holds precisely when ν(cpj,α1 ,...,αd ) = ν(cpk,β1 ,...,βd ) and αi = βi ,
for all i = 1, . . . , d.
For ease of notation, let us use α as a shorthand for α1 , . . . , αd , and xα for xα1 1 . . . xαd d .
Then for a fixed non-zero term cpj1 ,α yj1 xα , consider the set
{cpj1 ,α yj1 xα , cpj2 ,α yj2 xα , . . . , cpji ,α yji xα }
8

P
P
of all non-zero terms of fj=1 0≤βi ≤p−1 cpj,β1,...,βd yj xβ1 1 . . . xβd d having the same valuation as
cpj1 ,α yj1 xα . In particular, by the above reasoning we also have
ν(cpj1 ,α ) = ν(cpj2 ,α ) = · · · = ν(cpji ,α ).

Adding these terms of equal valuation, in the valuation ring Rν one can write

where

cpj1 ,α yj1 xα + cpj2 ,α yj2 xα + · · · + cpji ,α yji xα =






cj2 ,α p
cji ,α p
yj1 +
yj2 + · · · +
yji cpj1 ,α xα ,
cj1 ,α
cj1 ,α
yj1 +



cj2 ,α
cj1 ,α

p

yj2 + · · · +



cji ,α
cj1 ,α

p

yji

is a unit in Rν by the κpν -linear independence of the images of yj1 , . . . , yji in κν and the fact
that (cj2 ,α /cj1 ,α )p , . . . , (cji ,α /cj1 ,α )p are units in Rνp . Thus, the valuation of the sum
cpj1 ,α yj1 xα + · · · + cpji ,α yji xα
P
P
equals the valuation of any of its terms. Now rewriting fj=1 0≤βi ≤p−1 cpj,β1,...,βd yj xβ1 1 . . . xβd d
by collecting non-zero terms having the same valuation, (3.0.4.2), hence also the claim, follows.

Examples 3.0.5.
(a) A valuation ring of a function field of a curve over an F -finite ground field is always
Frobenius split. Indeed, such a valuation ring is always centered on some normal affine model
of dimension 1 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 {e1 , . . . , en }
denotes the standard basis of Z⊕n , then e1 > e2 > · · · > en . There exists a unique valuation
νlex on Fp (x1 , . . . , xn )/Fp such that for all i ∈ {1, . . . , n},
νlex (xi ) = ei .
The valuation νlex is clearly Abhyakar since dimQ (Q ⊗Z Z⊕n ) = n, which coincides with
the transcendence degree of Fp (x1 , . . . , xn )/Fp . One can also show that the valuation ring
Rνlex has Krull dimension n and residue field Fp . The valuation is centered on the regular
local ring Fp [x1 , . . . , xn ](x1 ,...,xn ) such that the valuations of the obvious regular system of
parameters freely generate Z⊕n and the residue field coincides with the residue field of νlex .
Then a Frobenius splitting of Rνlex → Rνplex is obtained by extending the canonical splitting
on Fp [x1 , . . . , xn ](x1 ,...,xn ) with respect to the basis
{xβ1 1 . . . xβn : 0 ≤ βi ≤ p − 1}.
This splitting of Fp [x1 , . . . , xn ](x1 ,...,xn ) maps
(
xα1 1 . . . xαnn
xα1 1 . . . xαnn 7→
0

if p|αi for all i,
otherwise.

(c) Let Γ = Z ⊕ Zπ ⊂ R. There exists a valuation ν on Fp (x, y, z)/Fp given by
ν(x) = ν(y) = 1, ν(z) = π.
9

Then dimQ (Q ⊗Z Γ) = 2, and the transcendence degree of the residue field κν /Fp is at least 1
since the image of y/x in the residue field is transcendental over Fp . Therefore the fundamental
inequality (2.1.0.1) implies that ν is Abhyankar. Although the valuation ν is centered on the
regular local ring Fp [x, y, z](x,y,z) , no regular system of parameters can freely generate the
value group because the center has dimension 3, whereas the value group is free of rank 2.
However, blowing up the origin in A3Fp , we see that ν is now centered on the regular local ring
Fp [x, y/x, z/x](x,z/x) , and the valuations of the regular system of paramaters {x, z/x} freely
generate Γ. Furthermore, the residue field of Fp [x, y/x, z/x](x,z/x) can be checked to coincide
with the residue field of the valuation ring. Relabelling y/x and z/x as u, w respectively, a
Frobenius splitting on Rν is obtained by extending the Frobenius splitting of Fp [x, u, w](x,w)
given by the same rule as in (a) with respect to the algebraically independent elements x, u, w
over Fp .
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
[Γν : pΓν ][κν : κpν ] = [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 easily
verified to be pΓν , and the residue field κν p can be identified with κpν . Thus, [Γν : pΓν ] is the
ramification index, and [κν : κpν ] 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 inseparable
extension of K p .
In terms of the theory of extensions of valuations, (4.0.0.1) can be reinterpreted as saying
that a necessary and sufficient condition for ν to be Abhyankar is for the unique extension of
valuations ν/ν p to be defectless (see [FV11, Page 281] for the definition of defect). There is
a natural generalization of the notion of an Abhyankar valuation for valuations of arbitrary
fields. The goal of this section is to introduce this more general notion, and investigate to
what 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 > 0 (not necessarily a function
field), and ν a valuation of K with valuation ring (V, mν , κν ) centered on a Noetherian local
ring (R, mR , κR ) such that
dim(R) < ∞.
Recall that centers, by convention, always have the same fraction field as the valuation ring.
Let Γν be the value group of ν.
Abhyankar greatly generalized (2.1.0.1) in [Abh56, Theorem 1], establishing that in the
above setup,
dimQ (Q ⊗Z Γν ) + tr. deg κν /κR ≤ dim(R).
(4.0.0.2)
Moreover, he showed that if equality holds in the above inequality, then Γν is a free abelian
group and κν is a finitely generated extension of κR .
When K/k is a function field, there is a close relationship between Abhyankar valuations
of K/k, and those valuations of K/k that admit a Noetherian center with respect to which
equality holds in (4.0.0.2). Indeed, if equality holds in (4.0.0.2) for an arbitrary valuation
10

ν of K/k with respect to a center R which is essentially of finite type over k, then ν is an
Abhyankar valuation of K/k. To see this, let n = tr. deg K/k. Then
tr. deg κR /k = n − dim(R),
because R is essentially of finite type over k with fraction field K, and so
tr. deg κν /k = tr. deg κν /κR + n − dim(R).
This implies
dimQ (Q ⊗Z Γν ) + tr. deg κν /k = (dimQ (Q ⊗Z Γν ) + tr. deg κν /κR ) + 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 satisfies
equality in (4.0.0.2) is not an intrinsic property of the valuation, but also depends on the
center R. In contrast, the property of being an Abhyankar valuation is intrinsic to valuations
of function fields. To better illustrate this difference, we construct a valuation of Fp (X, Y )
with two different Noetherian centers such that equality in (4.0.0.2) is satisfied with respect
to one center, but not the other. In our example we work over a base field of characteristic
p > 0, but the construction goes through when the ground field has characteristic 0.
Example 4.0.1. (see also [DS16, Example 4.0.5]) Consider the laurent series field Fp ((t))
with the canonical t-adic valuation, νt , whose corresponding valuation ring is the DVR Fp [[t]].
Choose an embedding of fields
i : Fp (X, Y ) ֒→ Fp ((t))
by mapping X 7→ t and Y 7→ p(t), where
p(t) ∈ Fp [[t]]
such that {t, p(t)} are algebraically independent over Fp . Such a power-series exists because
Fp ((t)) is uncountable, but Fp (t) is countable. Moreover, multiplying p(t) by t, we may even
assume that t|p(t). Then we get a new valuation ν on Fp (X, Y ), given by the composition
i

ν

t
ν := Fp (X, Y )× −
→ Fp ((t))× −→
Z.

The corresponding valuation ring V is a DVR with maximal ideal generated by X. Since
ν(X) = νt (t), ν(Y ) = νt (p(t)) ≥ 1,
(p(t) was scaled so that t|p(t)), we see that ν is centered on Fp [X, Y ](X,Y ) . Furthermore, ν is
also trivially centered on its own valuation ring. As Fp [[t]] dominates V and has residue field
Fp ,
κν = Fp .
Clearly ν satisfies equality in (4.0.0.2) with respect to its valuation ring as a center (this is true
more generally for any discrete valuation), but not with respect to the center Fp [X, Y ](X,Y ) .
Note that ν is not an Abhyankar valuation of Fp (X, Y )/Fp , since
dimQ (Q ⊗Z Z) + tr. deg κν /Fp = 1 6= tr. deg Fp (X, Y )/Fp .
11

Moreover, the valuation ring of ν is not an F -finite DVR. This follows from results stated in
the next section, but we include the justification here. Indeed, since the maximal ideal m of
V is principal, by [Section 5, (3)(i)]
dimκpν (V /m[p] ) = p[κν : κpν ] = p 6= [Fp (X, Y ) : Fp (X, Y )p ],
and so V is not F -finite by [Section 5, (2)]. It turns out that V is also not excellent [Section
5, (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
dimQ (Q ⊗Z Γν ) + tr. deg κν /κR = dim(R).
To summarize our observations, the property of being an Abhyankar valuation of a function
field is intrinsic to a valuation, while whether ν admits Abhyankar centers depends on the
centers. 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 possessing
these more restrictive Abhyankar centers becomes intrinsic to ν.
The interplay between Abhyankar valuations and valuations admitting Abhyankar centers
raises the natural question: does (4.0.0.1) have an analogue for valuations of fields that are
not necessarily function fields? Moreover, can the feature of possessing Abhyankar centers
become intrinsic to a valuation if we restrict the class of admissible centers? The next result
provides an affirmative answer for a broad class of Noetherian centers.
Theorem 4.0.3. Let (R, mR , κR ) be a Noetherian, F -finite local domain of characteristic
p > 0 and fraction field K. Suppose ν is a non-trivial valuation of K centered on R with
value group Γν and valuation ring (V, mν , κν ). Then R is an Abhyankar center of ν if and
only if
[Γν : pΓν ][κν : κpν ] = [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 > 0. If ν admits a
Noetherian, F -finite center which is Abhyankar, then any other Noetherian, F -finite center
of ν is also an Abhyankar center of ν.
In other words, the property of possessing Noetherian, F -finite, Abhyankar centers is intrinsic to a valuation.
Proof of Corollary 4.0.4. The proof follows easily from Theorem 4.0.3 using the observation that the identity [Γν : pΓν ][κν : κpν ] = [K : K p ] is independent of the choice of a
center.

Corollary 4.0.5. Let ν be a valuation of a function field K/k over a perfect field k of characteristic p > 0 (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.
12

Proof of Corollary 4.0.5. For (1) ⇒ (2), any center of the Abhyankar valuation ν which
is essentially of finite type over k, hence also excellent, is an Abhyankar center of ν. For
the 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 Rp is excellent (it is isomorphic to R), its integral
closure S in K is module finite over Rp , because K is a finite extension of K p . But R is an
Rp -submodule of S, and submodules of finitely generated modules over Noetherian rings are
finitely generated. So R is also module finite over Rp , that is, R is F -finite. Thus ν satisfies
the identity [Γν : pΓν ][κν : κpν ] = [K : K p ] by Theorem 4.0.3, and since K/k is a function
field, (4.0.0.1) implies that ν is an Abhyankar valuation of K/k.

We will prove Theorem 4.0.3 by first developing a connection between the inequality
dimQ (Q ⊗Z Γν ) + tr. deg κν /κR ≤ dim(R)
and the quantities [Γν : pΓν ] and [κν : κpν ]. 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-free
abelian 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 a
torsion-free abelian group such that dimQ (Q ⊗Z Γ) is finite. We have the following:
(1) If L is an algebraic extension of K, then
[L : Lp ] ≤ [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 : Lp ] ≤ pt [K : K p ],
with equality if L is finitely generated over K.
(3) If s = dimQ (Q ⊗Z Γ), then
[Γ : pΓ] ≤ ps ,
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 : Lp ] = [K : K p ] when K ⊆ L is finite is easy (see [DS16, Section 4.6]). So
suppose K ⊆ L is algebraic, and [K : K p ] < ∞. It suffices to show that if a1 , . . . , an ∈ L are
linearly independent over Lp , then
n ≤ [K : K p ].
Let
e := K(a1 , . . . , an ).
L

e is a finite extension K, and so by what we already established,
Since L is algebraic over K, L
e:L
e p ] = [K : K p ].
[L
13

e p ⊆ Lp , it follows
On the other hand, since a1 , . . . , an are linearly independent over Lp , and L
p
e
that a1 , . . . , an are also linearly independent over L . Thus,
e:L
e p ] = [K : K p ],
n ≤ [L
as desired.



Using the previous lemma, we can now relate the ramification index (i.e. [Γν : pΓν ]) and
residue degree (i.e. [κν : κpν ]) 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 > 0 with valuation
ring (V, mν , κν ), centered on Noetherian local domain (R, mR , κR ) such that
[κR : κpR ] < ∞.
We have the following:
(1) [Γν : pΓν ][κν : κpν ] ≤ pdim(R) [κR : κpR ].
(2) R is an Abhyankar center of ν if and only if [Γν : pΓν ][κν : κpν ] = pdim(R) [κR : κpR ].
Proof of Proposition 4.0.7. Throughout the proof, let
s := dimQ (Q ⊗Z Γν ) and t := tr. deg κν /κR .
(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
[Γν : pΓν ] ≤ ps .
On the other hand, since κR is F -finite by hypothesis, and κν has transcendence degree t over
κR , Lemma 4.0.6(2) shows
[κν : κpν ] ≤ pt [κR : κpR ].
Thus,
[Γν : pΓν ][κν : κpν ] ≤ ps+t [κR : κpR ] ≤ pdim(R) [κR : κpR ].
(4.0.7.1)
(2) Suppose R is an Abhyankar center of ν, that is,
s + t = dim(R).
By [Abh56, Theorem 1], Γν is a free abelian group of rank s, and κν is a finitely generated
field extension of κR of transcendence degree t. Again using Lemma 4.0.6, we get
[Γν : pΓν ] = ps and [κν : κpν ] = pt [κR : κpR ],
and so

[Γν : pΓν ][κν : κpν ] = ps+t [κR : κpR ] = pdim(R) [κR : κpR ],
proving the forward implication.
Conversely, if

[Γν : pΓν ][κν : κpν ] = pdim(R) [κR : κpR ]

then

pdim(R) [κR : κpR ] = [Γν : pΓν ][κν : κpν ] ≤ ps+t [κR : κpR ] ≤ pdim(R) [κR : κpR ],
where the inequalities follow from (4.0.7.1). Thus, dim(R) = s + t, which by definition means
that R is an Abhyankar center of ν.

14

Theorem 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 fraction
field K. Then
[κR : κpR ] and [K : K p ] < ∞.
In particular, R satisfies the hypotheses of Proposition 4.0.7, and so Theorem 4.0.3 follows if
we can show that
[K : K p ] = pdim(R) [κR : κpR ].
(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 to
show that when R is F -finite, then for any minimal prime ideal P of the mR -adic completion
b
R,
b
[K : K p ] = pdim(R/P) [κR : κpR ].
b
b is equidimensional.
This shows that dim(R/P)
is independent of P, or in other words that R
However, since P is minimal, we then have
b
b = dim(R),
dim(R/P)
= dim(R)
which confirms (4.0.7.2).



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 > 0 and fraction field K, then
[K : K p ] = pdim(R) [κR : κpR ].

(4.0.8.1)

A careful analysis of the proof of [Kun76, Proposition 2.1] reveals that (4.0.8.1) holds
for any Noetherian, local domain R such that Ω1R/Z is a finitely generated R-module
b is reduced, that is, if R is analytically unramified. We note
and the completion R
that when R is F -finite, it satisfies both these properties. Indeed, since R is a finitely
generated Rp -module,
Ω1R/Z = Ω1R/Rp
b is reduced by [Kun69, Lemma 2.4].
is then a finitely generated R-module, and R
Theorem 4.0.3, hence Corollary 4.0.4, clearly hold more generally for the class of
Noetherian centers of any F -finite field that satisfy (4.0.8.1). While such centers are
quite 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 -finite
DVR of Fp (X, Y ) constructed in Example 4.0.1. In particular, since regular local
rings are analytically unramified, our observations imply that the module of absolute
Kähler 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, then
any valuation ν of K/k admits an F -finite, Noetherian center. For instance, by the
valuative criterion of properness, ν is centered on a proper k-variety with function
field K, and the local ring of this variety at the center is F -finite. Since we observed
that ν is an Abhyankar valuation if and only if any center of ν which is essentially
of finite type over k is an Abhyankar center, it follows by Theorem 4.0.3 that ν is
Abhyankar precisely when [Γν : pΓν ][κν : κpν ] = [K : K p ].
15

(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 is
defectless.
(iv) When [K : K p ] < ∞, the condition [Γν : pΓν ][κν : κpν ] = [K : K p ] does not imply that
the valuation ν admits a Noetherian, F -finite center. See Example 5.0.5 for a counterexample, which shows that counter-examples can be constructed even for valuations
of fields that are not perfect.
(v) For a Noetherian domain R with F -finite fraction field K, the following are all equivalent:
(a) R is F -finite.
(b) R is excellent.
(c) R is a Japanese/N-2 ring.
(d) The integral closure of Rp in K is a finite Rp -module.
We already saw (b) ⇒ (c) ⇒ (d) ⇒ (a) in the proof of Corollary 4.0.5. The hard part
is to show (a) ⇒ (b), which follows from [Kun76, Theorem 2.5]. As a consequence, in
Theorem 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 field
of a valuation is F -finite, the property of admitting excellent Abhyankar centers is
intrinsic to the valuation. In particular, this is true for valuations of function fields
over perfect ground fields of prime characteristic.
(vi) The analogue of (v) is false for valuations of function fields over ground fields of
characteristic 0, that is, whether a valuation admits an excellent Abhyankar center
depends on the excellent center. For instance, any DVR of characteristic 0 is automatically excellent [Sta17, Tag 07QW], and by imitating the construction of Example
4.0.1 using the fields C(X, Y ) and C((t)) instead, one can show that there exists a
discrete valuation ν of C(X, Y )/C centered on C[X, Y ](X,Y ) such that the latter is not
an 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, Abhyankar
center of ν because ν is discrete. The same example also shows that Corollary 4.0.5
is false over ground fields of characteristic 0.
This remark and (v) indicate that excellent rings in characteristic p > 0 behave very
differently from excellent rings in characteristic 0, and that the notion of excellence
in prime characteristic is more restrictive than in characteristic 0.

5. Summary of F -singularities for valuation rings
We summarize all known results on F-singularities of valuation rings, grouping them according 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 > 0 with associated valuation
ν, singularities of V defined using the Frobenius map are intimately related to properties of
the 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 identified
with κp . 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.
16

In what follows Γ or Γν will always denote the value group of a valuation (ring), and κ or
κν its residue field.
Flatness of Frobenius and F-purity:
(1) [DS16, Theorem 3.1] The Frobenius endomorphism on any valuation ring of prime
characteristic is always faithfully flat. Hence a valuation ring of prime characteristic
is F -pure, and so close to being Frobenius split.
Remark 5.0.1. It is not difficult to construct valuation rings for which Frobenius
is pure but not split. For example, the non F -finite DVR of Example 4.0.1 is not
Frobenius split, because any Frobenius split Noetherian domain with F -finite fraction
field 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 necessary
condition for V to be F -finite is that [K : K p ] < ∞, that is, K is F -finite. So we implicitly
assume in our discussion of F -finiteness of V that K is F -finite to begin with. Note F finiteness of K also implies
[κ : κp ] < ∞,
that is, the residue field is always F -finite. This follows by observing that [κ : κp ] is the
residue degree of the extension of valuations ν/ν p and then using [Bou89, VI, §8.1, Lemma
2].
(2) The
(a)
(b)
(c)

following are equivalent:
V is F -finite.
V is a free V p -module of rank [K : K p ].
dimκp (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 equivalence
of (c) to (a) and (b) is not explicitly stated in [DS16, DS17]. Thus, we include a
complete proof here.
Let n := [K : K p ]. We show (b) and (c) are equivalent. Suppose V is a free module
of rank n over the subring V p , which is a valuation ring of K p . If η is the maximal
ideal of V p , we see that
∼ V ⊗V p V p /η
V /m[p] =
is a free κp = V p /η-module of rank n, which proves (b) ⇒ (c). For the converse,
suppose dimκp (V /m[p] ) = [K : K p ] = n. Choose x1 , . . . , xn ∈ V such that the images
of xi in V /m[p] form a κp -basis, and let
L := V p x1 + · · · + V p xn .
Note L is a finitely generated, torsion free V p -module, hence free over V p since finitely
generated torsion-free modules over valuation rings are free. To prove (b), it suffices
to show that
L = V.
The rank of L equals dimκp L/ηL, and it is easy to see that the images of x1 , . . . , xn
in L/ηL form a κp -basis of L/ηL. Thus, L is a free V p -module of rank n, and so
{x1 , . . . , xn } is a V p -basis of L.
17

Observe that {x1 , . . . , xn } is also linearly independent over K p , and since [K :
= n, this means that {x1 , . . . , xn } is a K p -basis of K. Let s ∈ V be a non-zero
element, and r1 , . . . , rn ∈ K p such that

K p]

s = r1 x1 + · · · + rn xn .
Clearly V = L, if we can prove that all the ri are elements of V p . By renumbering
the xi , and using the fact that V p is a valuation ring of K p , we may assume without
loss of generality that r1 6= 0 and
ri r1−1 ∈ V p ,
for all i ≥ 2. If r1 ∈ V p , then all the ri are already in V p . If not, then r1−1 is an
element of the maximal ideal, η, of V p . Thus,
r1−1 s = x1 + r2 r1−1 x2 + · · · + rn r1−1 xn ,
which contradicts κp -linear independence of the images of x1 , . . . , xn in V /ηV =
V /m[p] . Hence all the ri are elements of V p , and we are done.

(3)

(i) [DS17, Erratum, Lemma
( 2.2] For (V, m, κ) and K as above,
[κ : κp ]
if m is not finitely generated,
dimκp (V /m[p] ) =
p
p[κ : κ ]
if m is finitely generated.
Sketch of proof of (i). The assertion follows from the short exact sequence of
κp -vector spaces
0 → m/m[p] → V /m[p] → κ → 0,
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
dimκp (m/m[p] ) = (p − 1)[κ : κp ].

(ii) For an F-finite, valuation ring V with value group Γ 6= 0:
(a) [K : K p ] = [Γ : pΓ][κ : κp ] = dimκp (V /m[p] ) ([DS17, corrected Thm 4.3.1]
and (2)).
(b) The value group Γ satisfies [Γ : pΓ] = 1 or [Γ : pΓ] = 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 : pΓ][κ : κp ] = [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 : κp ], 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 : pΓ][κ : κp ] = [K : K p ].
Proof of (iv). If V is F -finite, then [Γ : pΓ][κ : κp ] = [K : K p ] by (ii)(a) above.
For the converse, we have
[K : K p ] = [Γ : pΓ][κ : κp ] = [Z : pZ][κ : κp ] = p[κ : κp ] = dimκp (V /m[p] ),
where we use (3)(i) for the final equality. So V is F -finite by (2).



Remark 5.0.3. The proof of (iv) shows more generally that if V is a valuation
ring with principal maximal ideal m and such that [Γ : pΓ] = p, then [Γ : pΓ][κ :
κp ] = [K : K p ] implies V is F -finite.
18

F -finiteness and finite field extensions:
(4) [DS17, Section 3] Let K ⊆ L be a finite extension of F -finite fields. Let ν be a
valuation on K and w an extension of ν to L. Then the valuation ring of ν is F -finite
if 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
[Γν : pΓν ][κν : κpν ] = [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 associated
valuation 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, its
valuation ν is divisorial. However, (3)(ii)(a) and (5) imply that ν is Abhyankar, and
so has a finitely generated value group. Using (3)(ii)(d) one then concludes V is a
DVR. Now a classical result of Zariski shows that any Abhyankar DVR is divisorial
[SZ60, VI, §14, Theorem 31].

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
Fp (X, Y )/Fp with value group Z⊕2 (ordered lexicographically) is Abhyankar (see
Example 3.0.5(a)), but not F -finite because
dimκp (V /m[p] ) = p[κ : κp ] = p[Fp : Fp ] = p 6= [Fp (X, Y ) : Fp (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 Fp .
(ii) When not in a function field, it is easy to construct non-Noetherian
F -finite
S
e
∞
valuation rings. For example, the perfection Fp [[t1/p ]] := e∈N Fp [[t1/p ]] of the
power series ring Fp [[t]] is a non-Noetherian, F -finite valuation ring of its fraction
∞
field Fp ((t1/p )). More generally, a non-trivial valuation ring of any perfect field
of prime characteristic is not Noetherian, but F -finite because Frobenius is an
isomorphism for such a ring. Rings of prime characteristic for which Frobenius
is an isomorphism are called perfect rings. Such rings have been extensively
investigated of late since finding applications in Scholze’s work on perfectoid
spaces (see [Sch12], [BS16]). While perfect rings are trivially F -finite, there exist
non-Noetherian, F -finite valuation rings that are not perfect.
Suppose L is a perfect field of prime characteristic equipped with a non-trivial
valuation ν with value group Γν . For instance L can be a perfectoid field, or the
algebraic closure of a field which has non-trivial valuations. Then the residue
field κν of the associated valuation ring is also perfect. Now consider the group
Γ′ := Γν ⊕ Z
19

ordered lexicographically, and the field L(X), where X is an indeterminate. There
exists a unique extension w ofPthe valuation ν to L(X) with value group Γ′ such
that for any polynomial f = ni=0 ai X i in L[X], we have
w(f ) = inf{(ν(ai ), i) : i = 0, . . . , n}.

The residue field κw of w equals the residue field κν [Bou89, VI, §10.1, Proposition
1], hence is also perfect. Also, Γ′ has a smallest element > 0 in the lex order,
namely (0, 1). Thus, if (Rw , mw ) is the valuation ring of w, the maximal ideal
mw is principal, and in fact generated by X. Using (3)(i), we see that
p
p
dimκpw (Rw /m[p]
w ) = p[κw : κw ] = p = [L(X) : L(X) ].

(5.0.4.1)

Then Rw is F -finite by (2), not Noetherian because Γ′ = Γν ⊕ Z has rational rank
at least 2, and not perfect because the field L(X) is not perfect.
(iii) Curiously, if instead of taking Γ′ = Γν ⊕ Z ordered lexicographically we take
Γ′ = Z ⊕ Γν ordered lexicographically in the above construction, the resulting
extension w of ν to L(X) (with obvious modifications to the definition of w) does
not have an F -finite valuation ring Rw . Indeed, then the maximal ideal of Rw is
not finitely generated, while the residue field κw still coincides with κν , which is
[p]
perfect. Thus dimκpw (Rw /mw ) = [κw : κpw ] = 1 6= [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 an
Abhyankar center of ν.
Proof of (7). For the forward implication, if V is F-finite, then [Γν : pΓν ][κν : κpν ] =
[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, Γν is a non-trivial finitely generated
abelian 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
[Γν : pΓν ][κν : κpν ] = [K : K p ].
Since V is also a DVR by hypothesis, it is F -finite by (3)(iv).



The above result has the following interesting consequence that we would like to highlight
separately.
(8) Suppose ν is a valuation of an F -finite field K with valuation ring V that satisfies
either of the following conditions:
(a) V is F -finite, but not Noetherian.
(b) dim(V ) > s, where [K : K p ] = ps .
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 is
excellent if and only if it is F -finite (see Remark 4.0.8(v)). Thus, it suffices to show
that ν is not centered on any F -finite, Noetherian local domain if it satisfies (a) or
(b).
20

Suppose ν satisfies (a). As V is not Noetherian, (7) implies that ν cannot be
centered 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 we
have the identity
pdim(R) [κR : κpR ] = [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
dimQ (Q ⊗Z Γν ) ≤ dim(R) ≤ s.
However, it is well-known that the Krull dimension of V is at most dimQ (Q ⊗Z Γν )
[Bou89, § 10.2, Corollary to Proposition 3]. Then dim(V ) ≤ s, which contradicts the
hypothesis of (b). Hence ν cannot be centered on any F -finite, Noetherian domain
with fraction field K.

Example 5.0.5. Let w be the valuation of L(X) (where L is a perfect field) constructed in Remark 5.0.4(ii). The valuation ring Rw satisfies conditions (a) and (b) of
(8). We have already observed that Rw satisfies (a). To see that Rw satisfies (b), note
that the value group of w has a proper, non-trivial isolated/convex subgroup. Thus
Rw has Krull dimension at least 2 [Bou89, §4.5], while [L(X) : L(X)p ] = p.
Although Rw is a valuation ring of a function field, it does not contain the ground
field L. So even though w/wp is defectless, this example does not contradict (5), or
the 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 with
valuation ring V , then (5) shows that V cannot satisfy (8)(a), while (2.1.0.1) shows
that V cannot satisfy (8)(b). Thus, the pathologies of (8) do not arise for valuations
of 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 follows
from (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 which
is 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 ring
is a splinter. We want to show that valuation rings are also splinters. So let V be a
valuation ring (of any characteristic), and
ϕ:V →S
a module finite ring extension. Choose a prime ideal P of S that lies over the zero
ideal of V . Then the composition
ϕ

V −
→ S ։ S/P
is also a module finite ring extension, and it suffices to show this composite extension
splits. Thus we may assume S is a domain, which makes S a finitely generated torsionfree V module. But finitely generated torsion free modules over valuation rings are
21

free. Nakayama’s lemma implies there exists a basis of S over V containing 1, and
then 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.
• dimκp (V /mp ) = [K : K p ].
• [Γ : pΓ][κ : κp ] = [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).

(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,
∼ κ[[t]],
V =
as rings, and clearly κp [[tp ]] is a direct summand of κ[[t]], even when [κ : κp ] is not
finite.

(13) [Section 3] For an Abhyankar valuation ring V of an F -finite function field K/k, if the
residue field κ is separable over k, then V is Frobenius split. In particular Abhyankar
valuation rings over perfect ground fields of prime characteristic are always Frobenius
split.
(14) We have seen in (5) that Abhyankar valuations in F -finite function fields are characterized as those valuations ν such that ν/ν p is defectless, that is, [Γν : pΓν ][κν : κpν ] =
[K : K p ]. At the opposite extreme is a valuation ν such that the extension ν/ν p is
totally unramified, which means that
[Γν : pΓν ][κν : κpν ] = 1.

(5.0.7.1)

ν/ν p

We want to show that when
is totally unramified and K is not perfect, the
valuation ring V of ν cannot be Frobenius split. Note that (5.0.7.1) implies Γν = pΓν
and κν = κpν , that is, the value group is p-divisible and the residue field is perfect.
The p-divisibility of Γν shows that
m = m[p].
Then any Frobenius splitting ϕ : V → V p maps the maximal ideal m of V into the
maximal ideal of V p , thereby inducing a Frobenius splitting of residue fields
ϕ
e : κν → κpν .

However, κν is perfect, so that ϕ
e is just the identity map. Since K is not perfect, ϕ has
a non-trivial kernel, that is, some non-zero x ∈ V gets mapped to 0. By p-divisibility,
one can write x = uy p , for a unit u in V , and y 6= 0. Then 0 = ϕ(x) = y p ϕ(u), which
shows that ϕ(u) = 0. But this contradicts injectivity of ϕ,
e proving that no Frobenius
splitting of V exists.

Remark 5.0.8. The author thanks Ray Heitmann for showing him a proof of (14) in
the special case of Q-valuations of Fp (X, Y ). Our argument above is a generalization
of Heitmann’s argument to all totally unramified extensions ν/ν p .
22

F -regularity:
Two notions of F -regularity were introduced in [DS16], generalizing strong F -regularity
to 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 and
Noetherian hypotheses from the definition of strong F -regularity, while F -pure regularity
replaces splitting of certain maps by purity and seems to be the better notion for rings that
are 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. The
following 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 singularity only for Noetherian rings.
Open Questions: Just as is the case in geometry, our results indicate that Frobenius splitting
is 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 Ffinite ground fields uses the local monomialization result of Knaf and Kuhlmann (Theorem
1), hence also the hypothesis that the residue field of the valuation ring is separable over the
ground field. However, it is probably the case that any Abhyankar valuation ring of an F -finite
function field is Frobenius split, and this will from our proof if one can remove the separability
hypothesis from Theorem 1. Moreover, a natural question is if one can generalize our result on
Frobenius 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 corresponding extension of valuations ν/ν p . For example, (5) shows that at least for function
fields over perfect ground fields, when ν/ν p is defectless, ν is Abhyankar and its valuation
ring 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 two
possible extremes.
For instance, suppose K/k is an F -finite function field. Is there a non-Abhyankar valuation
of K/k whose valuation ring is Frobenius split? We can use (16) to conclude that such
valuations, if they exist, cannot be discrete. On the other hand, Example 5.0.5 shows that
if we relax the condition that the valuation is trivial on the ground field, then there exists a
valuation w of K, not trivial on k, whose valuation ring is Frobenius split. Moreover, w has
23

the feature that it is not centered on any excellent domain whose fraction field is K. However,
even in this example, the extension w/wp is defectless.
There are interesting open questions pertaining to Frobenius splitting even for Noetherian
valuation rings. As far as we know, it is not known if every excellent DVR of prime characteristic is Frobenius split. (10) (resp. (12)) provides an affirmative answer when the fraction
field of a DVR is F -finite (resp. when the DVR is complete). At the same time it is worth
recalling 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.
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530 Church Street, East Hall 4080, Ann Arbor, MI 48109
E-mail address: rankeya@umich.edu

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