arXiv:1309.0135v1 [] 31 Aug 2013

RAMIFICATION OF LOCAL RINGS ALONG VALUATIONS
STEVEN DALE CUTKOSKY AND PHAM AN VINH
Abstract. In this paper we discuss stable forms of extensions of algebraic local rings
along a valuation in all dimensions over a field k of characteristic zero, and generalize
a formula of Ghezzi, Hà and Kashcheyeva describing the extension of associated graded
rings along the valuation for stable extensions of regular algebraic local rings of dimension
two to arbitrary ground fields k of characteristic zero. We discuss the failure of this result
in positive characteristic.

1. Introduction
Suppose that k is a field, K is an algebraic function field over k and ν is a valuation
of K (which is trivial on k). Let Vν be the valuation ring of ν, with maximal ideal mν .
Let Γν be the value group of ν. Important invariants associated to ν are its rank (one
less than the number of prime ideals in Vν ), rational rank (dimQ Γν ⊗Z Q) and dimension
(dim ν = trdegk Vν /mν ). We have that rank ν ≤ rational rank ν and by Abhyankar’s
inequality ([1] and Appendix 2 [12]),
(1)

rational rank ν + dim ν ≤ dim K

∼ Zrr as an unordered group,
where dim K = trdegk K, and if equality holds, then Γν =
where rr = rational rank ν. Such valuations are called Abhyankar valuations.
An algebraic local ring of K is a local ring R which is essentially of finite type over k
and whose quotient field is K. R is dominated by ν if R ⊂ Vν and mν ∩ R = mR is the
maximal ideal of R.
A monoidal transform R → R1 of R is a local ring R1 of the blowup of a regular prime
ideal P of R (R/P is regular). R → R1 is a quadratic transform if R1 is a local ring of
the blow up of the maximal ideal of R. R → R1 is a monoidal transform along ν if Vν
dominates R1 .
For each γ ∈ Γν , let
Pγ (R) = {f ∈ R | ν(f ) ≥ γ} and Pγ+ (R) = {f ∈ R | ν(f ) > γ}.
We define the associated graded algebra of ν on R (as in [10]) as
M
Pγ (R)/Pγ+ (R).
grν (R) =
γ∈Γν

If f ∈ R and ν(f ) = γ, we define the initial form inν (f ) of f in grν (R) as f + Pγ+ (R) ∈
Pγ /Pγ+ . A sequence {Pi }i≥0 in R is called a generating sequence of ν in R if {inν (Pi )}i≥0
generate grν (R) as an R/mR -algebra.
The semigroup of R is
S R (ν) = {ν(f ) | f ∈ R \ {0}}.
The first author was partially supported by NSF.
1

Now suppose that K ∗ is an algebraic function field over k such that K ∗ is finite separable
over K, and ν ∗ is a valuation of K ∗ which is an extension of ν. Let
n = trdegk K ∗ − trdegk Vν ∗ /mν ∗ .
Let
e = [Γν ∗ : Γν ] and f = [Vν ∗ /mν ∗ : Vν /mν ]
be the reduced ramification index and relative degree of ν ∗ over ν.
Suppose that R and S are algebraic local rings for K and K ∗ such that S dominates R
and ν ∗ dominates S (so that ν dominates R).
Lemma 1.1. Suppose that Vν ∗ /mν ∗ = (Vν /mν )(S/mν ). Then [S/mS : R/mR ] = f if and
only if Vν /mν and S/mS are linearly disjoint in Vν ∗ /mν ∗ over R/mR .
Proof. Suppose that [S/mS : R/mR ] = f . Let h1 , . . . , hs ∈ S/mS be linearly independent
over R/mR . Extend this set to a basis h1 , . . . , hf of S/mS over R/mR . Then h1 , . . . , hf
span Vν ∗ /mν ∗ over Vν /mν , so they are linearly independent over Vν /mν .
Now suppose that Vν /mν and S/mS are linearly disjoint over R/mR . There exist
α1 , . . . , αf ∈ S/mS which are a basis of Vν ∗ /mν ∗ over Vν /mν . Then α1 , . . . , αf are linearly
independent over R/mR , so [S/mS : R/mR ] ≥ f . However, a basis of S/mS over R/mR
is linearly independent over Vν /mν , so [S/mS : R/mR ] = f .

We will say that R → S is monomial if R and S are n-dimensional regular local rings and
there exist regular parameters x1 , . . . , xn in R, y1 , . . . , yn in S, an n × n matrix A = (aij )
of natural numbers with Det(A) 6= 0 and units δi ∈ S such that
(2)

xi = δi

n
Y

a

yj ij

j=1

for 1 ≤ i ≤ n. In Theorem 1.1 [3] it is proven that when the ground field k has characteristic
zero, there exists a commutative diagram
(3)

R0 → S 0
↑
↑
R → S

such that the vertical arrows are products of monoidal transforms along ν ∗ and R0 →
S0 is monomial. It is shown in Theorem 5.1 [3] and Theorem 4.8 [4] that the matrix
A0 describing R0 → S0 (with respect to regular parameters x1 (0), . . . , xn (0) in R0 and
y1 (0), . . . , yn (0) in S0 ) can be required to take a very special block form, which reflects
the rank and rational rank of ν ∗ . We will say that R0 → S0 is strongly monomial if it is
monomial and the matrix A0 has this special block form.
In Theorem 6.1 [4] it is shown (assuming that k has characteristic zero) that we can
always find a diagram (3) such that the following conditions hold:
1) R0 → S0 is strongly monomial.
2) If
R1 → S 1
↑
↑
R0 → S 0
is such that R1 → S1 is strongly monomial with respect to regular parameters
x1 (1), . . . , xn (1) in R1 and y1 (1), . . . , yn (1) in S1 , and the vertical arrows are products of monoidal transforms, then
2

2a) The natural group homomorphism
Zn /At1 Zn → Γν ∗ /Γν
defined by
(b1 , . . . , bn ) 7→ [b1 ν ∗ (y1 (1)) + · · · + bn ν ∗ (yn (1))]
is an isomorphism (where A1 is the matrix of exponents of R1 → S1 with
respect to our given systems of parameters).
2b) Vν ∗ /mν ∗ is the join Vν ∗ /mν ∗ = (Vν /mν )(S1 /mS1 ).
2c) Vν /mν and S1 /mS1 are linearly disjoint over R1 /mR1 in Vν ∗ /mν ∗ .
Theorem 6.1 [4] and Lemma 1.1 implies that, given R → S, there exists a monomial
extension R0 → S0 as in (3) satisfying 1) and 2) above. In Theorem 6.3 [4] it is shown
that the extension V → V ∗ can naturally be understood as a direct limit of R0 → S0 as
above.
We will say that R0 → S0 is stable if the conclusions 1) and 2) above hold.
If R → S is stable, we have that
e = Det(A) and f = [S/mS : R/mR ].
where e is the reduced ramification index and f is the relative degree of ν ∗ over ν.
The simplest valuations are the Abhyankar valuations (defined at the beginning of this
section). In this case, we easily obtain a very strong statement comparing the associated
graded rings of the valuations. We have the following proposition.
Proposition 1.2. Suppose that k has characteristic zero, ν is an Abhyankar valuation
and R → S is stable. Then we have a natural isomorphism of graded rings

grν ∗ (S) ∼
= grν (R) ⊗R/mR S/mS [y 1 , . . . , y n ]
where y 1 , . . . , y n are the initial forms of y1 , . . . , yn , with the only relations being
[xi ] = [δi ]y 1ai1 · · · y anin 1 ≤ i ≤ n
obtained from (2) ([δi ] is the class of δi in S/mS ). The degree of the extension of quotient
fields of
grν (R) → grν ∗ (S)
is ef .
Proof. Since R → S is stable, and ν ∗ and ν are Abhyankar valuations, we have that
ν ∗ (y1 ), . . . , ν ∗ (yn ) is a Z-basis of Γν ∗ and ν(x1 ), . . . , ν(xn ) is a Z-basis of Γν .
By Hensel’s lemma, R̂ ∼
= R/mR is a coefficient field of R̂.
= k′ [[x1 , . . . , xn ]] where k′ ∼
Since ν(x1 ), . . . , ν(xn ) are rationally independent, ν has a unique extension to a valuation
ν̂ of the quotient field of R̂, defined by
ν̂(f ) = min{i1 ν(x1 ) + · · · + in ν(xn ) | ai1 ,...,in 6= 0}
if f =
ai1 ,...,in xi11 · · · xinn ∈ k′ [[x1 , . . . , xn ]] (with ai1 ,...,in ∈ k′ ). Since distinct monomials
have distinct values, we have an isomorphism of residue fields Vν /mν ∼
= R/mR .
Hence grν (R) ∼
= R/mR [x1 , . . . , xn ], is a polynomial ring, where xi is the class of xi ,
with the grading deg xi = ν(xi ). Further grν ∗ (S) ∼
= S/mS [y 1 , . . . , y n ], is a polynomial
ring, where y i is the class of yi . The proposition follows.

P

3

If ν is an Abhyankar valuation, and R → S is quasi-finite, we have that S S (ν ∗ ) is finitely
generated as a module over the semigroup S R (ν) by Proposition 1.2.
It is natural to ask if an analog of Proposition 1.2 holds for more general valuations.
We have the essential difference that the valuation groups Γν are not finitely generated
in general. There even exist examples where R → S is quasi-finite but S S (ν ∗ ) is not a
finitely generated module over S R (ν). In Theorem 9.4 [6] an example is given of a finite
monomial extension of two dimensional regular algebraic local rings (over any ground
field) such that S S (ν ∗ ) is not a finitely generated module over S R (ν). This example is
necessarily not stable. Some other examples are given in [5] showing bad behavior of
S S (ν ∗ ) over S R (ν).
However, the conclusions of Proposition 1.2 always hold for stable mappings R → S
when R and S have dimension two (n = 2). By Abhyankar’s inequality, when n = 2,
ν is an Abhyankar valuation unless ν is rational (the value group is order isomorphic to
a subgroup of the rational numbers). We have the following theorem, which generalizes
Proposition 1.2 to this case. This surprising theorem was proven when k is algebraically
closed of characteristic zero and dim K = 2 by Ghezzi, Hà and Kashcheyeva in [7]. If
n = 2, ν is rational and R → S is stable, then R has regular parameters u, v, S has
regular parameters x, y and there exist a unit γ in S such that
u = γxe , v = y,

(4)

where e = |Γν ∗ /Γν | is the reduced ramification index.
Theorem 1.3. Suppose that k is a field of characteristic zero, ν ∗ is a rational 0-dimensional
valuation, n = 2 and R → S is stable. Then

gr ∗ (S) ∼
= gr (R) ⊗R/m S/mS [Z]/(Z e − [γ0 ]−1 [u]),
ν

ν

R

and the degree of the extension of quotient fields of gr ν (R) → grν ∗ (S) is ef .

The remaining sections of this paper are devoted to the proof of this theorem. Our
proof requires the construction of generating sequences for valuations in arbitrary regular
local rings of dimension two in [6]. Theorem 1.3 is proven in Section 4, as a consequence
of Proposition 4.1, which shows that a generating sequence in R is almost a generating
sequence in S if R → S is stable.
In contrast to the fact that finite generation may not hold even for a monomial mapping,
when ν ∗ is a rational 0-dimensional valuation with n = 2 (Example 9.4 [6]), we have finite
generation if R → S is stable.
Corollary 1.4. Suppose that k is a field of characteristic zero, ν ∗ is a rational 0-dimensional
valuation, n = 2 and R → S is stable. Then the semigroup S S (ν ∗ ) is a finitely generated
S R (ν)-module.
An interesting question is if an analogue of the conclusions of Proposition 1.2 holds in
general for any n and arbitrary valuations for stable mappings over fields k of characteristic
zero. It would be remarkable if this were true.
With some small modification in the definition of strongly monomial (in (3)), strong
monomialization holds for Ahhyankar valuations in positive characteristic, as follows from
[9], (a strong form of local uniformization is proven for Abhyankar valuations by Knaf
and Kuhlmann), and thus Proposition 1.2 holds in positive characteristic. A description
of grν (R) for ν an Abhyankar valuation dominating a (singular) local ring R, over an
algebraically closed field of arbitrary characteristic, and a proof of local uniformization for
4

Abhyankar valuations derived from this construction, has been recently given by Teissier
in [11].
Over fields of positive characteristic, it is shown in Section 7.11 of [4] that the strong
monomialization theorem is not true, even when n = 2, k is algebraically closed and ν is
rational and zero dimensional. It is not known if monomialization holds, although it seems
unlikely. Stable forms are given in [4] for mappings in dimension two over an algebraically
closed field of positive characteristic which are much more complicated than in the characteristic zero case. The fundamental obstruction to obtaining strong monomialization is
the defect. It is shown in [4] that strong monomialization holds in dimension two over
algebraically closed fields k for extensions of valuations for which there is no defect. In
[8], Ghezzi and Kashcheyeva prove Theorem 1.3 when k is algebraically closed of positive
characteristic, dim K = 2 and the extension has no defect.
In the example of Section 7.11 of [4], the stable forms Ri → Si satisfy
(5)

grν (Ri ) → grν ∗ (Si )

is integral but not finite, in contrast to the case of Proposition 1.2 and Theorem 1.3. In fact,
grν (Ri ) = grν ∗ (Si )p . Further, S Si (ν ∗ ) is not a finitely generated S Ri (ν)-module for any
∗
i. In this example, the degree of the extension of quotient fields of (5) is ef pδ(ν /ν) = p2 ,
where δ(ν ∗ /ν) = 2 is the defect of ν ∗ over ν. The defect is always zero in characteristic
zero, and for Abhyankar valuations.
2. A modification of the algorithm of [6] to construct a generating
sequence
Suppose that k is a field of characteristic zero and K is a two dimensional algebraic
function field over k. Suppose that ν is a rational 0-dimensional valuation of K (the
value group is isomorphic as an ordered group to a subgroup of Q and trdegk Vν /mν = 0).
Suppose that R is a regular algebraic local ring of K such that ν dominates R.
Let
R → T1 → T2 → · · ·
be the sequence of quadratic transforms of R along ν, so that Vν = ∪Ti . Suppose that
x, y are regular parameters in R. There exists a smallest value i such that the divisor of
xy in spec(Ti ) has only one component.
(6)

Define R1 = Ti .

We consider the algorithm of Theorem 4.2 [6] to construct a generating sequence in R
with Remark 4.3 [6] and the following observation: We can replace Ui with a unit τi ∈ R
times Ui in the algorithm. The algorithm (which we will call the modified algorithm to
construct a generating sequence) iterates in the following way. Suppose that for i ≥ 0 we
have constructed the first i + 1 terms
P0 = x, P1 = y, P2 , . . . , Pi
of a generating sequence by the (modified) algorithm. To produce the next term Pi+1 , the
algorithm proceeds as follows. First we compute
ni = [G(ν(P0 ), . . . , ν(Pi )) : G(ν(P0 ), . . . , ν(Pi−1 ))].
This allows us to find a suitable element
(7)

ω (i)

ω (i)

Ui = P0 0 P1 1

5

ω

(i)

i−1
· · · Pi−1
τi

with τi ∈ R an arbitary unit, such that ν(Pini ) = ν(Ui ). Let
"
#
Pini
(8)
αi =
∈ Vν /mν ,
Ui
and
fi (z) = z di + bi,di −1 z di −1 + · · · + bi,0
be the minimal polynomial of αi over R/mR (α1 , . . . , αi−1 ). Then the algorithm produces
an element Pi+1 ∈ R of the form
!
dX
λt
i −1
X
j (s,t)
ji−1 (s,t)
as,t P0 0
· · · Pi−1
Pitni
(9)
Pi+1 = Pini di +
t=0

s=1

where as,t ∈ R are units, j0 (s, t), . . . , ji−1 (s, t) ∈ N with 0 ≤ jk (s, t) < nk for k ≥ 1 and
0 ≤ t < di such that
j (s,t)

ν(P0 0

j

i−1
· · · Pi−1

(s,t)

Pitni ) = ni di ν(Pi )

for all s, t, and
(10)

bi,t =

"

λt
X

j (s,t)

as,t

P0 0

s=1

j

i−1
· · · Pi−1

(s,t)

Uidi −t

#

∈ Vν /mν .

Then P0 , P1 , . . . , Pi , Pi+1 are the first i + 2 terms of a generating sequence for ν in R.
The observation of Remark 4.3 [6] is that any choice of (9) such that (10) holds gives
an extension Pi+1 to the next term in a generating sequence, satisfying the conclusions of
Theorem 4.2 [6].
We will consider the (modified) algorithm of Theorem 4.2 [6] in various rings R with
given regular parameters x, y. We will denote
Pi (R) = Pi so P0 (R) = x, P1 (R) = y,
ni (R) = ni , Ui (R) = Ui , αi (R) = αi , fiR (z) = fi (z), di (R) = di , ni (R) = ni = di ni .
These calculations not only depend on R, but on the previous terms P0 , P1 , . . . , Pi−1
constructed in the algorithm.
We will also consider the algorithm of Theorem 7.1 [6] in different rings R, with given
regular parameters x, y, and a generating sequence
x = P0 , y = P1 , P2 , . . . , Pi , . . .
constructed by the (modified) algorithm 4.2 of [6]. This algorithm considers the birational
extension ring R1 of R defined by (6).
The positive integers n1 and ω0 (1) of Theorem 4.2 [6] are defined by the conditions that
n1 ν(y) = ω0 (1)ν(x) and gcd(n1 , ω0 (1)) = 1. Choose a, b ∈ N so that n1 b − ω0 (1)a = 1.
Define
(11)

x1 =

xb
y n1
,
y
=
.
1
ya
xω0 (1)

Let
(12)

σ = [y1 ] ∈ Vν /mν ,
6

which is nonzero. Then (as is shown in Theorem 7.1 [6]) R1 /mR1 = R/mR [σ]. Theorem
7.1 [6] shows that
(13)

Q 0 = x1 , Q 1 =

P2
ω0 (1)n1
x1

are regular parameters in R1 , and taking
(14)

Qi =

Pi+1
ω0 (1)n1 ···ni
Q0

for 1 ≤ i, the Qi are a generating sequence for ν in R1 produced by the algorithm of
Theorem 4.2 [6] (as interpreted by Remark 4.3 [6]).
We will consider the algorithm of Theorem 7.1 [6] in different rings R, and will denote
Qi (R1 ) = Qi , β̂i (R1 ) = β̂i , Vi (R1 ) = Vi , α̂i (R1 ) = α̂i
in the notation of the proof of Theorem 7.1 [6].
We have that the Vi (R1 ) constructed in the the proof of Theorem 7.1 [6] for R → R1
are actually the Ui (R1 ) as constructed by the algorithm of Theorem 4.2 [6].
Let L0 ∼
= R/mR be a coefficient field of R̂, so that R̂ = L0 [[x, y]].
R1 = R[x1 , y1 ]mν ∩R[x1 ,y1 ] .
ν(x1 ) > 0 and ν(y1 ) = 0. We have that
R1 /mR1 ∼
= L0 [σ],
where σ is the class of y1 in R1 /mR1 . Let L1 ∼
= L0 (σ) be a coefficient field of R̂1 containing
L0 (this is possible since k has characteristic zero, by Hensel’s Lemma). Let
y1∗ = y1 − σ ∈ mRˆ1 .

(15)

Thus x1 , y1∗ are regular parameters in R̂1 and hence
R̂1 = L1 [[x1 , y1∗ ]].
We have an expression
ω (1)

x = xn1 1 (y1∗ + σ)a , y = x1 0

(y1∗ + σ)b

in R̂1 .
3. Monomial forms under sequences of quadratic transforms
Suppose that k is a field of characteristic zero, and K → K ∗ is an extension of two
dimensional algebraic function fields over k. Suppose that ν ∗ is a rational 0-dimensional
valuation of K ∗ which restricts to ν. Suppose that R and S are regular algebraic local
rings of K and K ∗ respectively such that ν ∗ dominates S and S dominates R.
By Theorem 5.1 [3] and Theorem 4.8 [4] (summarized after (3)), there exists a sequence
of quadratic transforms along ν ∗
R′ → S ′
↑
↑
R → S
7

such that R′ → S ′ is strongly monomial. For this type of valuation, this means that R′
has a regular system of parameters u, v and S ′ has a regular system of parameters x, y
giving an expression
u = γ0 x t , v = y

(16)

where γ0 is a unit in S ′ . For the rest of this section, we will assume that R → S is strongly
monomial (so R, S have regular parameters satisfying (16)), but we do not assume that
R → S is stable.
Lemma 3.1. Suppose that R has regular parameters u, v and S has regular parameters
x, y giving an expression
u = γ0 x t , v = y
where γ0 is a unit in S. Let R → R1 be the sequence of quadratic transforms along ν
defined by (6) and Let S → S1 be the sequence of quadratic transforms along ν ∗ defined
by (6). Then R1 has regular parameters u1 , ṽ1 and S1 has regular parameters x1 , ỹ1 such
that
u1 = γ1 xt11 , ṽ1 = ỹ1
where γ1 is a unit in S1 .
Proof. We use the notation of the previous section. We have that
R1 = R[u1 , v1 ]mν ∩R[u1 ,v1 ] ,
n (R) a(R)
v1 , v

u = u1 1

ω (1)(R) b(R)
v1

= u1 0

with
n1 (R)b(R) − ω0 (1)(R)a(R) = 1.
We have that

v n1 (R)
uω0 (1)(R)
so that ν(v1 ) = 0 and [v1 ] = σ(R) in Vν /mν . We have
v1 =

u1 =

ub(R)
.
v a(R)

Further,
S1 = S[x1 , y1 ]mν ∗ ∩S[x1,y1 ]
where

n (S) a(S)

ω (1)(S) b(S)
y1

x = x1 1 y 1 , y = x1 0
with n1 (S)b(S) − ω0 (1)(S)a(S) = 1. We have that

y n1 (S)
xω0 (1)(S)
so that ν ∗ (y1 ) = 0, and [y1 ] = σ(S) in Vν ∗ /mν ∗ . We have
y1 =

(17)
Substitute

x1 =

xb(S)
.
y a(S)

b(R)

u1 = ub(R) v −a(R) = γ0 xtb(R) y −a(R)
b(R) n (S) a(S)
ω (1)(S) b(S) −a(R)
= γ0 (x1 1 y1 )tb(R) (x1 0
y1 )
b(R) n (S)tb(R)−ω0 (1)(S)a(R) a(S)tb(R)−a(R)b(S)
y1
.
= γ0 x1 1
8

Set t1 = n1 (S)tb(R) − ω0 (1)(S)a(R). Since ν ∗ (u1 ) > 0, ν ∗ (x1 ) > 0 and ν ∗ (γ0 ) = ν ∗ (y1 ) =
0, we have t1 > 0.
v1 = u−ω0 (1)(R) v n1 (R) = (γ0 xt )−ω0 (1)(R) y n1 (R)
−ω (1)(R) n1 (S) a(S) −tω0 (1)(R) ω0 (1)(S) b(S) n1 (R)
= γ0 0
(x1
y1 )
(x1
y1 )
−ω0 (1)(R) ω0 (1)(S)n1 (R)−tω0 (1)(R)n 1 (S) b(S)n1 (R)−a(S)tω0 (1)(R)
y1
.
= γ0
x1
ν ∗ (v1 ) = ν ∗ (y1 ) = ν ∗ (γ0 ) = 0 and ν ∗ (x1 ) > 0 implies
ω0 (1)(S)n1 (R) − tω0 (1)(R)n1 (S) = 0.
Since n1 (S)b(S) − ω0 (1)(S)a(S) 6= 0, we have that


Thus

n1 (S) ω0 (1)(S)
a(S) b(S)



−tω0 (1)(R)
n1 (R)



6=



0
0



.

m := b(S)n1 (R) − a(S)tω0 (1)(R) 6= 0.

(18)

We have that u1 , v1 ∈ S1 , so that
R1 = R[u1 , v1 ]mν ∩R[u1 ,v1 ] ⊂ S1 .
We have a commutative diagram
R̂1 = L1 [[u1 , v1∗ ]] → Sˆ1 = M1 [[x1 , y1∗ ]]
↑
↑
R̂ = L[[u, v]]
→
Ŝ = M [[x, y]]
where L, M, L1 , M1 are coefficient fields of R̂, Ŝ, R̂1 , Sˆ1 such that there are inclusions
L1 → M1
↑
↑
L → M
This is possible (by Hensel’s Lemma) since R, S, R1 , S1 have equicharacteristic zero.
y1∗ = y1 − σ(S), v1∗ = v1 − σ(R) are constructed as in (15). We compute in M1 [[x1 , y1∗ ]],
y1m = (y1∗ + σ(S))m = σ(S)m + mσ(S)m−1 y1∗ +

m(m − 1)
σ(S)m−2 (y1∗ )2 + · · ·
2!

−ω0 (1)(R)

γ0

= β + x1 Ω with 0 6= β ∈ M1 and Ω ∈ Sˆ1 .
In Sˆ1 we have an expression
v1 = (β + x1 Ω)(σ(S)m + mσ(S)m−1 y1∗ + (y1∗ )2 Λ)
= βσ(S)m + βmσ(S)m−1 y1∗ + x1 Ω′ + (y1∗ )2 Λ′

for some Λ ∈ Sˆ1 , Ω′ , Λ′ ∈ Sˆ1 . Thus x1 , v1 − βσ(S)m are regular parameters in Sˆ1 , (and
βσ(S)m = σ(R)). Hence if u1 , v1′ are regular parameters in R1 , then x1 , y1′ = v1′ are regular
parameters in S1 , and we have an expression:
u1 = γ1 xt11
v1′ = y1′
with γ1 a unit in S1 .


9

By iteration of Lemma 3.1 and (6), we obtain an infinite sequence
..
..
.
.
↑
↑
R2 → S 2
↑
↑
R1 → S 1
↑
↑
R → S
where each Ri has regular parameters ui , ṽi and each Si has regular parameters xi , ỹi such
that
ui = γi xtii , ṽi = ỹi
where γi is a unit in Si .
Let e = |Γν ∗ /Γν | and f = [Vν ∗ /mν ∗ : Vν /mν ]. If R → S is stable, then
(19)

ti = e and [Si /mSi : Ri /mRi ] = f

for i ≥ 0.
4. Construction of a generating sequence in S from that of R
In this section, we continue to have the assumptions of Section 3. We further assume
that R → S is stable. Let
P0 (R) = u, P1 (R) = v, P2 (R), . . .
be a generating sequence in R, constructed by the algorithm of Theorem 4.2 [6].
Let P0 (S) = x, P1 (S) = y.
Then we have that the t and t1 in Lemma 3.1 satisfy
(20)

t = |Γν ∗ /Γν | = t1 ,

and
(21)

[S/mS : R/mR ] = [Vν ∗ /mν ∗ : Vν /mν ] = [S1 /mS1 : R1 /mR1 ].

By the calculations in the previous section, we have that



 

b(R)
−a(R)
t 0
n1 (S)
a(S)
t1 ∗
(22)
=
.
−ω0 (1)(R) n1 (R)
0 1
ω0 (1)(S) b(S)
0 m
Taking determinants and using the fact that t1 = t gives t = tm so that m = 1.
Multiplying (22) by


n1 (R)
a(R)
,
ω0 (1)(R) b(R)
we obtain
(23)

n1 (S) = n1 (R), ω0 (1)(S) = tω0 (1)(R).

Since
P1 (S)n1 (S) = P1 (R)n1 (R) ,
we can take U1 (S) to be U1 (R) = uω0 (1)(R) , so
ω (1)(R) tω0 (1)(R)

U1 (S) = uω0 (1)(R) = γ0 0

x

10

ω (1)(R) ω0 (1)(S)

= γ0 0

x

.

ω (1)(R)

That is, we take τ1 = γ0 0
in (7). Thus
# "
# "
"
#
P1 (R)n1 (R)
v n1 (R)
P1 (S)n1 (S)
=
=
α1 (S) =
= α1 (R),
U1 (S)
U1 (R)
uω0 (1)(R)
with the notation of (8). We have that R1 /mR1 = R/mR [σ(R)] and S1 /mS1 = S/mS [σ(S)]
(with notation of (12)).
#
"
v n1 (R)
= σ(R)
α1 (R) =
uω0 (1)(R)
and
α1 (S) =
=

h

y n1 (R)



i



y n1 (R)
= ω0 (1)(R)
tω (1)(R)
uω0 (1)(R)
γ
h 0n (S) ix 0
1
y
−ω
(1)(R)
[γ0 ] 0
= [γ0 ]−ω0 (1)(R) σ(S).
xω0 (1)(S)

Thus R1 /mR1 = R/mR (α1 (R)) and S1 /mS1 = S/mS (α1 (R)).
By (21), we have that

[S1 /mS1 : S/mS ] = [R1 /mR1 : R/mR ]
and thus
d1 (S) = [S/mS (α1 (R)) : S/mS ] = [R/mR (α1 (R)) : R/mR ] = d1 (R),
and the minimal polynomial f1S (z) of α1 (S) over S/mS is the minimal polynomial f1R (z)
of α1 (R) over R/mR . Thus
x, y = P1 (R), P2 (R)
are the first terms of a generating sequence in S, obtained by the (modified) algorithm of
Theorem 4.2 [6].
Proposition 4.1. Suppose that i ≥ 2 and
P0 (S) = x, P1 (S) = y, P2 (S) = P2 (R), . . . , Pi (S) = Pi (R)
are the first i + 1 terms of a generating sequence in S produced by the modified algorithm
of Theorem 4.2 [6]. Then
P0 (S) = x, P1 (S) = y, P2 (S) = P2 (R), . . . , Pi (S) = Pi (R), Pi+1 (S) = Pi+1 (R)
are the first i + 2 terms of a generating sequence in S produced by the modified algorithm
of Theorem 4.2 [6].
Proof. With the assumption, we have that for j ≤ i − 1,
nj (S) = nj (R), αj (S) = αj (R),
dj (S) = [S/mS (α1 (S), . . . , αj (S)) : S/mS (α1 (S), . . . , αj−1 (S))]
= [R/mR (α1 (R), . . . , αj (R)) : R/mR (α1 (R), . . . , αj−1 (R))] = dj (R)
and the minimal polynomial fjS (z) of αj (S) = αj (R) over S/mS (α1 (S), . . . , αj−1 (S)) is
the minimal polynomial fjR (z) of αj (R) over R/mR (α1 (R), . . . , αj−1 (R)).
Theorem 7.1 [6] produces a generating sequence Q0 (R1 ) = u1 , Q1 (R1 ), Q2 (R1 ), . . . in R1
from P0 (R), P1 (R), P2 (R), . . .. The generating sequence Q0 (R1 ) = u1 , Q1 (R1 ), Q2 (R1 ), . . .
in R1 can be produced by the algorithm of Theorem 4.2 from the regular system of
parameters u1 , Q1 (R1 ) in R1 (as shown in Theorem 7.1 [6] and recalled in (22) and (14)).
Since R → S is stable, we have that
(24)

u1 = γ1 xt1
11

for some unit γ1 ∈ S1 , and recalling (23), we have that
(25)

ω0 (1)(S) = tω0 (1)(R).

By the induction hypothesis applied to the stable map R1 → S1 , we have that
x1 , Q1 (R1 ), . . . , Qi (R1 )
are the first i+1 terms of a generating sequence in S1 , produced by the modified algorithm
of Theorem 4.2 [6] in S1 .
For j ≥ 1, let
nj (R1 ), Uj (R1 ), αj (R1 ), dj (R1 ), fjR1 (z)
be the calculations of the algorithm of Theorem 4.2 [6] in R1 , obtained in the construction
of the generating sequence Q0 (R1 ) = u1 , Q1 (R1 ), Q2 (R1 ), . . ..
For j ≤ i, let
nj (S1 ), Uj (S1 ), αj (S1 ), dj (S1 ), fjS1 (z)
be the calculations of the modified algorithm of Theorem 4.2 [6] in S1 , obtained in the construction of the first i+ 1 terms of the generating sequence x1 , Q1 (R1 ), Q2 (R1 ), . . . , Qi (R1 )
in S1 . We have that for j ≤ i − 1,
(26)

nj (S1 ) = nj (R1 ), Uj (S1 ) = Uj (R1 ), αj (S1 ) = αj (R1 ),
dj (S1 ) = dj (R1 ), fjS1 (u) = fjR1 (u).

Since

−ω0 (1)(R)

Q0 (S1 )ω0 (1)(S) = Q0 (R1 )ω0 (1)(R) γ1
we have from (14) that for j ≤ i − 1,
ω (1)(R)n1 (S)···nj (S)

Qj (S1 ) = γ1 0

ω (1)(R)n1 (S)···nj (S)

= γ1 0

,

Pj+1 (S)
Q0 (R1 )ω0 (1)(R)n1 (S)···nj (S)

Qj (R1 ).

For j ≤ i − 1, we have by (14) and (26), and then by (24) and (25), that
ν ∗ (Qj (R1 )) = ν ∗ (Pj+1 (R)) − ω0 (1)(R)n1 (R) · · · nj (R)ν ∗ (u1 )
= ν ∗ (Pj+1 (R)) − n1 (R) · · · nj (R)ω0 (1)(S)ν ∗ (x1 ).
Thus
G(ν ∗ (x1 ), ν(Q1 (R)), . . . , ν(Qj (R))) = G(ν ∗ (x1 ), ν(P2 (R)), . . . , ν(Pj+1 (R)))
= G(ν ∗ (x), ν ∗ (y), ν(P2 (R)), . . . , ν(Pj+1 (R)))
= G(ν ∗ (x), ν(P1 (R)), . . . , ν(Pj+1 (R)))
since G(ν ∗ (x1 )) = G(ν ∗ (x), ν ∗ (y)), as calculated before (55) in the proof of Theorem 7.1 [6].
Thus ni−1 (S1 ) = ni (S). We have that ni−1 (S1 ) = ni−1 (R1 ) by (26), and ni−1 (R1 ) = ni (R)
by (55) and (54) in the proof of Theorem 7.1 [6]. Thus
ni (S) = ni (R).
In applying the modified algorithm of Theorem 4.2 [6] to extend x, P1 (R), . . . , Pi (R) to a
generating sequence in S, we can thus take Ui (S) = Ui (R), and then
# "
#
"
Pi (R)ni (R)
Pi (S)ni (S)
=
= αi (R).
αi (S) =
Ui (S)
Ui (R)
We have from (11) that
y1 =

n1 (R)
y n1 (S)
ω0 (1)(R) v
ω (1)(R)
=
γ
= γ0 0
v1 .
0
xω0 (1)(S)
uω0 (1)(R)
12

Thus
σ(S1 ) = [y1 ] = [γ0 ]ω0 (1)(R) [v1 ] = [γ0 ]ω0 (1)(R) α1 (R)
in Vν ∗ /mν ∗ , and
S1 /mS1 = S/mS [α1 (R)] and R1 /mR1 = R/mR [α1 (R)].
For 1 ≤ j ≤ i − 1, by (60) of [6], we have that
αj (S1 ) = αj (R1 ) = α̂j (R1 ) = αj+1 (R)α1 (R)a(R)ω0 (i+1)(R)+b(R)ω1 (i+1)(R) .
Thus
di−1 (S1 ) = [S1 /mS1 (α1 (S1 ), . . . , αi−1 (S1 )) : S1 /mS1 (α1 (S1 ), . . . , αi−2 (S1 ))]
= [S/mS (α1 (R), α2 (R), . . . , αi (R)) : S/mS (α1 (R), . . . , αi−1 (R))] = di (S)
and
di−1 (R1 ) = [R1 /mR1 (α1 (R1 ), . . . , αi−1 (R1 )) : R1 /mR1 (α1 (R1 ), . . . , αi−2 (R1 ))]
= [R/mR (α1 (R), α2 (R), . . . , αi (R)) : R/mR (α1 (R), . . . , αi−1 (R))] = di (R).
We thus have that di (S) = di (R) since di−1 (S1 ) = di−1 (R1 ) by (26). Thus the minimal
polynomial fiS (z) of αi (S) = αi (R) over S/mS (α1 (R), . . . , αi−1 (R)) is the minimal polynomial fiR (z) of αi (R) over R/mR (α1 (R), . . . , αi−1 (R)). Thus we can take Pi+1 (S) = Pi+1 (R)
in the modified algorithm of Theorem 4.2 [6].

We obtain the following theorem (Theorem 1.3 from the Introduction of this paper).
Theorem 4.2 is proven by Ghezzi, Hà and Kashcheyeva in [7] when k is algebraically closed
of characteristic zero.
Theorem 4.2. Suppose that k is a field of characteristic zero, ν ∗ is a rational 0-dimensional
valuation, n = 2 and R → S is stable. Then

gr ∗ (S) ∼
= gr (R) ⊗R/m S/mS [Z]/(Z e − [γ0 ]−1 [u]),
ν

ν

R

and the degree of the extension of quotient fields of gr ν (R) → grν ∗ (S) is ef .

Proof. We have an inclusion of graded algebras grν (R) → grν ∗ (S). The classes [Pi (R)] for
i ≥ 0 generate grν (R) as a grν (R)0 = R/mR -algebra and the classes [P0 (S)] and [Pi (R)] for
i ≥ 1 generate grν ∗ (S) as a grν (S)0 = S/mS -algebra by Theorem 4.11 [6] and Proposition
4.1. We have the relation
(27)

[P0 (S)]t [γ0 ] = [P0 (R)]

in grν ∗ (S). Further,
(28)

ni (R) = ni (S) for i ≥ 1

by Proposition 4.1.
Since grν (R) ⊗R/mR S/mS → grν ∗ (S) is homogeneous, to verify that it is 1-1, it suffices
to show that the homomorphism of S/mS -vector spaces
(29)

grν (R)λ ⊗R/mR S/mS → grν ∗ (S)λ

is 1-1 for all λ ∈ S R (ν). By 2) of Theorem 4.2 [6], the set of all monomials
(30)

[P0 (R)]i0 [P1 (R)]i1 · · · [Pr (R)]ir

such that r ∈ N, ik ∈ N, 0 ≤ ik < nk (R) for 1 ≤ k ≤ r and
i0 ν(P0 (R)) + · · · + ir ν(Pr (R)) = λ
13

is an R/mR -basis of grν (R)λ , and the set of all
(31)

[P0 (S)]j0 [P1 (S)]j1 · · · [Ps (S)]js

such that s ∈ N, jk ∈ N, 0 ≤ jk < nk (S) for 1 ≤ k ≤ s and
j0 ν ∗ (P0 (S)) + · · · + js ν ∗ (Ps (S)) = λ
is an S/mS -basis of grν ∗ (S)λ .
By (27), (28), (30) and (31), we have that (29) is 1-1, so
grν (R) ⊗R/mR S/mS → grν ∗ (S)
is 1-1.
We have established that [P0 (S)] generates grν ∗ (S) as a grν (R) ⊗R/mR S/mS -algebra
and that the relation (27) holds. To establish that the conclusions of the theorem hold,
we must show that if there is a relation
(32)

h0 + [P0 (S)]h1 + · · · + [P0 (S)]t−1 ht−1 = 0

in grν ∗ (S), with hi ∈ grν (R) ⊗R/mR S/mS , then h0 = h1 = · · · = ht−1 = 0. We may
assume that each [P0 (S)]j hj is homogeneous of the same degree λ. Since R → S is stable,
we have that t = [Γν ∗ : Γν ] and iν(P0 (S)) 6∈ Γν for 1 ≤ i ≤ t − 1. Thus there can be at
most one nonzero expression in (32), so all terms are zero.

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[2] S. Abhyankar, Ramification theoretic method in Algebraic Geometry, Princeton University
Press, Princeton, New Jersey, 1959.
[3] S.D. Cutkosky, Local factorization and monomialization of morphisms, Astérisque 260, 1999.
[4] S.D. Cutkosky and O. Piltant, Ramification of valuations, Advances in Math. 183 (2004),
1-79.
[5] S.D. Cutkosky and B. Teissier, Semigroups of valuations on local rings, Mich. Math. J. 57
(2008), 173 - 193.
[6] S.D. Cutkosky and Pham An Vinh, Valuation semigroups of two dimensional local rings, to
appear in the Proceedings of the London Math. Soc.
[7] L. Ghezzi, Huy Tài Hà and O. Kashcheyeva, Toroidalization of generating sequences in
dimension two function fields, J. Algebra 301 (2006) 838-866.
[8] L. Ghezzi and O. Kashcheyeva, Toroidalization of generating sequences in dimension two
function fields of positive characteristic, J. Pure and Applied Algebra 209 (2007), 631 - 649.
[9] H. Knaf and F.-V. Kuhlmann, Abhyankar places admit local uniformization in any characteristic, Ann. Scient. Ec. Norm. Sup 30 (2005), 833 - 846.
[10] B. Teissier, Valuations, Deformations and Toric geometry, Proceedings of the Saskatoon
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Jersey, 1960.
Steven Dale Cutkosky, Department of Mathematics, University of Missouri, Columbia,
MO 65211, USA
E-mail address: cutkoskys@missouri.edu
Pham An Vinh, Department of Mathematics, University of Missouri, Columbia, MO 65211,
USA
E-mail address: vapnnc@mizzou.edu
14

