Monomialization and Ramification of Valuations - · PDF filecase there exists a regular system...

Monomialization and Ramification of Valuations

Steven Dale Cutkosky

July 14, 2012

Steven Dale Cutkosky Monomialization and Ramification of Valuations

Suppose that K is an algebraic function field over a base field k . Avaluation ν of K/k is a surjective group homomorphism

ν : K× → Γν

where Γν is a totally ordered abelian group such that

ν(a + b) ≥ minν(a), ν(b)

for a, b ∈ K×, and ν(c) = 0 for c ∈ k×.Set ν(0) =∞. The valuation ring of ν is

Vν = f ∈ K | ν(f ) ≥ 0.

Vν is a (generally non noetherian) local ring with maximal ideal

mν = f ∈ K | ν(f ) > 0.


Suppose that (R,mR) is a local ring contained in K . R is analgebraic local ring of K if R is essentially of finite type over k andthe quotient field of R is K . Vν dominates R if R ⊂ Vν andmν ∩ R = mR .


Suppose that p ⊂ R is a regular prime (R/p is a regular local ring).If f ∈ p is an element of minimal value then R[ p

f ] is contained inVν , and if q = R[ p

f ] ∩mν , then

R1 = R[pf

]q

is an algebraic local ring of K which is dominated by Vν . We saythat R → R1 is a monoidal transform along ν.

If R is a regular local ring, then R1 is a regular local ring. In thiscase there exists a regular system of parameters (x1, . . . , xn) in Rsuch that if height(p) = r , then

R1 = R

[x2

x1, . . . ,

xr

x1

]q

.


Suppose that K → K ∗ is a finitely generated extension of algebraicfunction fields over k and V ∗ is a valuation ring of K ∗/k of avaluation ν∗ with value group Γ∗. Then the restriction ν = ν∗|K ofν∗ to K is a valuation of K/k with valuation ring V = K ∩ V ∗.Let Γ be the value group of ν. There is a commutative diagram

K → K ∗

↑ ↑V = K ∩ V ∗ → V ∗


The fact that the valuation ring V = V ∗ ∩K has the quotient fieldK is a highly desirable property of valuations rings. There existalgebraic regular local rings R∗ in a finite field extension of K ∗/Ksuch that K ∩ R∗ = k; geometrically this means that there doesnot exist a germ of a (localization) of a finite map from spec(R∗)to a variety with function field K .


Theorem. (local monomialization, [C1], [C5]) Let notation be asabove, and assume that k has characteristic zero. Suppose that S∗

is an algebraic local ring of K ∗ which is dominated by ν∗ and R∗ isan algebraic local ring of K which is dominated by S∗, so thatthere is a commutative diagram:

K → K ∗

↑ ↑V = K ∩ V ∗ → V ∗

↑ ↑R∗ → S∗


Then there exist sequences of monoidal transforms R∗ → R0 andS∗ → S such that ν∗ dominates S and S dominates R0 so thatthere is a commutative diagram

V → V ∗

↑ ↑R0 → S↑ ↑R∗ → S∗,

and there are regular parameters (x1, . . . , xm) in R0, (y1, . . . , yn) inS , units δ1, . . . δm in S and a matrix A = (aij) of nonnegativeintegers such that rank(A) = m such that

xi = δi

n∏j=1

yaij

j for 1 ≤ i ≤ m.


The significance of the rank(A) = m condition is that formally,even in an appropriate etale extension S of S , R0 → S is truly amonomial mapping, as there exist regular parameters (y1, . . . , yn)in S such that

xi =n∏

j=1

yaj

j for 1 ≤ i ≤ m.

In the special case that K = k , The local monomialization theoremrecovers the local uniformization theorem of Zariski [Z].


The starting point of our theory of ramification of generalvaluations is the following theorem, which proves “weaksimultaneous local resolution”, which was conjectured byAbhyankar [Ab], for fields of characteristic zero (Abhyankar gave acounterexample to “strong simultaneous local resolution” [Ab]).

Theorem. (weak simultaneous local resolution, [C2]) Let k be afield of characteristic zero, K an algebraic function field over k , K ∗

a finite algebraic extension of K , ν∗ a valuation of K ∗/k ,withvaluation ring V ∗. Suppose that S∗ is an algebraic local ring of K ∗

which is dominated by ν∗ and R∗ is an algebraic local ring of Kwhich is dominated by S∗.


Then there exists a commutative diagram

R0 → R → S ⊂ V ∗

↑ ↑R∗ → S∗

where S∗ → S and R∗ → R0 are sequences of monoidal transformsalong ν∗ such that R0 → S have regular parameters of the form ofthe conclusions of the local monomialization theorem, R is anormal algebraic local ring of K with toric singularities which is thelocalization of the blowup of an ideal in R0, and the regular localring S is the localization at a maximal ideal of the integral closureof R in K ∗.


proof. By resolution of singularities, we first reduce to the casewhere R∗ and S∗ are regular, and then construct, by the localmonomialization theorem, a sequence of monoidal transformsalong ν∗,

R0 → S ⊂ V ∗

↑ ↑R∗ → S∗

so that R0 is a regular local ring with regular parameters(x1, . . . , xn), S is a regular local ring with regular parameters(y1, . . . , yn), there are units δ1, . . . , δn in S , and a matrix A = (aij)of natural numbers with nonzero determinant d such that

xi = δiyai11 · · · y

ainn

for 1 ≤ i ≤ n. After possibly reindexing the yi , we many assumethat d > 0


Let B = (bij) be the adjoint matrix of A. Set

fi =n∏

j=1

xbij

j =

n∏j=1

δbij

j

ydi

for 1 ≤ i ≤ n. Let R be the integral closure of R0[f1, . . . , fn] in K ,localized at the center of ν∗. Since

√mRS = mS , Zariski’s Main

Theorem shows that R is a normal local ring of K such that S liesover R.


The global version of weak simultaneous resolution is:

Suppose that f : X → Y is a proper, generically finite morphism ofk-varieties. Does there exist a commutative diagram

X1f1→ Y1

↓ ↓X

f→ Y

such that f1 is finite, X1 and Y1 are proper k-varieties such that X1

is nonsingular, Y1 is normal and the vertical arrows are birational?

We give an example showing that the answer is “no” [C4].


Our main theorem on ramification of general valuations is jointwork with Olivier Piltant.

Theorem. ([CP]) Suppose that k has characteristic zero, and weare given a diagram where K ∗ is finite over K and S∗ is analgebraic local ring of K ∗:

K → K ∗

↑ ↑V = V ∗ ∩ K → V ∗

↑S∗.

Let k ′ be an algebraic closure of the residue field k(V ∗) of V ∗. LetΓ be the value group of ν and Γ∗ be the value group of V ∗. Letk(V ), k(V ∗) be the respective residue fields.


Then there exists a regular algebraic local ring R1 of K such that ifR0 is a regular algebraic local ring of K which contains R1 suchthat

R0 → R → S ⊂ V ∗

↑S∗

is a diagram satisfying the conclusions of the weak simultaneouslocal resolution theorem, then

1. There is a natural isomorphism Zn/AtZn ∼= Γ∗/Γ.

2. Γ∗/Γ acts on S⊗k(S)k′ and the invariant ring is

R⊗k(R)k′ ∼= (S⊗k(S)k

′)Γ∗/Γ.

3. The reduced ramification index is e = |Γ∗/Γ| = |Det(A)|.4. The relative degree is f = [k(V ∗) : k(V )] = [k(S) : k(R)].


By assumption, we have regular parameters (x1, . . . , xn) in R0 and(y1, . . . , yn) in S which satisfy the equations

xi = δi

n∏j=1

yaij

j for 1 ≤ i ≤ n,

where Det(A) 6= 0. We have relations

ν(xj) =n∑

j=1

aijν∗(yj) ∈ Γ

for 1 ≤ i ≤ n. Thus there is a group homomorphismZn/AtZn → Γ∗/Γ defined by

(b1, . . . , bn) 7→ b1ν∗(y1) + · · ·+ bnν

∗(yn).

We have that R⊗k(R)k′ is a quotient singularity, by a group whose

invariant factors are determined by Γ∗/Γ.


Characteristic p

Are the conclusions of the local monomialization theorem true incharacteristic p > 0? This is not known, even in dimension two.The only case where local monomialization is not known to hold indimension 2 is for rational rank 1 valuations (the usual troublecase). Here everything is OK unless we have a defect extension.In [CP] we give stable forms of mappings which can be attainedafter enough blowing up in the image and domain, but we do notknow if these will eventually be monomial. We give an exampleshowing failure of “strong monomialization”. In characteristic 0, inthe case of rational rank 1, after enough blowing up we get stableforms

x1 = δyn1 , x2 = y2

where δ is a unit. We give a characteristic p example where such agood form cannot be attained.


Suppose that K → K ∗ is a finite and separable Galois extension,and V ∗ is a valuation ring of K ∗, V = V ∗ ∩ K is the inducedvaluation ring of K . The defect pδ(V ∗/V ) is defined by the equality

|G s(V ∗/V )| = pδ(V ∗/V )[k(V ∗) : k(V )]|Γ∗/Γ|

where G s(V ∗/V ) is the splitting (decomposition) group.

If K ∗ is finite and separable, but not Galois, then the defect isdefined by taking a Galois closure K ′ of K ∗ over K , and anextension of V ∗ to a valuation ring V ′ of K ′. Define

δ(V ∗/V ) = δ(V ′/V )− δ(V ′/V ∗).


In the case where K → K ∗ is a separable extension of twodimensional algebraic function fields and R → S are regularalgebraic local rings of these fields, we construct a diagram ofregular local rings where the horizontal arrows are products ofquadratic transforms along V ∗,

S → S1 → · · · → Sn → · · ·↑ ↑ ↑R → R1 → · · · → Rn → · · ·

such that Rn, Sn have regular parameters un, vn and xn, yn suchthat for n 0,

un = γnxanpαn

n , vn = xbnn (τny

dnpβn

n + xnΩn)

where an, dn are relatively prime to p, γn and τn are units, αn + βn

is a constant and the defect pδ(V ∗/V ) satisfies

βn ≤ δ(V ∗/V ) ≤ αn + βn.


Global Monomialization

Suppose that Φ : X → Y is a dominant morphism of nonsingularvarieties, over a field k of characteristic zero. Φ is monomial atp ∈ X if there exist regular parameters (y1, . . . , ym) in OY ,Φ(p) andan etale cover U of an affine neighborhood of p, uniformizingparameters (x1, . . . , xn) on U and a matrix (aij) of nonnegativeintegers such that

y1 = xa111 · · · xa1n

n...

ym = xam11 · · · xamn

n


As a consequence of the local monomialization theorem, we obtain:

Theorem. Suppose that Φ : X → Y is a dominant morphism ofproper k-varieties where k is a field of characteristic zero. Supposethat ν is a valuation of the function field of X . Then there exists acommutative diagram

X1Ψ→ Y1

α ↓ ↓ βX

Φ→ Y

such that α and β factor as products of blowups of nonsingularsubvarieties, and Ψ is monomial at the center of ν on X1.


Theorem.([C3], [C6], [C7]) Suppose that k is an algebraicallyclosed field of characteristic zero, X is a 3-fold over k , Y is ak-variety, and Φ : X → Y is a dominant morphism. Then thereexists a commutative diagram

X1Ψ→ Y1

α ↓ ↓ βX

Φ→ Y

such that α and β factor as products of blowups of nonsingularsubvarieties, and Ψ is monomial at all points of X1. In fact, we canmake Ψ to be a “toroidal morphism”.


References

[Ab] S. Abhyankar, Simultaneous Resolution for AlgebraicSurfaces, Amer. J. Math 78 (1956), 761 - 790.[C1] S.D. Cutkosky, Local Monomialization and Factorization ofMorphisms, Asterisque 260, 1999.[C2] Simultaneous Resolution of Singularities, Proc. Amer. Math.Soc 128 (2000), 1905 - 1910.[C3] S.D. Cutkosky, Monomialization of Morphisms from 3-folds toSurfaces, LNM 186, Springer Verlag, (2002).[C4] Generically Finite Morphisms and Simultaneous Resolution ofSingularities, Contemp. Math. 331 (2003), 75 - 99.[C5] S.D. Cutkosky, Local Monomialization of TranscendentalExtensions, Annales de L’Institut Fourier 55 (2005), 1517 - 1586.[C6] S.D. Cutkosky, Toroidalization of Dominant Morphisms of3-folds, Memoirs of the AMS 890 (2007).


[C7] S.D. Cutkosky, A Simpler Proof of Toroidalization ofMorphisms From 3-folds to Surfaces, to appear in Annales deL’Institut Fourier, arXiv.1006.5369.[CP] S.D. Cutkosky and O. Piltant, Ramification of Valuations,Advances in Math. 183 (2004), 1 - 79.[Z] O. Zariski, Local Uniformization of Algebraic Varieties, Annalsof Math 41 (1940), 852 - 896.


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