The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation
aa r X i v : . [ m a t h . AG ] J a n THE ROLE OF DEFECT AND SPLITTING IN FINITE GENERATIONOF EXTENSIONS OF ASSOCIATED GRADED RINGS ALONG AVALUATION
STEVEN DALE CUTKOSKY
Abstract.
Suppose that R is a 2 dimensional excellent local domain with quotient field K , K ∗ is a finite separable extension of K and S is a 2 dimensional local domain withquotient field K ∗ such that S dominates R . Suppose that ν ∗ is a valuation of K ∗ such that ν ∗ dominates S . Let ν be the restriction of ν ∗ to K . The associated graded ring gr ν ( R )was introduced by Bernard Teissier. It plays an important role in local uniformization.We show in Theorem 0.1 that the extension ( K, ν ) → ( K ∗ , ν ∗ ) of valued fields is withoutdefect if and only if there exist regular local rings R and S such that R is a local ringof a blow up of R , S is a local ring of a blowup of S , ν ∗ dominates S , S dominates R and the associated graded ring gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra.We also investigate the role of splitting of the valuation ν in K ∗ in finite generationof the extensions of associated graded rings along the valuation. We will say that ν doesnot split in S if ν ∗ is the unique extension of ν to K ∗ which dominates S . We showin Theorem 0.5 that if R and S are regular local rings, ν ∗ has rational rank 1 and isnot discrete and gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra, then ν does not split in S . We give examples showing that such a strong statement is not true when ν does notsatisfy these assumptions. As a consequence of Theorem 0.5, we deduce in Corollary 0.6that if ν has rational rank 1 and is not discrete and if R → R ′ is a nontrivial sequenceof quadratic transforms along ν , then gr ν ( R ′ ) is not a finitely generated gr ν ( R )-algebra. Suppose that K is a field. Associated to a valuation ν of K is a value group Φ ν and avaluation ring V ν with maximal ideal m ν . Let R be a local domain with quotient field K .We say that ν dominates R if R ⊂ V ν and m ν ∩ R = m R where m R is the maximal idealof R . We have an associated semigroup S R ( ν ) = { ν ( f ) | f ∈ R } , as well as the associatedgraded ring along the valuation(1) gr ν ( R ) = M γ ∈ Φ ν P γ ( R ) / P + γ ( R ) = M γ ∈ S R ( ν ) P γ ( R ) / P + γ ( R )which is defined by Teissier in [44]. Here P γ ( R ) = { f ∈ R | ν ( f ) ≥ γ } and P + γ ( R ) = { f ∈ R | ν ( f ) > γ } . This ring plays an important role in local uniformization of singularities ([44] and [45]).The ring gr ν ( R ) is a domain, but it is often not Noetherian, even when R is.Suppose that K → K ∗ is a finite extension of fields and ν ∗ is a valuation which is anextension of ν to K ∗ . We have the classical indices e ( ν ∗ /ν ) = [Φ ν ∗ : Φ ν ] and f ( ν ∗ /ν ) = [ V ν ∗ /m ν ∗ : V ν /m ν ]as well as the defect δ ( ν ∗ /ν ) of the extension. Ramification of valuations and the defectare discussed in Chapter VI of [49], [21] and Kuhlmann’s papers [33] and [35]. A survey partially supported by NSF. s given in Section 7.1 of [16]. By Ostrowski’s lemma, if ν ∗ is the unique extension of ν to K ∗ , we have that(2) [ K ∗ : K ] = e ( ν ∗ /ν ) f ( ν ∗ /ν ) p δ ( ν ∗ /ν ) where p is the characteristic of the residue field V ν /m ν . From this formula, the defectcan be computed using Galois theory in an arbitrary finite extension. If V ν /m ν hascharacteristic 0, then δ ( ν ∗ /ν ) = 0 and p δ ( ν ∗ /ν ) = 1, so there is no defect. Further, ifΦ ν = Z and K ∗ is separable over K then there is no defect.If K is an algebraic function field over a field k , then an algebraic local ring R of K isa local domain which is essentially of finite type over k and has K as its field of fractions.In [10], it is shown that if K → K ∗ is a finite extension of algebraic function fields over afield k of characteristic zero, ν ∗ is a valuation of K ∗ (which is trivial on k ) with restriction ν to K and if R → S is an inclusion of algebraic regular local rings of K and K ∗ suchthat ν ∗ dominates S and S dominates R then there exists a commutative diagram(3) R → S ↑ ↑ R → S where the vertical arrows are products of blowups of nonsingular subschemes along thevaluation ν ∗ (monoidal transforms) and R → S is dominated by ν ∗ and is a monomialmapping; that is, there exist regular parameters x , . . . , x n in R , regular parameters y , . . . , y n in S , units δ i ∈ S , and a matrix A = ( a ij ) of natural numbers with Det( A ) = 0such that(4) x i = δ i n Y j =1 y a ij for 1 ≤ j ≤ n. In [16], it is shown that this theorem is true, giving a monomial form of the mapping (4)after appropriate blowing up (3) along the valuation, if K → K ∗ is a separable extensionof two dimension algebraic function fields over an algebraically closed field, which has nodefect. This result is generalized to the situation of this paper, that is when R is a twodimensional excellent local ring, in [14]. However, it may be that such monomial forms donot exist, even after blowing up, if the extension has defect, as is shown by examples in[12].In the case when k has characteristic zero and for separable defectless extensions of twodimensional algebraic function fields in positive characteristic, it is further shown in [16]that the expressions (3) and (4) are stable under further simple sequences of blow upsalong ν ∗ and the form of the matrix A stably reflects invariants of the valuation.We always have an inclusion of graded domains gr ν ( R ) → gr ν ∗ ( S ) and the index of theirquotient fields is(5) [QF(gr ν ∗ ( S )) : QF(gr ν ( R ))] = e ( ν ∗ /ν ) f ( ν ∗ /ν )as shown in Proposition 3.3 [13]. Comparing with Ostrowski’s lemma (2), we see that thedefect has disappeared in equation (5).Even though QF(gr ν ∗ ( S )) is finite over QF(gr ν ( R )), it is possible for gr ν ∗ ( S ) to notbe a finitely generated gr ν ( R )-algebra. Examples showing this for extensions R → S oftwo dimensional algebraic local rings over arbitrary algebraically closed fields are given inExample 9.4 of [17].It was shown by Ghezzi, H`a and Kashcheyeva in [23] for extensions of two dimensionalalgebraic function fields over an algebraically closed field k of characteristic zero and later y Ghezzi and Kashcheyeva in [24] for defectless separable extensions of two dimensionalalgebraic functions fields over an algebraically closed field k of positive characteristic thatthere exists a commutative diagram (3) such that gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra. Further, this property is stable under further suitable sequences of blow ups.In Theorem 1.6 [13], it is shown that for algebraic regular local rings of arbitrary dimen-sion, if the ground field k is algebraically closed of characteristic zero, and the valuationhas rank 1 and is zero dimensional ( V ν /m ν = k ) then we can also construct a commutativediagram (3) such that gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra and this propertyis stable under further suitable sequences of blow ups.An example is given in [8] of an inclusion R → S in a separable defect extension of twodimensional algebraic function fields such that gr ν ∗ ( S ) is stably not a finitely generatedgr ν ( R )-algebra in diagram (3) under sequences of blow ups. This raises the question ofwhether the existence of a finitely generated extension of associated graded rings alongthe valuation implies that K ∗ is a defectless extension of K .We find that we must impose the condition that K ∗ is a separable extension of K toobtain a positive answer to this question, as there are simple examples of inseparable defectextensions such that gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra, such as in the followingexample, which is Example 8.6 [33]. Let k be a field of characteristic p > k (( x )) bethe field of formal power series over k , with the x -adic valuation ν x . Let y ∈ k (( x )) betranscendental over k ( x ) with ν x ( y ) >
0. Let ˜ y = y p , and K = k ( x, ˜ y ) ⊂ K ∗ = k ( x, y ).Let ν ∗ = ν x | K ∗ and ν = ν x | K . Then we have equality of value groups Φ ν = Φ ν ∗ = ν ( x ) Z and equality of residue fields of valuation rings V ν /m ν = V ν ∗ /m ν ∗ = k , so e ( ν ∗ /ν ) = 1and f ( ν ∗ /ν ) = 1. We have that ν ∗ is the unique extension of ν to K ∗ since K ∗ is purelyinseparable over K . By Ostrowski’s lemma (2), the extension ( K, ν ) → ( K ∗ , ν ∗ ) is a defectextension with defect δ ( ν ∗ /ν ) = 1. Let R = k [ x, ˜ y ] ( x, ˜ y ) → S = k [ x, y ] ( x,y ) . Then we haveequality gr ν ( R ) = k [ t ] = gr ν ∗ ( S )where t is the class of x .In this paper we show that the question does have a positive answer for separableextensions in the following theorem. Theorem 0.1.
Suppose that R is a 2 dimensional excellent local domain with quotient field K . Further suppose that K ∗ is a finite separable extension of K and S is a 2 dimensionallocal domain with quotient field K ∗ such that S dominates R . Suppose that ν ∗ is a valuationof K ∗ such that ν ∗ dominates S . Let ν be the restriction of ν ∗ to K . Then the extension ( K, ν ) → ( K ∗ , ν ∗ ) is without defect if and only if there exist regular local rings R and S such that R is a local ring of a blow up of R , S is a local ring of a blowup of S , ν ∗ dominates S , S dominates R and gr ν ∗ ( S ) is a finitely generated gr ν ( R ) -algebra. We immediately obtain the following corollary for two dimensional algebraic functionfields.
Corollary 0.2.
Suppose that K → K ∗ is a finite separable extension of two dimensionalalgebraic function fields over a field k and ν ∗ is a valuation of K ∗ with restriction ν to K .Then the extension ( K, ν ) → ( K ∗ , ν ∗ ) is without defect if and only if there exist algebraicregular local rings R of K and S of K ∗ such that ν ∗ dominates S , S dominates R and gr ν ∗ ( S ) is a finitely generated gr ν ( R ) -algebra. We see from Theorem 0.1 that the defect, which is completely lost in the extension ofquotient fields of the associated graded rings along the valuation (5), can be recovered rom knowledge of all extensions of associated graded rings along the valuation of regularlocal rings R → S within the field extension which dominate R → S and are dominatedby the valuation.The fact that there exists R → S as in the conclusions of the theorem if the as-sumptions of the theorem hold and the extension is without defect is proven within 2-dimensional algebraic function fields over an algebraically closed field in [23] and [24], andin the generality of the assumptions of Theorem 0.1 in Theorems 4.3 and 4.4 of [14]. Fur-ther, if the assumptions of the theorem hold and the defect δ ( ν ∗ /ν ) = 0, then the valuegroup Φ ν ∗ is not finitely generated by Theorem 7.3 [16] in the case of algebraic functionfields over an algebraically closed field. With the full generality of the hypothesis of Theo-rem 0.1 , the defect is zero by Corollary 18.7 [21] in the case of discrete, rank 1 valuationsand the defect is zero by Theorem 3.7 [14] in the case of rational rank 2 valuations, so byAbhyankar’s inequality, Proposition 2 [1] or Appendix 2 [49], if the defect δ ( ν ∗ /ν ) = 0,then the value group Φ ν ∗ has rational rank 1 and is not discrete and V ν ∗ /m ν ∗ is alge-braic over S/m S . Thus to prove Theorem 0.1, we have reduced to proving the followingproposition, which we establish in this paper. Proposition 0.3.
Suppose that R is a 2 dimensional excellent local domain with quotientfield K . Further suppose that K ∗ is a finite separable extension of K and S is a 2 dimen-sional local domain with quotient field K ∗ such that S dominates R . Suppose that ν ∗ is avaluation of K ∗ such that ν ∗ dominates S . Let ν be the restriction of ν ∗ to K .Suppose that ν ∗ has rational rank 1 and ν ∗ is not discrete. Further suppose that thereexist regular local rings R and S such that R is a local ring of a blow up of R , S is alocal ring of a blowup of S , ν ∗ dominates S , S dominates R and gr ν ∗ ( S ) is a finitelygenerated gr ν ( R ) -algebra. Then the defect δ ( ν ∗ /ν ) = 0 . Another factor in the question of finite generation of extensions of associated gradedrings along a valuation is the splitting of ν in K ∗ . We will say that ν does not split in S if ν ∗ is the unique extension of ν to K ∗ such that ν ∗ dominates S . After a little blowingup, we can always obtain non splitting, as the following lemma shows. Lemma 0.4.
Given an extension R → S as in the hypotheses of Theorem 0.1, there existsa normal local ring R ′ which is a local ring of a blow up of R such that ν dominates R ′ and if R → S ↑ ↑ R → S is a commutative diagram of normal local rings, where R is a local ring of a blow up of R and S is a local ring of a blow up of S , ν ∗ dominate S and R dominates R ′ , then ν does not split in S . Lemma 0.4 will be proven in Section 1.We have the following theorem.
Theorem 0.5.
Suppose that R is a 2 dimensional excellent regular local ring with quo-tient field K . Further suppose that K ∗ is a finite separable extension of K and S is a 2dimensional regular local ring with quotient field K ∗ such that S dominates R . Supposethat ν ∗ is a valuation of K ∗ such that ν ∗ dominates S . Let ν be the restriction of ν ∗ to K .Further suppose that ν ∗ has rational rank 1 and ν ∗ is not discrete. Suppose that gr ν ∗ ( S ) is a finitely generated gr ν ( R ) -algebra. Then S is a localization of the integral closure of R in K ∗ , the defect δ ( ν ∗ /ν ) = 0 and ν ∗ does not split in S . e give examples showing that the condition rational rank 1 and discrete on ν ∗ inTheorem 0.5 are necessary.As an immediate consequence of Theorem 0.5, we obtain the following corollary. Corollary 0.6.
Suppose that R is a 2 dimensional excellent regular local ring with quotientfield K . Suppose that ν is a valuation of K such that ν dominates R . Further suppose that ν has rational rank 1 and ν is not discrete. Suppose that R → R ′ is a nontrivial sequenceof quadratic transforms along ν . Then gr ν ( R ′ ) is not a finitely generated gr ν ( R ) -algebra. In [47], Michel Vaqui´e extends MacLane’s theory of key polynomials [37] to show thatif (
K, ν ) → ( K ∗ , ν ∗ ) is a finite extension of valued fields with δ ( ν ∗ /ν ) = 0 and ν ∗ is theunique extension of ν to K ∗ , then ν ∗ can be constructed from ν by a finite sequence ofaugmented valuations. This suggests that a converse of Theorem 0.5 may be true.We thank Bernard Teissier for discussions on the topics of this paper.1. Local degree and defect
We will use the following criterion to measure defect, which is Proposition 3.4 [14]. Thisresult is implicit in [16] with the assumptions of Proposition 0.3.
Proposition 1.1.
Suppose that R is a 2 dimensional excellent local domain with quotientfield K . Further suppose that K ∗ is a finite separable extension of K and S is a 2 dimen-sional local domain with quotient field K ∗ such that S dominates R . Suppose that ν ∗ isa valuation of K ∗ such that ν ∗ dominates S , the residue field V ν ∗ /m ν ∗ of V ν ∗ is algebraicover S/m S and the value group Φ ν ∗ of ν ∗ has rational rank 1. Let ν be the restriction of ν ∗ to K . There exists a local ring R ′ of K which is essentially of finite type over R , isdominated by ν and dominates R such that if we have a commutative diagram (6) V ν → V ν ∗ ↑ ↑ R → S ↑ R ′ ↑↑ R → S where R is a regular local ring of K which is essentially of finite type over R and dominates R , S is a regular local ring of K ∗ which is essentially of finite type over S and dominates S , R has a regular system of parameters u, v and S has a regular system of parameters x, y such that there is an expression u = γx a , v = x b f where a > , b ≥ , γ is a unit in S , x f in S and f is not a unit in S , then (7) ad [ S /m S : R /m R ] = e ( ν ∗ /ν ) f ( ν ∗ /ν ) p δ ( ν ∗ /ν ) where d = ν ( f mod x ) with ν being the natural valuation of the DVR S/xS . We now prove Lemma 0.4 from the introduction. Let ν = ν ∗ , ν , . . . , ν r be the exten-sions of ν to K ∗ . Let T be the integral closure of V ν in K ∗ . Then T = V ν ∩ · · · ∩ V ν r isthe integral closure of V ν ∗ in K ∗ (by Propositions 2.36 and 2.38 [3]). Let m i = m ν i ∩ T e the maximal ideals of T . By the Chinese remainder theorem, there exists u ∈ T suchthat u ∈ m and u m i for 2 ≤ i ≤ r . Let u n + a u n − + · · · + a n = 0be an equation of integral dependence of u over V ν . Let A be the integral closure of R [ a , . . . , a n ] in K and let R ′ = A A ∩ m ν . Let T ′ be the integral closure of R ′ in K ∗ . Wehave that u ∈ T ′ ∩ m i if and only if i = 1. Let S ′ = T ′ T ′ ∩ m . Then ν does not split in S ′ and R ′ has the property of the conclusions of the lemma.2. Generating Sequences
Given an additive group G with λ , . . . , λ r ∈ G , G ( λ , . . . , λ r ) will denote the sub-group generated by λ , . . . , λ r . The semigroup generated by λ , . . . , λ r will be denoted by S ( λ , . . . , λ r ).In this section, we will suppose that R is a regular local ring of dimension two, withmaximal ideal m R and residue field R/m R . For f ∈ R , let f or [ f ] denote the residue of f in R/m R .The following theorem is Theorem 4.2 of [17], as interpreted by Remark 4.3 [17]. Theorem 2.1.
Suppose that ν is a valuation of the quotient field of R dominating R .Let L = V ν /m ν be the residue field of the valuation ring V ν of ν . For f ∈ V ν , let [ f ] denote the class of f in L . Suppose that x, y are regular parameters in R . Then thereexist Ω ∈ Z + ∪ {∞} and P i ( ν, R ) ∈ m R for i ∈ Z + with i < min { Ω + 1 , ∞} such that P ( ν, R ) = x , P ( ν, R ) = y and for ≤ i < Ω , there is an expression (8) P i +1 ( ν, R ) = P i ( ν, R ) n i ( ν,R ) + λ i X k =1 c k P ( ν, R ) σ i, ( k ) P ( ν, R ) σ i, ( k ) · · · P i ( ν, R ) σ i,i ( k ) with n i ( ν, R ) ≥ , λ i ≥ , (9) 0 = c k units in R for ≤ k ≤ λ i , σ i,s ( k ) ∈ N for all s, k , ≤ σ i,s ( k ) < n s ( ν, R ) for s ≥ . Further, n i ( ν, R ) ν ( P i ( ν, R )) = ν ( P ( ν, R ) σ i, ( k ) P ( ν, R ) σ i, ( k ) · · · P i ( ν, R ) σ i,i ( k ) ) for all k .For all i ∈ Z + with i < Ω , the following are true: ν ( P i +1 ( ν, R )) > n i ( ν, R ) ν ( P i ( ν, R )) . Suppose that r ∈ N , m ∈ Z + , j k ( l ) ∈ N for ≤ l ≤ m and ≤ j k ( l ) < n k ( ν, R ) for ≤ k ≤ r are such that ( j ( l ) , j ( l ) , . . . , j r ( l )) are distinct for ≤ l ≤ m , and ν ( P ( ν, R ) j ( l ) P ( ν, R ) j ( l ) · · · P r ( ν, R ) j r ( l ) ) = ν ( P ( ν, R ) j (1) · · · P r ( ν, R ) j r (1) ) for ≤ l ≤ m . Then , " P ( ν, R ) j (2) P ( ν, R ) j (2) · · · P r ( ν, R ) j r (2) P ( ν, R ) j (1) P ( ν, R ) j (1) · · · P r ( ν, R ) j r (1) , . . . , " P ( ν, R ) j ( m ) P ( ν, R ) j ( m ) · · · P r ( ν, R ) j r ( m ) P ( ν, R ) j (1) P ( ν, R ) j (1) · · · P r ( ν, R ) j r (1) are linearly independent over R/m R . Let n i ( ν, R ) = [ G ( ν ( P ( ν, R )) , . . . , ν ( P ( ν, R ) i )) : G ( ν ( P ( ν, R )) , . . . , ν ( P i − ( ν, R )))] . Then n i ( ν, R ) divides σ i,i ( k ) for all k in (8). In particular, n i ( ν, R ) = n i ( ν, R ) d i ( ν, R ) with d i ( ν, R ) ∈ Z + 6 ) There exists U i ( ν, R ) = P ( ν, R ) w ( i ) P ( ν, R ) w ( i ) · · · P i − ( ν, R ) w i − ( i ) for i ≥ with w ( i ) , . . . , w i − ( i ) ∈ N and ≤ w j ( i ) < n j ( ν, R ) for ≤ j ≤ i − such that ν ( P i ( ν, R ) n i ) = ν ( U i ( ν, R )) and setting α i ( ν, R ) = " P i ( ν, R ) n i ( ν,R ) U i ( ν, R ) then b i,t = (cid:20)P σ i,i ( k )= tn i ( ν,R ) c k P ( ν,R ) σi, k ) P ( ν,R ) σi, k ) ··· P i − ( ν,R ) σi,i − k ) U i ( ν,R ) ( di ( ν,R ) − t ) (cid:21) ∈ R/m R ( α ( ν, R ) , . . . , α i − ( ν, R )) for ≤ t ≤ d i ( ν, R ) − and f i ( u ) = u d i ( ν,R ) + b i,d i ( ν,R ) − u d i ( ν,R ) − + · · · + b i, is the minimal polynomial of α i ( ν, R ) over R/m R ( α ( ν, R ) , . . . , α i − ( ν, R )) .The algorithm terminates with Ω < ∞ if and only if either (10) n Ω ( ν, R ) = [ G ( ν ( P ( ν, R )) , . . . , ν ( P Ω ( ν, R ))) : G ( ν ( P ( ν, R )) , . . . , ν ( P Ω − ( ν, R )))] = ∞ or (11) n Ω ( ν, R ) < ∞ (so that α Ω ( ν, R ) is defined as in 4)) and d Ω ( ν, R ) = [ R/m R ( α ( ν, R ) , . . . , α Ω ( ν, R )) : R/m R ( α ( ν, R ) , . . . , α Ω − ( ν, R ))] = ∞ .If n Ω ( ν, R ) = ∞ , set α Ω ( ν, R ) = 1 . Let notation be as in Theorem 2.1.The following formula is formula B ( i ) on page 10 of [17].(12) Suppose that M is a Laurent monomial in P ( ν, R ) , P ( ν, R ) , . . . , P i ( ν, R )and ν ( M ) = 0. Then there exist s i ∈ Z such that M = Q ij =1 (cid:20) P j ( ν,R ) nj U j ( ν,R ) (cid:21) s j , so that[ M ] ∈ R/m R [ α ( ν, R ) , . . . , α i ( ν, R )] . Define β i ( ν, R ) = ν ( P i ( ν, R )) for 0 ≤ i .Since ν is a valuation of the quotient field of R , we have that(13) Φ ν = ∪ ∞ i =1 G ( β ( ν, R ) , β , . . . , β i ( ν, R ))and(14) V ν /m ν = ∪ ∞ i =1 R/m R [ α ( ν, R ) , . . . , α i ( ν, R )]The following is Theorem 4.10 [17]. Theorem 2.2.
Suppose that ν is a valuation dominating R . Let P ( ν, R ) = x, P ( ν, R ) = y, P ( ν, R ) , . . . e the sequence of elements of R constructed by Theorem 2.1. Suppose that f ∈ R andthere exists n ∈ Z + such that ν ( f ) < nν ( m R ) . Then there exists an expansion f = X I a I P ( ν, R ) i P ( ν, R ) i · · · P r ( ν, R ) i r + X J ϕ J P ( ν, R ) j · · · P r ( ν, R ) j r + h where r ∈ N , a I are units in R , I, J ∈ N r +1 , ν ( P ( ν, R ) i P ( ν, R ) i · · · P r ( ν, R ) i r ) = ν ( f ) for all I in the first sum, ≤ i k < n k ( ν, R ) for ≤ k ≤ r , ν ( P ( ν, R ) j · · · P r ( ν, R ) j r ) >ν ( f ) for all terms in the second sum, ϕ J ∈ R and h ∈ m nR . The terms in the first sum areuniquely determined, up to the choice of units a i , whose residues in R/m R are uniquelydetermined. Let σ ( ν, R ) = 0 and inductively define(15) σ i +1 ( ν, R ) = min { j > σ i ( ν, R ) | n j ( ν, R ) > } . In Theorem 2.2, we see that all of the monomials in the expansion of f are in terms ofthe P σ i .We have that S ( β ( ν, R ) , β ( ν, R ) , . . . , β σ j ( ν,R ) ) = S ( β σ ( ν, R ) , β σ ( ν,R ) , . . . , β σ j ( ν,R ) )for all j ≥ R/m R [ α ( ν, R ) , α ( ν, R ) , . . . , α σ j ( ν,R ) ( ν, R )]= R/m R [ α σ ( ν,R ) ( ν, R ) , α σ ( ν,R ) ( ν, R ) , . . . , α σ j ( ν,R ) ( ν, R )]for all j ≥ R is a regular local ring of dimension two which is dominated by a valua-tion ν . The quadratic transform T of R along ν is defined as follows. Let u, v be a systemof regular parameters in R , Then R [ vu ] ⊂ V ν if ν ( u ) ≤ ν ( v ) and R [ uv ] ⊂ V ν if ν ( u ) ≥ ν ( v ).Let T = R h vu i R [ vu ] ∩ m ν or T = R h uv i R [ uv ] ∩ m ν , depending on if ν ( u ) ≤ ν ( v ) or ν ( u ) > ν ( v ). T is a two dimensional regular local ringwhich is dominated by ν . Let(16) R → T → T → · · · be the infinite sequence of quadratic transforms along ν , so that V ν = ∪ i ≥ T i (Lemma 4.5[3]) and L = V ν /m ν = ∪ i ≥ T i /m T i .For f ∈ R and R → R ∗ a sequence of quadratic transforms along ν , we define a stricttransform of f in R ∗ to be f if f ∈ R ∗ is a local equation of the strict transform in R ∗ of the subscheme f = 0 of R . In this way, a strict transform is only defined up tomultiplication by a unit in R ∗ . This ambiguity will not be a difficulty in our proof. Wewill denote a strict transform of f in R ∗ by st R ∗ ( f ).We use the notation of Theorem 2.1 and its proof for R and the { P i ( ν, R ) } . Recallthat U = U w (1) . Let w = w (1). Since n ( ν, R ) and w are relatively prime, there exist a, b ∈ N such that ε := n ( ν, R ) b − wa = ± . Define elements of the quotient field of R by(17) x = ( x b y − a ) ε , y = ( x − w y n ( ν,R ) ) ε . We have that(18) x = x n ( ν,R )1 y a , y = x w y b . ince n ( ν, R ) ν ( y ) = wν ( x ), it follows that n ( ν, R ) ν ( x ) = ν ( x ) > ν ( y ) = 0 . We further have that(19) α ( ν, R ) = [ y ] ε ∈ V ν /m ν . Let A = R [ x , y ] ⊂ V ν and m A = m ν ∩ A .Let R = A m A . We have that R is a regular local ring and the divisor of xy in R has only one component ( x = 0). In particular, R → R is “free” (Definition 7.5 [16]). R → R factors (uniquely) as a product of quadratic transforms and the divisor of xy in R has two distinct irreducible factors in all intermediate rings.The following is Theorem 7.1 [17]. Theorem 2.3.
Let R be a two dimensional regular local ring with regular parameters x, y . Suppose that R is dominated by a valuation ν . Let P ( ν, R ) = x , P ( ν, R ) = y and { P i ( ν, R ) } be the sequence of elements of R constructed in Theorem 2.1. Suppose that Ω ≥ . Then there exists some smallest value i in the sequence (16) such that the divisor of xy in Spec ( T i ) has only one component. Let R = T i . Then R /m R ∼ = R/m R ( α ( ν, R )) ,and there exists x ∈ R and w ∈ Z + such that x = 0 is a local equation of the exceptionaldivisor of Spec ( R ) → Spec ( R ) , and Q = x , Q = P x wn are regular parameters in R .We have that P i ( ν, R ) = P i +1 ( ν, R ) P ( ν, R ) wn ( ν,R ) ··· n i ( ν,R ) for ≤ i < max { Ω , ∞} satisfy the conclusions of Theorem 2.1 for the ring R . We have that G ( β ( ν, R ) , . . . , β i ( ν, R )) = G ( β ( ν, R ) , . . . , β i +1 ( ν, R ))for i ≥ n i ( ν, R ) = n i +1 ( ν, R ) for i ≥ R /m R [ α ( ν, R ) , . . . , α i ( ν, R )] = R/m R [ α ( ν, R ) , . . . , α i +1 ( ν, R )] for i ≥ d i ( ν, R ) = d i +1 ( ν, R ) and n i ( ν, R ) = n i +1 ( ν, R ) for i ≥ . Let σ ( ν, R ) = 0 and inductively define σ i +1 ( ν, R ) = min { j > σ i (1) | n j ( ν, R ) > } . We then have that σ ( ν, R ) = 0 and for i ≥ σ i ( ν, R ) = σ i +1 ( ν, R ) − n ( ν, R ) > σ i ( ν, R ) = σ i ( ν, R ) − n ( ν, R ) = 1, and for all j ≥ S ( β ( ν, R ) , β ( ν, R ) , . . . , β σ j +1 ( ν,R ) ( ν, R )) = S ( β σ (1) ( ν, R ) , β σ ( ν,R ) , . . . , β σ j ( ν,R ) ( ν, R ))Iterating this construction, we produce a sequence of sequences of quadratic transformsalong ν , R → R → · · · → R σ ( ν,R ) . Now x, y = P σ ( ν,R ) are regular parameters in R . By (17) (with y replaced with y ) wehave that R σ ( ν,R ) has regular parameters(20) x = ( x b y − a ) ε , y = ( x − ω y n σ ν,R ) ( ν,R ) ) ε where ω, a, b ∈ N satisfy ε = n σ ( ν,R ) ( ν, R ) b − ωa = ± urther, R σ ( ν,R ) has regular parameters x σ ( ν,R ) , y σ ( ν,R ) where x = δx n σ ν,R ) ( ν,R ) σ ( ν,R ) and y σ ( ν,R ) = st R σ ( ν,R ) P σ ( ν,R ) ( ν, R ) with δ ∈ R σ ( ν,R ) a unit.For the remainder of this section, we will suppose that R is a two dimensional regularlocal ring and ν is a non discrete rational rank 1 valuation of the quotient field of R with valuation ring V ν , so that V ν /m ν is algebraic over R/m R . Suppose that f ∈ R and ν ( f ) = γ . We will denote the class of f in P γ ( R ) / P + γ ( R ) ⊂ gr ν ( R ) by in ν ( f ). ByTheorem 2.2, we have that gr ν ( R ) is generated by the initial forms of the P i ( ν, R ) as an R/m R -algebra. That is,gr ν ( R ) = R/m R [in ν ( P ( ν, R )) , in ν ( P ( ν, R )) , . . . ]= R/m R [in ν ( P σ ( ν,R ) ( ν, R )) , in ν ( P σ ( ν,R ) ( ν, R )) , . . . ] . Thus the semigroup S R ( ν ) = { ν ( f ) | f ∈ R } is equal to S R ( ν ) = S ( β ( ν, R ) , β ( ν, R ) , . . . ) = S ( β σ ( ν,R ) ( ν, R ) , β σ ( ν,R ) ( ν, R ) , . . . )and the value group Φ ν = G ( β ( ν, R ) , β ( ν, R ) . . . )and the residue field of the valuation ring V ν /m ν = R/m R [ α ( ν, R ) , α ( ν, R ) , . . . ] = R/m R [ α σ ( ν, R ) , α σ ( ν, R ) , . . . ]By 1) of Theorem 2.1, every element β ∈ S R ( ν ) has a unique expression β = r X i =0 a i β i ( ν, R )for some r with a i ∈ N for all i and 0 ≤ a i < n i ( ν, R ) for 1 ≤ i . In particular, if a i = 0 inthe expansion then β i ( ν, R ) = β σ j ( ν,R ) ( ν, R ) for some j . Lemma 2.4.
Let σ i = σ i ( ν, R ) , β i = β i ( ν, R ) , P i = P i ( ν, R ) , n i = n i ( ν, R ) , n i = n i ( ν, R ) ,σ i (1) = σ i ( ν, R σ ) , β i = β i ( ν, R σ ) , P i (1) = P i ( ν, R σ ) , n i (1) = n i ( ν, R σ ) , n i (1) = n i ( ν, R σ ) . Suppose i ∈ N , r ∈ N and a j ∈ N for j = 0 , . . . , r with ≤ a j < n σ j for j ≥ are suchthat ν ( P a σ · · · P a r σ r ) > ν ( P σ i ) or r < i and ν ( P a σ · · · P a r σ r ) = ν ( P σ i ) . By (18) and Theorem 2.3, we have expressions in R σ = R [ x , y ] m ν ∩ R [ x ,y ] where x , y are defined by (20) P a σ · · · P a r σ r = y aa + ba P σ (1) (1) a · · · P σ r − (1) (1) a r P σ (1) (1) t where t = n σ a + ωa + ωn σ a + · · · + ωn σ · · · n σ r − a r and P σ i = y a P σ (1) (1) n σ if i = 0 y b P σ (1) (1) ω if i = 1 P σ i − (1) (1) P σ (1) (1) ωn σ ··· n σi − if i ≥ . et λ = n σ if i = 0 ω if i = 1 ωn σ · · · n σ i − if i ≥ . Then t > λ, except in the case where i = 1 , P a σ · · · P a r σ r = P σ , and n σ = ω = 1 . In this case λ = t .Proof. First suppose that i ≥ r ≥ i . Then t − λ = ( n σ a + ωa + ωn σ a + · · · + ωn σ · · · n σ r − a r ) − ωn σ · · · n σ i − > . Now suppose that i ≥ r < i . We have that( n a + ωa + . . . + ωn · · · n r − a r − ωn ( ν, R ) · · · n i − ) β σ (1) (1) ≥ β σ i − (1) (1) − a β σ (1) (1) − . . . − a r β σ r − (1) (1) > n σ j (1) (1) = n σ j +1 for all j , and so n σ j +1 β σ j (1) (1) < β σ j +1 (1) (1) for all j .Now suppose that i = 1. As in the proof for the case i ≥ t − λ > r ≥
1, so suppose that i = 1 and r = 0. Then n σ β σ = ωβ σ . From our assumption a ν ( P ) ≥ ν ( P ) we obtain t − λ = n σ a − ω ≥ a = ω = n σ = 1 since gcd( ω, n σ ) = 1.Now suppose i = 0. As in the previous cases, we have t − λ > r > t − λ > r = 1 except possibly if P a · · · P a r r = P a . We then have that ν ( P a σ ) > ν ( P σ ), and so a β σ β σ > . Since β σ β σ = ωn σ , we have that t − λ = ωa − n σ > (cid:3) Lemma 2.5.
Let notation be as in Lemma 2.4. Suppose that f ∈ R , with ν ( f ) = ν ( P σ i ) for some i ≥ , and that f has an expression of the form of Theorem 2.2, f = cP σ i + s X j =1 c i P a ( j ) σ P a ( j ) σ · · · P a r ( j ) σ r + h where s, r ∈ N , c, c j are units in R , with ≤ a k ( j ) < n k for ≤ k ≤ r for ≤ j ≤ s , ν ( f ) = ν ( P σ i ) ≤ ν ( P a ( j ) σ P a ( j ) σ · · · P a r ( j ) σ r ) for ≤ j ≤ s , a k ( j ) = 0 for k ≥ i if ν ( f ) = ν ( P a p ( j ) σ · · · P a r ( j ) σ r ) and h ∈ m nR with n > ν ( f ) .Then st R σ ( f ) is a unit in R σ if i = 0 or 1 and if i > , there exists a unit c in R σ and Ω ∈ R σ such that st R σ ( f ) = cP σ i − (1) (1) + x Ω with ν (st R σ ( f )) = ν ( P σ i − (1) (1)) and ν ( P σ i − (1) (1)) ≤ ν ( x Ω) . roof. Let λ = n if i = 0 ω if i = 1 ωn σ · · · n σ r − if i ≥ f = cH i + s X j =1 c j ( y ) aa ( j )+ ba ( j ) P σ (1) (1) t j P σ (1) a ( j ) · · · P σ r − (1) (1) a r ( j ) + P σ (1) (1) t h ′ with H i = ( y ) a P σ (1) (1) n if i = 0( y ) b P σ (1) (1) ω if i = 1 P σ (1) (1) ωn ··· n i − P σ i − (1) (1) if i ≥ t j = n a ( j ) + ωa ( j ) + ωn σ a + · · · + ωn σ · · · n σ r − a r ( j )for 1 ≤ j ≤ s , t > λ and h ′ ∈ R σ . By Lemma 2.4, if i ≥ i = 0, we have that t j > λ for all j . Thus f = P σ (1) (1) λ f where f = cG i + s X j =1 c j P σ (1) (1) t j − λ P σ (1) (1) a ( j ) · · · P σ r − (1) (1) a r ( j ) + P σ (1) (1) t − λ h ′ with G i = ( y ) a if i = 0( y ) b if i = 1 P σ i − (1) (1) if i ≥ f = st R ( f ) of f in R .If i = 1, then by Lemma 2.4, t j > λ for all j , except possibly for a single term (that wecan assume is t ) which is P σ , and we have that ω = n σ = 1. In this case t = λ . Then (cid:20) P σ P σ (cid:21) = α σ ( ν, R ) ∈ V ν /m ν which has degree d σ ( ν, R ) = n σ > R/m R . By (18), x = x , y = x y and f = x [ c + c y + x Ω]with Ω ∈ R σ . We have that c + c y is a unit in R σ since[ y ] = (cid:20) P σ P σ (cid:21) R/m R . (cid:3) Finite generation implies no defect
Suppose that R is a two dimensional regular local ring of K and S is a two dimensionalregular local ring such that S dominates R Let K be the quotient field of R and K ∗ be the quotient field of S . Suppose that K → K ∗ is a finite separable field extension.Suppose that ν ∗ is a non discrete rational rank 1 valuation of K ∗ such that V ν ∗ /m ν ∗ isalgebraic over S/m S and that ν ∗ dominates S . Then we have a natural graded inclusiongr ν ( R ) → gr ν ∗ ( S ), so that for f ∈ R , we have that in ν ( f ) = in ν ∗ ( f ). Let ν = ν ∗ | K . Let L = V ν ∗ /m ν ∗ . Suppose that gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra.Let x, y be regular parameters in R , with associated generating sequence to ν , P = P ( ν, R ) = x, P = P ( ν, R ) = y, P = P ( ν, R ) , . . . in R as constructed in Theorem 2.1, ith U i = U i ( ν, R ), β i = β i ( ν, R ) = ν ( P i ), γ i = α i ( ν, R ), m i = m i ( ν, R ), m i = m i ( ν, R ), d i = d i ( ν, R ) and σ i = σ i ( ν, R ) defined as in Section 2.Let u, v be regular parameters in S , with associated generating sequence to ν ∗ , Q = P ( ν ∗ , S ) = u, Q = P ( ν ∗ , S ) = v, Q = P ( ν ∗ , S ) , . . . in S as constructed in Theorem 2.1,with V i = U i ( ν ∗ , S ), γ i = β i ( ν ∗ , S ) = ν ∗ ( Q i ), δ i = α i ( ν ∗ , S ), n i = n i ( ν ∗ , S ), n i = n i ( ν ∗ , S ), e i = α i ( ν ∗ , S ) and τ i = σ i ( ν ∗ , S ) defined as in Section 2.With our assumption that gr ν ∗ ( S ) is a finitely generated gr ν ( R )-algebra, we have thatfor all sufficiently large l ,(21) gr ν ∗ ( S ) = gr ν ( R )[in ν ∗ Q τ , . . . , in ν ∗ Q τ l ] . Proposition 3.1.
With our assumption that gr ν ∗ ( S ) is a finitely generated gr ν ( R ) -algebra,there exist integers s > and r > such that for all j ≥ , β σ r + j = γ τ s + j , m σ r + j = n τ s + j , d σ r + j = e τ s + j , m σ r + j = n τ s + j ,G ( β σ , . . . , β σ r + j ) ⊂ G ( γ τ , . . . , γ τ s + j ) , [ G ( γ τ , . . . , γ τ s + j ) : G ( β σ , . . . , β σ r + j )] = e ( ν ∗ /ν ) ,R/m R [ δ σ , . . . , δ σ r + j ] ⊂ S/m S [ ε τ , . . . , ε τ s + j ] and [ S/m S [ ε τ , . . . , ε τ s + j ] : R/m R [ δ σ , . . . , δ σ r + j ]] = f ( ν ∗ /ν ) . Proof.
Let l be as in (21). For s ≥ l , define the sub algebra A τ s of gr ν ∗ ( S ) by A τ s = S/m S [in ν ∗ Q τ , . . . , in ν ∗ Q τ s ] . For s ≥ l , let r s = max { j | in ν ∗ P σ j ∈ A τ s } ,λ s = [ G ( γ τ , . . . , γ τ s ) : G ( β σ , . . . , β σ rs )] , and χ s = [ S/m S [ ε τ , . . . , ε τ s ] : R/m R [ δ σ , . . . , δ σ rs ]] . To simplify notation, we will write r = r s .We will now show that β σ r +1 = γ τ s +1 . Suppose that β σ r +1 > γ τ s +1 . We have thatin ν ∗ Q τ s +1 ∈ gr ν ( R )[in ν ∗ Q τ , . . . , in ν ∗ Q τ s ] . Since β σ r +1 < β σ r +2 < · · · we then have that in ν ∗ Q τ s +1 ∈ A τ s which is impossible. Thus β σ r +1 ≤ γ τ s +1 . If β σ r +1 <γ τ s +1 , then since γ τ s +1 < γ τ s +2 < · · · and in ν ∗ P σ r +1 ∈ gr ν ∗ ( S ), we have that in ν ∗ P σ r +1 ∈ A τ s , which is impossible. Thus β σ r +1 = γ τ s +1 .We will now establish that either we have a reduction λ s +1 < λ s or(22) λ s +1 = λ s , β σ r +1 = γ τ s +1 and m σ r +1 = n τ s +1 . Let ω be a generator of the group G ( γ τ , . . . , γ τ s ), so that G ( γ τ , . . . , γ τ s ) = Z ω . We havethat G ( γ τ , . . . , γ τ s +1 ) = 1 n τ s +1 Z ω nd G ( β σ , . . . , β σ r +1 ) = 1 m σ r +1 Z ( λ s ω ) . There exists a positive integer f with gcd( f, n τ s +1 ) = 1 such that γ τ s +1 = fn τ s +1 ω There exists a positive integer g with gcd( g, m σ r +1 ) = 1 such that β σ r +1 = gm σ r +1 λ s ω. Since β σ r +1 = γ τ s +1 , we have gλ s n τ s +1 = f m σ r +1 . Thus n τ s +1 divides m σ r +1 and m σ r +1 divides λ s n τ s +1 , so that a = m σ r +1 n τ s +1 is a positive integer and defining λ = λ s a , we have that λ is a positive integer with λ s m σ r +1 = λn τ s +1 and λ = [ G ( γ τ , . . . , γ τ s +1 ) : G ( β σ , . . . , β σ r +1 )] . Since λ s +1 ≤ λ , either λ s +1 < λ s or λ s +1 = λ s and m σ r +1 = n τ s +1 .We will now suppose that s is sufficiently large that (22) holds. Sincein ν ∗ Q τ s +1 ∈ gr ν ∗ ( S ) = gr ν ( R )[in ν ∗ Q τ , . . . , in ν ∗ Q τ s ] , if n τ s +1 > ν ∗ P σ r +1 = in ν ∗ ( α )in ν ∗ Q τ s +1 in P γ τs +1 ( S ) / P + γ τs +1 ( S ) with α a unit in S and if n τ s +1 = 1, since in ν ∗ P σ r +1 A τ s , we havean expression(24) in ν ∗ P σ r +1 = in ν ∗ ( α )in ν ∗ Q τ s +1 + X in ν ∗ ( α J )(in ν ∗ Q τ ) j · · · (in ν ∗ Q τ s ) j s in P γ τs +1 ( S ) / P + γ τs +1 ( S ) with α a unit in S and the sum is over certain J = ( j i , . . . , j s ) ∈ N s such that the α J are units in S , and the terms in ν ∗ Q τ s +1 and the (in ν ∗ Q τ ) j · · · (in ν ∗ Q τ s ) j s are linearly independent over S/m S .The monomial U σ r +1 in P σ , . . . , P σ r and the monomial V τ s +1 in Q τ , . . . , Q τ s both havethe value n τ s +1 γ τ s +1 = m σ r +1 β σ r +1 , and satisfy ε τ s +1 = " Q n τs +1 τ s +1 V τ s +1 and δ σ r +1 = " P n τs +1 σ r +1 U σ r +1 . ince U σ r +1 , V τ s +1 ∈ A τ s and by (12) and 2) of Theorem 2.1, we have that (cid:20) V τ s +1 U σ r +1 (cid:21) ∈ S/m S [ ε τ , . . . , ε τ s ] . If n τ s +1 >
1, then by (23), we have " P n τs +1 σ r +1 U σ r +1 = (cid:20) V τ s +1 U σ r +1 (cid:21) [ α ] n τs +1 " Q n τs +1 τ s +1 V τ s +1 in L = V ν ∗ /m ν ∗ , and if n τ s +1 = 1, then by (24), we have (cid:20) P σ r +1 U σ r +1 (cid:21) = (cid:20) V τ s +1 U σ r +1 (cid:21) [ α ] (cid:20) Q τ s +1 V τ s +1 (cid:21) + X [ α J ] " Q j τ · · · Q j s τ s V τ s +1 . Thus by equation (12),(25)
S/m S [ ε τ , . . . , ε τ s ][ ε τ s +1 ] = S/m S [ ε τ , . . . , ε τ s ][ δ σ r +1 ] . We have a commutative diagram
S/m S [ ε τ , . . . , ε τ s ] → S/m S [ ε τ , . . . , ε τ s , ε τ s +1 ] = S/m S [ ε τ , . . . , ε τ s ][ δ σ r +1 ] ↑ ↑ R/m R [ δ σ , . . . , δ σ r ] → R/m R [ δ σ , . . . , δ σ r ][ δ σ r +1 ] . Let χ = [ S/m S [ ε τ , . . . , ε τ s , ε τ s +1 ] : R/m R [ δ σ , . . . , δ σ r , δ σ r +1 ]] . Since
S/m S [ ε τ , . . . , ε τ s , ε τ s +1 ] = S/m S [ ε τ , . . . , ε τ s ][ δ σ r +1 ] , we have that e τ s +1 | d σ r +1 . Further, d σ r +1 e τ s +1 χ = χ s , whence χ ≤ χ s . Thus χ s +1 ≤ χ s and if χ s +1 = χ s , then d σ r +1 = e τ s +1 and r s +1 = r s + 1since P σ r +2 ∈ A τ s +1 implies λ s +1 < λ s or χ s +1 < χ s .We may thus choose s sufficiently large that there exists an integer r > j ≥ β σ r + j = γ τ s + j , m σ r + j = n τ s + j , d σ r + j = e τ s + j , m σ r + j = n τ s + j ,G ( β σ , . . . , β σ r + j ) ⊂ G ( γ τ , . . . , γ τ s + j ) , there is a constant λ (which does not depend on j ) such that[ G ( γ τ , . . . , γ τ s + j ) : G ( β σ , . . . , β σ r + j )] = λR/m R [ δ σ , . . . , δ σ r + j ] ⊂ S/m S [ ε τ , . . . , ε τ s + j ]and there is a constant χ (which does not depend on j ) such that[ S/m S [ ε τ , . . . , ε τ s + j ] : R/m R [ δ σ , . . . , δ σ r + j ]] = χ. Then Φ ν ∗ = ∪ j ≥ n τ s +1 · · · n τ s + j Z ω where G ( γ τ , . . . , γ τ s ) = Z ω , andΦ ν = ∪ j ≥ m σ r +1 · · · m σ r + j λ Z ω = ∪ j ≥ n τ s +1 · · · n τ s + j λ Z ω o that λ = [Φ ν ∗ : Φ ν ] = e ( ν ∗ /ν ) . For i ≥
0, let K i = R/m R [ δ σ , . . . , δ σ r + i ] and M i = S/m S [ ε τ , . . . , ε τ s + i ]. We have that M i +1 = M i [ δ σ r + i +1 ] for i ≥ χ = [ M i : K i ] for all i . Further, ∪ ∞ i =0 M i = V ν ∗ /m ν ∗ and ∪ ∞ i =0 K i = V ν /m ν . Thus if g , . . . , g λ ∈ M form a basis of M as a K -vector space, then g , . . . , g λ form abasis of M i as a K i -vector space for all i ≥
0. Thus χ = [ V ν ∗ /m ν ∗ : V ν /m ν ] = f ( ν ∗ /ν ) . (cid:3) Let r and s be as in the conclusions of Proposition 3.1. There exists τ t with t ≥ s suchthat we have a commutative diagram of inclusions of regular local rings (with the notationintroduced in Section 2) R σ r → S τ t ↑ ↑ R → S. After possibly increasing s and r , we may assume that R ′ ⊂ R σ r , where R ′ is the localring of the conclusions of Proposition 1.1. Recall that R has regular parameters x = P , y = P and S has regular parameters u = Q , v = Q , R σ r has regular parameters x σ r , y σ r such that x = δx m σ ··· m σr σ r , y σ r = st R σr P σ r +1 where δ is a unit in R σ r and S τ t has regular parameters u τ t , v τ t such that u = εu n τ ··· n τt τ t , v τ t = st S τt Q τ t +1 where ε is a unit in S τ t . We may choose t ≫ x σ r = ϕu λτ t for some positive integer λ where ϕ is a unit in S τ t , since ∪ ∞ t =0 S τ t = V ν ∗ .We have expressions P i = ψ i x c i σ r in R σ r where ψ i are units in R σ r for i ≤ σ r so that P i = ψ ∗ i u c i λτ t in S τ t where ψ ∗ i are units in S τ t for i ≤ σ r by (26). Lemma 3.2.
For j ≥ we have st R σr ( P σ r + j ) = u λ j τ t st S τt ( P σ r + j ) for some λ j ∈ N , where we regard P σ r + j as an element of R on the left hand side of theequation and regard P σ r + j as an element of S on the right hand side.Proof. Using (26), we have P σ r + j = st R σr ( P σ r + j ) x f j σ r = st R σr ( P σ r + j ) u λf j τ t ϕ f j where f j ∈ N . Viewing P σ r + j as an element of S , we have that P σ r + j = st S τt ( P σ r + j ) u g j τ t for some g j ∈ N . Since u τ t st S τt ( P σ r + j ), we have that f j λ ≤ g j and so λ j = g j − f j λ ≥ (cid:3) y induction in the sequence of quadratic transforms above R and S in Lemma 2.5, andsince ν ∗ ( P σ r + j ) = β σ r + j = γ τ s + j by Proposition 3.1, we have by (23) and (24) an expression(27) st S τt ( P σ r + j ) = c st S τt ( Q τ s + j ) + u τ t Ωwith c ∈ S τ t a unit, Ω ∈ S τ t and ν ∗ ( u τ t Ω) ≥ ν ∗ (st S τt ( Q τ s + j )) if s + j > t and(28) S τ t ( P σ r + j ) is a unit in S τ t if s + j ≤ t . Thus P σ r + j = u d j τ t ϕ j in S τ t where d j is a positive integer and ϕ j is a unit in S τ t if s + j ≤ t .Suppose s < t . Then y σ r = st R σr ( P σ r +1 ) = ˜ ϕu hτ t where ˜ ϕ is a unit in S τ t and h is a positive integer. As shown in equation (20) of Section2, R σ r +1 = R σ r [ x , y ] m ν ∩ R σr [ x ,y ] where x = ( x bσ r y − aσ r ) ε , y = ( x − ωσ r y m σr σ r ) ε with ε = m σ r b − ωa = ± ν ( x ) > ν ( y ) = 0. Substituting x σ r = ϕu λτ t and y σ = ˜ ϕu hτ t we see that R σ r +1 is dominated by S τ t . We thus have a factorization R σ r → R σ r +1 → S τ t with x σ r +1 = x = ˆ ϕu λ ′ τ t where ˆ ϕ is a unit in S τ t and λ ′ is a positive integer. We may thusreplace s with s + 1, r with r + 1 and R σ r with R σ r +1 .Iterating this argument, we may assume that s = t (with r = r s ) so that by Lemma3.2, (28) and (27), y σ r = st R σr ( P σ r +1 ) = u µτ s st S τs ( P σ r +1 )where st S τs ( P σ r +1 ) = c st S τs ( Q τ s +1 ) + u τ s Ωwith c a unit in S τ s and Ω ∈ S τ s . Thus by (26), we have an expression x σ r = ϕu λτ s , y σ r = εu ατ s ( v τ s + u τ s Ω)where λ is a positive integer, α ∈ N , ϕ and ε are units in S τ s and Ω ∈ S τ s .We have that ν ∗ ( x σ r ) = λν ∗ ( u τ s ), ν ( x σ r ) Z = G ( ν ( x σ r )) = G ( β σ , . . . , β σ r ) and ν ∗ ( u τ s ) Z = G ( ν ∗ ( u τ s )) = G ( γ τ , . . . , γ τ s ) . Thus λ = [ G ( γ τ , . . . , γ τ s ) : G ( β σ , . . . , β σ r )] = e ( ν ∗ /ν )by Proposition 3.1.By Theorem 2.3, we have that R σ r /m R σr = R/m R [ δ σ , . . . , δ σ r ] and S τ s /m S τs = S/m S [ ε τ , . . . , ε τ s ] . Thus [ S τ s /m S τs : R σ r /m R σr ] = f ( ν ∗ /ν )by Proposition 3.1. ince the ring R ′ of Proposition 1.1 is contained in R σ r by our construction, we haveby Proposition 1.1 that ( K, ν ) → ( K ∗ , ν ∗ ) is without defect, completing the proofs ofProposition 0.3 and Theorem 0.1.4. non splitting and finite generation In this section, we will have the following assumptions. Suppose that R is a 2 dimen-sional excellent local domain with quotient field K . Further suppose that K ∗ is a finiteseparable extension of K and S is a 2 dimensional local domain with quotient field K ∗ such that S dominates R . Suppose that ν ∗ is a valuation of K ∗ such that ν ∗ dominates S . Let ν be the restriction of ν ∗ to K .Suppose that ν ∗ has rational rank 1 and ν ∗ is not discrete. Then V ν ∗ /m ν ∗ is algebraicover S/m S , by Abhyankar’s inequality, Proposition 2 [1]. Lemma 4.1.
Let assumptions be as above. Then the associated graded ring gr ν ∗ ( S ) is anintegral extension of gr ν ( R ) .Proof. It suffices to show that in ν ∗ ( f ) is integral over gr ν ( R ) whenever f ∈ S . Supposethat f ∈ S . There exists n > n ν ∗ ( f ) ∈ Φ ν . Let x ∈ m R and ω = ν ( x ).Then there exists a positive integer b and natural number a such that bn ν ∗ ( f ) = aω , so ν ∗ (cid:18) f bn x a (cid:19) = 0 . Let ξ = (cid:20) f bn x a (cid:21) ∈ V ν ∗ /m ν ∗ , and let g ( t ) = t r + a r − t r − + · · · + a with a i ∈ R/m R be the minimal polynomial of ξ over R/m R . Let a i be lifts of the a i to R . Then ν ∗ ( f b n r + a r − x a f bn ( r − + · · · + a x ar ) > ν ∗ ( f bn r ) = ν ∗ ( a r − x a f bn ( r − ) = · · · = ν ∗ ( a x ar ) . Thus in ν ∗ ( f ) b n r + in ν ( a r − x a )in ν ∗ ( f ) bn ( r − + · · · + in ν ( a x ar ) = 0in gr ν ∗ ( S ). Thus in ν ∗ ( f ) is integral over gr ν ∗ ( R ). (cid:3) We now establish Theorem 0.5. Recall (as defined after Proposition 0.3) that ν ∗ doesnot split in S if ν ∗ is the unique extension of ν to K ∗ which dominates S . Theorem 4.2.
Let assumptions be as above and suppose that R and S are regular localrings. Suppose that gr ν ∗ ( S ) is a finitely generated gr ν ( R ) -algebra. Then S is a localizationof the integral closure of R in K ∗ , the defect δ ( ν ∗ /ν ) = 0 and ν ∗ does not split in S .Proof. Let s and r be as in the conclusions of Proposition 3.1. We will first show that P σ r + j is irreducible in ˆ S for all j >
0. There exists a unique extension of ν ∗ to the quotientfield of ˆ S which dominates ˆ S ([43], [17], [22]). The extension is immediate since ν ∗ isnot discrete; that is, there is no increase in value group or residue field for the extendedvaluation. It has the property that if f ∈ ˆ S and { f i } is a a Cauchy sequence in ˆ S whichconverges to f , then ν ∗ ( f ) = ν ∗ ( f i ) for all i ≫ P σ r + j is not irreducible in ˆ S for some j >
0. We will derive a contradiction.With this assumption, P σ r + j = f g with f, g ∈ m ˆ S . Let { f i } be a Cauchy sequencein S which converges to f and let { g i } be a Cauchy sequence in S which converges to g . For i sufficiently large, f − f i , g − g i ∈ m n ˆ S where n is so large that nν ∗ ( m ˆ S ) = ν ∗ ( m S ) > ν ( P σ r + j ). Thus P σ r + j = f i g i + h with h ∈ m n ˆ S ∩ S = m nS , and so in ν ∗ ( P σ r + j ) =in ν ∗ ( f i )in ν ∗ ( g i ). Now ν ∗ ( f i ) , ν ∗ ( g i ) < ν ( P σ r + j ) = β σ r + j = γ τ s + j = ν ∗ ( Q τ s + j )so that in ν ∗ ( f i ) , in ν ∗ ( g i ) ∈ S/m S [in ν ∗ ( Q τ ) , . . . , in ν ∗ ( Q τ s + j − )]which implies in ν ∗ ( P σ r + j ) ∈ S/m S [in ν ∗ ( Q τ ) , . . . , in ν ∗ ( Q τ s + j − )] . But then (24) impliesin ν ∗ ( Q τ s + j ) ∈ S/m S [in ν ∗ ( Q τ ) , . . . , in ν ∗ ( Q τ s + j − )]which is impossible. Thus P σ r + j is irreducible in ˆ S for all j > S is not a localization of the integral closure of R in K ∗ , then by Zariski’s MainTheorem (Theorem 1 of Chapter 4 [41]), m R S = f N where f ∈ m S and N is an m S -primary ideal. Thus f divides P i in S for all i , which is impossible since we have shownthat P σ r + j is analytically irreducible in S for all j >
0; we cannot have P σ r + j = a j f where a j is a unit in S for j > ν ( P σ r + j ) = ν ∗ ( Q τ s + j ) by Proposition 3.1.Now suppose that ν ∗ is not the unique extension of ν to K ∗ which dominates S . Recallthat V ν is the union of all quadratic transforms above R along ν and V ν ∗ is the union ofall quadratic transforms above S along ν ∗ (Lemma 4.5 [3]).Then for all i ≫
0, we have a commutative diagram R σ i → T i ↑ ↑ R → T where T is the integral closure of R in K ∗ , T i is the integral closure of R σ i in K ∗ , S = T p for some maximal ideal p in T which lies over m R , and there exist r ≥ p ( i ) , . . . , p r ( i ) in T i which lie over m R σi and whose intersection with T is p . We mayassume that p ( i ) is the center of ν ∗ .There exists an m R -primary ideal I i in R such that the blow up of I i is γ : X σ i → Spec( R ) where X σ i is regular and R σ i is a local ring of X σ i . Let Z σ i be the integralclosure of X σ i in K ∗ . Let Y σ i = Z σ i × Spec ( T ) Spec( S ). We have a commutative diagramof morphisms Y σ i β → X σ i δ ↓ γ ↓ Spec( S ) α → Spec( R )The morphism δ is projective (by Proposition II.5.5.5 [25] and Corollary II.6.1.11 [25] andit is birational, so since Y σ i and Spec( S ) are integral, it is a blow up of an ideal J i in S (Proposition III.2.3.5 [26]), which we can take to be m S -primary since S is a regularlocal ring and hence factorial. Define curves C = Spec( R/ ( P σ i )) and C ′ = α − ( C ) =Spec( S/ ( P σ i )). Denote the Zariski closure of a set W by W . The strict transform C ∗ of C ′ in Y σ i is the Zariski closure(29) C ∗ = δ − ( C ′ \ m S ) = δ − α − ( C \ m R ) = β − γ − ( C \ m R )= β − ( γ − ( C \ m R )) since β is quasi finite= β − ( ˜ C ) here ˜ C is the strict transform of C in X σ i . We have that Z σ i × X σi Spec( R σ i ) ∼ = Spec( T i ),so Y σ i × X σi Spec( R σ i ) ∼ = Spec( T i ⊗ T S ) . Let x σ i be a local equation in R σ i of the exceptional divisor of Spec( R σ i ) → Spec( R ) andlet y σ i = st R σi ( P σ i ). Then x σ i , y σ i are regular parameters in R σ i . We have that q m R σi ( T i ⊗ T S ) = ∩ rj =1 p j ( i )( T i ⊗ T S ) . The blow up of J i ( S/ ( P σ i )) in C ′ is δ : C ∗ → C ′ , where δ is the restriction of δ to C ∗ Corollary II.7.15 [28]). Since y σ i is a local equation of ˜ C in R σ i , we have by (29) that p ( i ) , . . . , p r ( i ) ∈ δ − ( m S ) ⊂ C ∗ . Since δ is proper and C ′ is a curve, C ∗ = Spec( A ) for some excellent one dimensionaldomain A such that the inclusion S/ ( P σ i ) → A is finite (Corollary I.1.10 [39]). Let B = A ⊗ S/ ( P σi ) ˆ S/ ( P σ i ). Then C ∗ × Spec ( S/ ( P σi )) Spec( ˆ S/ ( P σ i )) = Spec( B ) → Spec( ˆ S/ ( P σ i ))is the blow up of J i ( ˆ S/ ( P σ i )) in ˆ S/ ( P σ i ). The extension ˆ S/ ( P σ i ) → B is finite since S/ ( P σ i ) → A is finite.Now assume that S/ ( P σ i ) is analytically irreducible. Then B has only one minimalprime since the blow up Spec( B ) → Spec( ˆ S/ ( P σ i )) is birational.Since a complete local ring is Henselian, B is a local ring (Theorem I.4.2 on page 32 of[39]), a contradiction to our assumption that r > (cid:3) As a consequence of the above theorem (Theorem 0.5), we now obtain Corollary 0.6.
Corollary 4.3.
Let assumptions be as above and suppose that R is a regular local ring.Suppose that R → R ′ is a nontrivial sequence of quadratic transforms along ν . Thengr ν ( R ′ ) is not a finitely generated gr ν ( R ) -algebra. The conclusions of Theorem 0.5 do not hold if we remove the assumption that ν ∗ isnot discrete, when V ν /m ν is finite over R/m R . We give a simple example. Let k be analgebraically closed field of characteristic not equal to 2 and let p ( u ) be a transcendentalseries in the power series ring k [[ u ]] such that p (0) = 1. Then f = v − up ( u ) is irreduciblein the power series ring k [[ u, v ]] and k [[ u, v ]] / ( f ) is a discrete valuation ring with regularparameter u . Let ν be the natural valuation of this ring. Let R = k [ u, v ] ( u,v ) and S = k [ x, y ] ( x,y ) . Define a k -algebra homomorphism R → S by u x and v y . The series f ( x , y ) factors as f = ( y − x p p ( x ))( y + x p p ( x )) in k [[ x, y ]]. Let f = y − x p p ( x )and f = y + x p p ( x ). The rings k [[ x, y ]] / ( f i ) are discrete valuation rings with regularparameter x . Let ν and ν be the natural valuations of these ring.Let ν be the valuation of the quotient field of R which dominates R defined by thenatural inclusion R → k [[ u, v ]] / ( f ) and let ν i for i = 1 , S which dominate S and are defined by the respective natural inclusions S → k [[ x, y ]] / ( f i ) . Then ν and ν are distinct extensions of ν to the quotient field of S whichdominate S . However, we have that gr ν ( R ) = k [in ν ( u )] and gr ν i ( S ) = k [in ν ∗ ( x )] within ν ∗ ( x ) = in ν ( u ). Thus gr ν i ( S ) is a finite gr ν ( R )-algebra.We now give an example where ν ∗ has rational rank 2 and ν splits in S but gr ν ∗ ( S )is a finitely generated gr ν ( R )-algebra. Suppose that k is an algebraically closed field ofcharacteristic not equal to 2. Let R = k [ x, y ] ( x,y ) and S = k [ u, v ] ( u,v ) . The substitutions = x and v = y make S into a finite separable extension of R . Define a valuation ν ofthe quotient field K ∗ of S by ν ( x ) = 1 and ν ( y − x ) = π +1 and define a valuation ν of thequotient field K ∗ by ν ( x ) = 1 and ν ( y + x ) = π +1. Since u = x and v − u = ( y − x )( y + x ),we have that ν ( u ) = ν ( u ) = 2 and ν ( v − u ) = ν ( v − u ) = π + 2. Let ν be the commonrestriction of ν and ν to the quotient field K of R . Then ν splits in S . However, gr ν ( S )is a finitely generated gr ν ( R )-algebra since gr ν ( S ) = k [in ν ( x ) , in ν ( y − x )] is a finitelygenerated k -algebra. Note that gr ν ( R ) = k [in ν ( u ) , in ν ( v − u )] with in ν ( x ) = in ν ( u ) andin ν ( v − u ) = 2in ν ( y − x )in ν ( x ). References [1] S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321- 348.[2] S. Abhyankar, Local uniformization of algebraic surfaces over ground fields of characteristic p = 0, Annals of Math. 63 (1956), 491 -526.[3] S. Abhyankar, Ramification theoretic methods in algebraic geometry, Princeton Univ Press,1959.[4] S. Abhyankar, Resolution of singularities of embedded algebraic surfaces, second edition,Springer Verlag, New York, Berlin, Heidelberg, 1998.[5] A. Benito, O. Villamayor U., Techniques for the study of singularities with application toresolution of 2-dim schemes, Math. Ann. 353 (2012), 1937 - 1068.[6] A. Bravo and O. Villamayor U., Singularities in positive characteristic, Stratification andsimplification of the singular locus, Advances in Math. 224, (2010), 1349 - 1418.[7] V. Cossart, U. Jannsen and S. Saito, Canonical embedded and non-embedded resolution ofsingularities for excellent two-dimensional schemes, arXiv:0905.2191[8] V. Cossart and O. Piltant, Resolution of singularities of threefolds in positive characteristicI, Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J.Algebra 320 (2008), 1051 - 1082.[9] V. Cossart, and O. Piltant, Resolution of singularities of threefolds in positive characteristicII, J. Algebra 321 (2009), 1836 - 1976.[10] S.D. Cutkosky, Local factorization and monomialization of morphisms, Ast´erisque 260, 1999.[11] S.D. Cutkosky, Resolution of Singularities for 3-folds in positive characteristic, Amer. J.Math. 131 (2009), 59 - 127.[12] S.D. Cutkosky, Counterexamples to local monomialization in positive characteristic, Math.Annalen 362 (2015), 321 - 334.[13] S.D. Cutkosky, A generalization of the Abhyankar Jung Theorem to associated graded ringsof valuations, to appear in Proc. Cambridge Phil.Soc.[14] S.D. Cutkosky, Ramification of valuations and local rings in positive characteristic, Commu-nications in Algebra 44 (2016), 2828 - 2866.[15] S.D. Cutkosky and L. Ghezzi, Completions of valuation rings, Contemp. math. 386 (2005),13 - 34.[16] S.D. Cutkosky and O. Piltant, Ramification of Valuations, Advances in Math. 183 (2004),1-79.[17] S.D. Cutkosky and Pham An Vinh, Valuation semigroups of two dimensional local rings,Proceedings of the London Mathematical Society 108 (2014), 350 - 384.[18] S.D. Cutkosky and Pham An Vinh, Ramification of local rings along valuations, Journal ofpure and applied algebra 219 (21015), 2489 - 2505.[19] S.D. Cutkosky and B. Teissier, Semigroups of valuations on local rings, Mich. Math. J. 57(2008), 173 - 193.[20] A.J. de Jong, Smoothness, semi-stablility and alterations, Inst. Hautes Etudes Sci. Publ.math. 83 (1996) 51 -93.[21] O. Endler, Valuation Theory, Springer Verlag, New York, Heidelberg, Berlin, 1972.[22] F.J. Herrera Govantes, M.A. Olalla Acosta, M. Spivakovsky, B. Teissier, Extending valuationsto formal completions, in Valuation Theory in Interaction , 252 - 265 EMS Ser. Congr. Rep,Eur. Math. Soc., Zurich, 2014.
23] L. Ghezzi, Huy T`ai H`a and O. Kashcheyeva, Toroidalization of generating sequences indimension two function fields, J. Algebra 301 (2006) 838-866.[24] L. Ghezzi and O. Kashcheyeva, Toroidalization of generating sequences in dimension twofunction fields of positive characteristic, J. Pure Appl. Algebra 209 (2007), 631-649.[25] A. Grothendieck, and A. Dieudonn´e, El´ements de g´eom´etrie alg´ebrique II, Publ. Math. IHES8 (1961).[26] A. Grothendieck, and A. Dieudonn´e, El´ements de g´eom´etrie alg´ebrique III, Publ. Math. IHES11 (1961).[27] A. Grothendieck, and A. Dieudonn´e, El´ements de g´eom´etrie alg´ebrique IV, vol. 2, Publ.Math. IHES 24 (1965).[28] R. Hartshorne, Algebraic Geometry, Springer, New York, Heidelberg, Berlin, 1977.[29] H. Hauser, On the problem of resolution of singularities in positive characteristic (or: a proofwe are waiting for) Bull. Amer. Math. Soc. 47 (2010), 1-30.[30] H. Hironaka, Three key theorems on infinitely near singularities, Singularit´es Franco-Japonaises, 87 - 126, S´emin. Congr. 10 Soc. Math. France, Paris 2005.[31] H. Knaf and F.-V. Kuhlmann, Every place admits local uniformization in a finite extensionof the function field, Adv. Math. 221 (2009), 428 - 453.[32] H. Knaf and F-V. Kuhlmann, Abhyankar places admit local uniformization in any charac-teristic, Ann. Sci. ´Ecole Norm. Sup. 38 (2005), 833 - 846.[33] F.-V. Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, inResolution of Singularities - A Research Textbook in Tribute to Oscar Zariski, H. Hauser, J.Lipman, F. Oort, A. Quiros (es.), Progress in Math. 181, Birkh¨auser (2000), 4559 - 4600.[34] F.-V. Kuhlmann, Value groups, residue fields, and bad places of algebraic function fields,Trans. Amer. Math. Soc. 356 (2004), 363 - 395.[35] F.-V. Kuhlmann, A classification of Artin Schreier defect extensions and a characterizationof defectless fields, Illinois J. Math. 54 (2010), 397 - 448.[36] J. Lipman, Desingularization of 2-dimensional schemes, Annals of Math. 107 (1978), 115 –207.[37] S. MacLane, A construction for absolute values in polynomial rings, Trans. Amer. Math.Soc. 40 (1936), 363 - 395.[38] S. MacLane and O. Schilling, Zero-dimensional branches of rank 1 on algebraic varieties,Annals of Math. 40 (1939), 507 - 520.[39] J.S. Milne, ´Etale cohomology, Princeton University Press, 1980.[40] M. Moghaddam, A construction for a class of valuations of the field K ( X , . . . , X d , Y ) withlarge value group, J. Algebra 319, 7 (2008), 2803-2829.[41] M. Raynaud, Anneaux Locaux, Hens´eliens, Springer Verlag, Berlin, Heidelberg, New York,1970.[42] J.P. Serre, Corps Locaux, Hermann, 1962.[43] M. Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), 107 -156.[44] B. Teissier, Valuations, deformations and toric geometry, Valuation theory and its applica-tions II, F.V. Kuhlmann, S. Kuhlmann and M. Marshall, editors, Fields Institute Commu-nications 33 (2003), Amer. Math. Soc., Providence, RI, 361 – 459.[45] B. Teissier, Overweight deformations of affine toric varieties and local uniformization, in Val-uation theory in interaction, Proceedings of the second international conference on valuationtheory, Segovia-El Escorial, 2011. Edited by A. Campillo, F-V- Kehlmann and B. Teissier.European Math. Soc. Publishing House, Congress Reports Series, Sept. 2014, 474 - 565.[46] M. Temkin, Inseparable local uniformization, J. Algebra 373 (2013), 65 - 119.[47] M. Vaqui´e, Famille admissible de valuations et d´efaut d’une extension, J. algebra 311 (2007),859 - 876.[48] O. Zariski and P. Samuel, Commutative Algebra Volume I, Van Nostrand, 1958.[49] O. Zariski and P. Samuel, Commutative Algebra Volume II, Van Nostrand, 1960. Steven Dale Cutkosky, Department of Mathematics, University of Missouri, Columbia,MO 65211, USA
E-mail address : [email protected]@missouri.edu