SEMIGROUPS OF VALUATIONS ON LOCAL RINGS, II

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1 SEMIGROUPS OF VALUATIONS ON LOCAL RINGS, II STEVEN DALE CUTKOSKY, BERNARD TEISSIER Let (R, m R ) be a oca domain, with fraction fied K. Suppose that ν is a vauation of K with vauation ring (V, m V ) which dominates R; that is, R V and m V R = m R. The vaue groups Γ of ν which can appear when K is an agebraic function fied have been extensivey studied and cassified, incuding in the papers MacLane [0], MacLane and Schiing [], Zariski and Samue [6], Kuhmann [9] and Moghaddam []. These groups are we understood. The most basic fact is that there is an order preserving embedding of Γ into R h with the ex order, where h is the rank of the vauation, which is ess than or equa to the dimension of R. The semigroups S R (ν) = {ν(f) f m R {0}}, which can appear when R is a noetherian domain with fraction fied K dominated by ν, are not we understood, athough they are known to encode important information about the idea theory of R and the geometry and resoution of singuarities of Spec R. In particuar, after [5], the toric resoutions of singuarities of the affine toric varieties associated to certain finitey generated subsemigroups of S R (ν) are cosey reated to the oca uniformizations of ν on R. In Zariski and Samue s cassic book on Commutative Agebra [6], two genera facts about semigroups S R (ν) of vauations on noetherian oca domains are proven (in Appendix 3 to Voume II).. For any vauation ν of K which is non negative on R, the semigroup S R (ν) is a we ordered subset of the positive part of the vaue group Γ of ν, of ordina type at most ω h, where ω is the ordina type of the we ordered set N, and h is the rank of the vauation.. If ν dominates R, the rationa rank of ν pus the transcendence degree of V/m V over R/m R is ess than or equa to the dimension of R. The second condition is the Abhyankar inequaity []. In [6], the authors give some exampes showing that some surprising semigroups of rank > can occur as semigroups of vauations on noetherian domains, and raise the genera question of finding new constraints on vaue semigroups and cassifying semigroups which occur as vaue semigroups. The ony semigroups which are reaized by a vauation on a one dimensiona reguar oca ring are isomorphic to the natura numbers. The semigroups which are reaized by a vauation on a reguar oca ring of dimension with agebraicay cosed residue fied are much more compicated, but are competey cassified by Spivakovsky in [4]. A different proof is given by Favre and Jonsson in [7], and we reformuated the theorem in the context of semigroups in [6]. However, very itte is known in higher dimensions. The cassification of semigroups of vauations on reguar oca rings of dimension two does suggest that there may be constraints on the rate of growth of the number of new generators on semigroups The first author was partiay supported by NSF and by the University Paris 7-Denis Diderot. AMS cassification: Primary: 3A8, 4 E5, 6W50. Secondary: 06F05.

2 of vauations dominating a noetherian domain. In [4], such a constraint is found for rank vauations. We prove in this paper that there is such a constraint for vauations of arbitrary rank. In [4], a very simpe poynomia bound is found on the growth of S R (ν) for a rank vauation ν. This bound aowed the construction in [4] of a we ordered subsemigroup of Q + of ordina type ω, which is not a vaue semigroup of a noetherian oca domain. Thus the above conditions and do not characterize vaue semigroups on oca domains. Uness otherwise stated, in this text a oca rings are assumed to be noetherian. A vauation of a oca domain is a vauation ν of its fied of fractions whose ring R ν contains R in such a way that m ν R m R. In Section of this paper, we describe a poynomia behavior of vauation ideas P ϕ (R) = {x R ν(x) ϕ} and P ϕ + (R) = {x R ν(x) > ϕ}. Given a vauation ν with center p on a oca domain R we find very genera poynomia bounds on the growth of the sums of the mutipicities of the finitey generated R/p-modues P ϕ (R)/P ϕ + (R) when ϕ runs through growing regions of the vaue group Γ of ν viewed as a subgroup of R h. These modues are nonzero precisey when ϕ is 0 or beongs to the semigroup S R (ν). Our resuts therefore aso bound the number of eements of S R (ν) in those regions. This ast resut generaizes to a ranks the bound given for rank vauations in [4] (restated as Theorem. in this paper). The statement and proof for higher rank vauations is significanty more compex. We give an exampe (Exampe.) of a rank semigroup T which satisfies a restrictions on the semigroup of a vauation on an s dimensiona oca domain imposed by our poynomia bounds for modues over the rank convex subgroup Φ of the group Γ generated by T, but is not a vauation semigroup on an s dimensiona oca domain. The proof uses our most genera bound, Theorem.7, in the case of rank vauations. Our poynomia bounds are estimates of sums over the intersection of S R (ν) with certain regions of Γ. These regions are defined by their intersections with the convex subgroups of Γ and depend on a certain function ϕ whose precise definition is given in Definition. of Section. Given a vauation ν i+ composed with ν and an eement ϕ in ν i+ (R \ {0}) Γ/Φ i, the vaue ϕ is the smaest eement in the semigroup S R (ν i ) which projects to ϕ; it is an eement of Γ/Φ i. A ower imit in the sum at eve i is determined by the vaues traced out by ϕ as ϕ varies in the semigroup ν i+ (R \ {0}), whie the upper imit is of the form ϕ + y i t i, y i N. It is interesting to consider how cose these regions are to being poydiscs. The most desirabe situation is when the vaue group can be embedded by an order preserving homomorphism into (R h ) ex so that a of these regions are poydiscs. In the first exampes that one is ikey to consider, this is in fact the case. However, the genera situation is not so simpe. In Section we give exampes showing that the tide function can exhibit a rather wid behavior. We show that we can make ϕ decrease arbitrariy fast, and that this is independent of the embedding. We aso show that ϕ can increase arbitrariy fast, and finay that ϕ can jump back and forth from negative numbers which decrease arbitrariy fast to positive numbers which increase arbitrariy fast. A these exampes are independent of the embedding of Γ into R h. In view of the resuts of Section, the behavior of ϕ is an interesting measure of the compexity of the vauation.

3 . Poynomia bounds on vauation ideas In this section, we derive some very genera bounds on the growth of the number of distinct vauation ideas corresponding to vaues ying in certain parts of the group Γ. If G is a totay ordered abeian group, then G + wi denote the positive eements of G, and G 0 wi denote the nonnegative eements. If R is a oca ring, m R wi denote its maxima idea, and ength R (N) wi denote the ength of an R-modue N. Suppose that R is a domain and ν is a vauation of R. Let Γ be the vaue group of ν. We wi denote the vaue semigroup of ν on R by S R (ν) = {ν(f) f m R {0}}. S R (ν) is a subsemigroup of the nonnegative part, Γ 0, of Γ, and if ν dominates R, so that a eements of the maxima idea m R of R have positive vaue, then S R (ν) is a subsemigroup of the semigroup Γ + of positive eements of Γ. Suppose that I R is an idea. We wi write ν(i) = min{ν(f) f I {0}}. Note that ν(i) Γ 0 exists since R is noetherian. Suppose that ϕ is an eement of the vaue group Γ. We wi denote by P ϕ (R) the idea {x R ν(x) ϕ} and by P + ϕ (R) the idea {x R ν(x) > ϕ}. When no confusion on the ring is possibe we wi write P ϕ, P + ϕ. We note that P ϕ (R)/P + ϕ (R) = 0 if and ony if ϕ / S R (ν) {0}. The associated graded ring of ν on R is gr ν (R) = ϕ Γ P ϕ (R)/P + ϕ (R). This (R/m ν R)-agebra is not in genera finitey generated; it is graded by the semigroup S R (ν), which is not finitey generated in genera. Our resuts can be seen as an extension to these agebras of the cassica resuts on N-graded finitey generated agebras. Suppose that Γ is a totay ordered abeian group, and a, b Γ. We set [a, b] = {x Γ a x b} and [a, b[= {x Γ a x < b} The concepts of rank of a vauation, the convex (isoated) subgroups of a vauation group and the corresponding composed vauations are discussed in detai in Chapter VI of [6]. Suppose that ν has rank n. Let 0 = Φ 0 Φ Φ n = Γ be the sequence of convex subgroups of Γ. Let ν i, for i n, be the vauations on the quotient fied of R with which ν is composed. We have ν = ν. Let p n p be the corresponding centers on R of ν i. We define p n+ = (0). The vaue group of ν i is Γ i = Γ/Φ i. For i n, set t i = ν i (p i ) Φ i /Φ i Γ/Φ i. Let λ i : Γ i = Γ/Φ i Γ i+ = Γ/Φ i be the corresponding maps from the vaue group of ν i to the vaue group of ν i+. If p i p i+, then t i is in the kerne of λ i, which is a rank group. When there is no ambiguity, we denote by ϕ i the image in Γ/Φ i of an eement ϕ Γ. 3

4 Definition.. Given ϕ i Γ/Φ i, denote by ϕ i Γ/Φ i the minimum of ν i (f) for f R such that ν i (f) = ϕ i. This minimum exists since the semigroup S R (ν i ) is we ordered. Note that λ i ( ϕ i ) = ϕ i. If p i p i, we remark that for y i N and ϕ i [ ϕ i, ϕ i + y i t i ], we have the incusions: p y i i P ϕ i P ϕi P ϕi, P ϕi = P ϕi and since Φ i /Φ i is of rank one, the number of eements of ν i (R \ {0}) in the interva [ ϕ i, ϕ i + y i t i ] is finite (see [6], oc. cit.). Lemma.. Suppose that p p. Then for any function A on R-modues with vaues in R which is additive on short exact sequences of finitey generated R-modues whose unique minima prime is p, we have for a y N: () ϕ [ ϕ, ϕ +y t [ A(P ϕ /P + ϕ ) A(M ϕ /p y M ϕ ), where M ϕ = P ϕ /P + ϕ, a finitey generated torsion free R/p -modue. Proof. For y N, [ ϕ, ϕ + t y [ intersects S R (ν) {0} in a finite set {τ,..., τ r }, with τ = ϕ < τ < < τ r < ϕ + t y. We have incusions of R modues whose unique minima prime is p, P τr /P ϕ +t y P τr /P ϕ +t y P τ /P ϕ +t y = P ϕ /P ϕ +t y. By the additivity of A we have A(P ϕ /P ϕ +t y ) = ϕ [ ϕ, ϕ +y t [ A(P ϕ /P + ϕ ). From the incusion p y P ϕ P ϕ +y t, we have an exact sequence of R-modues whose unique minima prime is p : 0 P ϕ +t y /(P ϕ + + p y P ϕ ) P ϕ /(P ϕ + + p y P ϕ ) P ϕ /P ϕ +t y 0. Since M ϕ /p y M ϕ = Pϕ /(P ϕ + + p y P ϕ ), we have that A(P ϕ /P ϕ +t y ) A(M ϕ /p y M ϕ ), and the concusions of the emma foow. Suppose that p 0 is a prime idea of R such that p p p 0. Let e mi (N) denote the mutipicity of an R pi modue N with respect to m i = p i R pi. From the above emma, we immediatey deduce the foowing resut. Theorem.. Let R be a oca domain and ν a rank vauation of R. Let p 0 be a prime idea idea of R containing the center p of ν. Then e m0 ((P ϕ /P ϕ + ) p0 ) e m0 ((R/p ) p0 )ength Rp (R p /p y R p ) ϕ [0,yt [ for a y N. Thus we have e m0 ((P ϕ /P ϕ + ) p0 ) e m0 ((R/p ) p0 )P Rp (y) ϕ [0,yt [ 4

5 for y 0, where P Rp (y) is the Hibert-Samue poynomia of the oca ring R p. Proof. This foows from Lemma., with A(N) = e m0 (N R p0 ) for R-modues N. We take ν = ν, Γ = 0, ϕ = 0, and p p 0, so that ϕ = 0 and M ϕ = R, to get for y N, e m0 ((P ϕ /P ϕ + ) p0 ) e m0 (R p0 /p y R p 0 ). ϕ [0,yt [ The concusions of the theorem now foow from the associativity formua for mutipicity, [], Section 7, no., Proposition 3, which shows that e m0 (R p0 /p y R p 0 ) = e m0 ((R/p ) p0 )e m (R p /p y R p ) = e m0 ((R/p ) p0 )ength Rp (R p /p y R p ). If ν has rank, and dominates R, so that p = p 0 = m R is the maxima idea of R, we obtain the inequaity of [4], () #(S R (ν) [0, yt [ ) < P R (y) for y N sufficienty arge. This foows from Theorem. since P ϕ /P + ϕ 0 if and ony if there exists f R such that ν(f) = ϕ, and since ϕ = 0 is not in S R (ν). As was shown in [4], we may now easiy construct a we ordered subsemigroup U of Q + such that U has ordina type ω and U S R (ν) for any vauation ν dominating a oca domain R. We et T be any subset of Q + which has as its smaest eement, and y y < #([y, y + [ ) T ) < for a y Z +. Let U = n= nt be the semigroup generated by T. Then U is we ordered by a resut of B. H. Neumann (see [7]), and the function #([0, y[ U) grows faster than y d for any d N. Since the Hibert-Samue poynomia of a noetherian oca domain R has degree d = dim R <, it foows from formua () that U cannot be the semigroup of a vauation dominating a noetherian oca domain. Suppose now that p 0 is a prime idea of R such that p p p 0. Let e mi (N) denote the mutipicity of an R pi modue N with respect to m i = p i R pi. If p = p, so that t is not in the kerne of λ, et us define ϕ + = min{ν (f) f R and ν (f) > ϕ }. By [6] (Appendix 3, Coroary to Lemma 4), the interva [ ϕ, ϕ + [ contains ony finitey many eements of S R (ν). Theorem.3. Let R be a oca domain and ν, ν two vauations of R such that ν is composed with ν and the difference of their ranks is equa to one. Let p 0 be a prime idea of R containing the centers p p of ν and ν. Then we have: a) Suppose that p p. Then there exists a function s(ε, ϕ ) such that for ϕ Γ, ε > 0 and y N such that y > s(ε, ϕ ), we have ϕ [ ϕ, ϕ +y t [ e m 0 (P ϕ (R p0 )/P + ϕ (R p0 )) ( + ε) em 0 ((R/p ) p0 ) (dim (R/p ) p )! e m (P ϕ (R p )/P + ϕ (R p ))y dim (R/p ) p. b) If p = p we have the equaities ϕ [ ϕ, g ϕ + [ e m 0 (P ϕ (R p0 )/P + ϕ (R p0 )) = e m0 (P ϕ (R p0 )/P + ϕ (R p0 )) = e m0 ((R/p ) p0 )e m (P ϕ (R p )/P + ϕ (R p )). 5

6 Proof. Assume first that p p. Taking A(N) = e m0 (N p0 ) in Lemma., and using the identities P ϕ (R p0 ) = (P ϕ ) p0, we obtain e m0 (P ϕ (R p0 )/P ϕ + (R p0 )) e m0 ((M ϕ ) p0 /p y (M ϕ ) p0 ). ϕ [ ϕ, ϕ +y t [ Since (p ) p0 is the unique minima prime of (M ϕ ) p0 /p y (M ϕ ) p0, by [], Section 7, no., Proposition 3, we have e m0 ((M ϕ ) p0 /p y (M ϕ ) p0 ) = ength Rp ((M ϕ ) p /p y (M ϕ ) p )e m0 ((R/p ) p0 ). Since p is the unique minima prime of M ϕ, there exists a function s(ϕ ) such that ength Rp ((M ϕ ) p /p y (M ϕ ) p ) = H (Mϕ ) p (y ) = em ((Mϕ )p ) (dim(r/p ) p )! ydim(r/p ) p + ower order terms in y for y s(ϕ ), where H (Mϕ ) p (y ) is the Hibert-Samue poynomia of (M ϕ ) p. This poynomia bound impies that there exists a function s(ε, ϕ ) such that for y s(ε, ϕ ). ength Rp ((M ϕ ) p /p y (M ϕ ) p ) ( + ε) e m ((M ϕ ) p ) (dim(r/p ) p )! ydim(r/p ) p If p = p, We have p (P ϕ /P + ϕ ) = p (P ϕ /P + ϕ ) = 0, so the first inequaity stated in this case foows directy from the additivity of the mutipicity e m0. The second equaity foows from the first and the associativity formua of [], Section 7, no., Proposition 3. Coroary.4. Suppose that p n = = p = p. Then e m0 ((P ϕ /P ϕ + ) e m0 ((R/p ) p0 )ength Rp (R p /p y R p ) ϕ n [0,yt n[ ϕ n [ ϕ n, ϕ + n [ ϕ [ ϕ, ϕ + [ for a y N. Thus we have ϕ n [0,yt n[ ϕ n [ ϕ n, ϕ + n [ ϕ [ ϕ, ϕ + [ e m0 ((P ϕ /P + ϕ ) e m0 ((R/p ) p0 )P Rp (y) for y 0, where P Rp (y) is the Hibert-Samue poynomia of R p. Proof. We wi prove the formua by induction on the rank n of the vauation. If n =, this is just the statement of Theorem.. We wi assume that the formua is true for vauations of rank < n, and derived the formua for a rank n vauation ν. Let ν be the rank n vauation which ν is composite with. Consider the chain of ideas (0) = q n q n = = q = q 0, where q n = p n,..., q = p are the centers on R of the successive vauations ν n,..., ν with which ν is composed, and q 0 = p. We obtain (3) e m (P ϕ /P ϕ + ) e m ((R/p ) p )P Rp (y) = P Rp (y) ϕ n [0,yt n[ ϕ n [ ϕ n, ϕ + n [ ϕ [ ϕ 3, ϕ + 3 [ 6

7 for y 0. We appy Theorem.3 to the vauations ν = ν and ν and p = p p 0 to obtain for ϕ [ ϕ 3, ϕ + 3 [ (or ϕ [ ϕ n, ϕ + n + t n y[ if n = 3), (4) e m0 ((P ϕ /P ϕ + ) p0 ) e m0 ((R/p ) p0 )e m ((P ϕ /P ϕ + ) p ). ϕ [ ϕ, ϕ + [ Now sum over (3) and (4) to obtain the formua for ν. Coroary.5. In the specia case where ν is a vauation of rank one and ν is the trivia vauation, we have p = 0, Γ = 0, ϕ = 0 Γ and the inequaity of Theorem.3 reduces to: ϕ [0,y t [ e m 0 (P ϕ (R p0 )/P ϕ + (R p0 )) (5) for y > s(ε). ( + ε)e m0 ((R/p ) p0 ) em (Rp ) (dimr p )! ydimrp Coroary.6. Taking p 0 = p, we deduce that when p p we have for y > s(ε, ϕ ) an inequaity #(λ (ϕ ) S R (ν) [ ϕ, ϕ + y t [) ( + ε) em (Pϕ (Rp )/P+ ϕ (R p )) y dim(r/p ) p (dim(r/p ) p )!. When p = p, we have #(λ (ϕ ) S R (ν)) ength Rp (P ϕ (R p )/P + ϕ (R p )) <. Let (0) = p n+ p n p be the centers of the vauations with which ν is composed. Define I = {i {,..., n}/p i p i+ } and note that n I. By [6] (Appendix 3) we know that if i / I, if we define ϕ + i = min{ν i (f) f R and ν i (f) > ϕ i }, the intersection S R (ν) [ ϕ i, ϕ + i [ is finite. Let us agree that in this case, for arge y i the interva [ ϕ i+, ϕ i+ + t i y i [ coincides with this intersection. Remember aso that if i / I we have dim(r/p i+ ) pi = 0 and e mi ((R/p i+ ) pi ) = (dim(r/p i+ ) pi )! =. Theorem.7. Let R be a oca domain and ν a vauation of R which is of rank n. There exist functions s n (ε) and s i (ε, y i+, y i+,..., y n ) for i n, such that, using the notations and conventions introduced above, we have ϕ n [0,t ny n[ ( + ε) ϕ n [ ϕ n, ϕ n+t n y n [ Q n i=0 em i ((R/p i+) pi ) n Q n i= (dim(r/p i+) pi )! i= ydim(r/p i+) pi i ϕ [ ϕ, ϕ +t y [ e m 0 ((P ϕ /P + ϕ ) p0 ) for y n, y n,..., y N satisfying y n s n (ε), y n s n (ε, y n ),..., y s (ε, y,..., y n ). 7

8 Proof. The proof of this formua is by induction on the rank n of the vauation ν. We first prove the formua in the case when n =. We appy (5) to the ring R p0 and observe that for ϕ Γ, P ϕ (R p0 ) = (P ϕ ) p0, to obtain ϕ [0,t y [ e m0 ((P ϕ /P ϕ + ) p0 ) ( + ε) e m 0 ((R/p ) p0 ) e m (R p )y dim Rp (dim R p )! for y s (ε), which is the formua for n =. We now assume that the formua is true for vauations of rank < n. We wi derive the formua for a rank n vauation ν. We appy the formua to the rank n vauation ν which ν is composite with, and the chain of prime ideas (0) = q n q n q q 0 where q n = p n,..., q = p are the centers on R of the successive vauations ν n,..., ν with which ν is composed, and q 0 = p is the new mute prime idea. We obtain the inequaity (6) ϕ n [0,t ny n[ for ϕ [ ϕ 3, ϕ 3 +t y [ n e m ((P ϕ /P ϕ + i= ) p ) (+ε ) e m i ((R/p i+ ) pi ) n i= (dim (R/p i+) pi )! y n s n (ε ), y n s n (ε, y n ),..., y s (ε, y 3,..., y n ). n i= y dim (R/p i+) pi i We appy Theorem.3 to the vauations ν = ν and ν, and p p p 0, to obtain for ϕ [ ϕ 3 + t y [, In the case where p p (7) e m0 ((P ϕ /P ϕ + ) p0 ) ( + ε ) e m 0 ((R/p ) p0 ) (dim (R/p ) p )! e m ((P ϕ /P ϕ + ) p )y dim (R/p ) p ϕ [ ϕ, ϕ +y t [ for y > s(ε, ϕ ). Since #(S R (ν) [ ϕ 3, ϕ 3 + t y [ ) <, we may define s (ε, y, y 3,..., y n ) = max{s(ε, ϕ ) ϕ [ ϕ 3, ϕ 3 +t y [, ϕ 3 [ ϕ 4, ϕ 4 +t 3 y 3 [,..., ϕ n [0, t n y n [}. In the case where p = p we have by theorem.3, b) the equaity + e m0 ((P ϕ /Pϕ ) p0 ) = e m0 ((P ϕ /P ϕ + ) p0 ) = e m0 ((R/p ) p0 )e m ((P ϕ /P ϕ + ) p ), ϕ [ ϕ, ϕ g+ [ and define s (ε, y ) =. Finay, we set ε = og (+ε), so that ( + ε ) = + ε, and sum over (7) and (6) after mutipication by the appropriate factor to obtain the desired formua for ν. As an immediate coroary, we obtain Coroary.8. The sum ϕ n [0,t ny n[ ϕ n [ ϕ n, ϕ n+t n y n [ ϕ [ ϕ, ϕ +t y [ 8 e m0 ((P ϕ /P + ϕ ) p0 )

9 is bounded for y y y n 0 by a function which behaves asymptoticay as n i=0 e m i ((R/p i+ ) pi ) n n i= (dim(r/p y dim (R/p i+) pi i. i+) pi )! Using the notations of Definition., and the conventions preceding Theorem.7, define the pseudo-boxes B Γ (y,..., y n ) = {ϕ Γ/ϕ [ ϕ, ϕ + t y [, ϕ [ ϕ 3, ϕ 3 + t y [,..., ϕ n [0, t n y n [}. Then we have: Coroary.9. For y y y n 0 the number #(S R (ν) B Γ (y,..., y n )) is bounded by the same function as in Coroary.8. Remarks.0. ) The ony centers p i which contribute to the right hand side of the inequaities are those for which the incusion p i+ p i is strict. ) The tota degree of the monomia appearing on the right hand side is dim R dim R/p, which is dim R in the case where ν is centered at the maxima idea m R. We now give an appication of Theorem.7. Suppose that ν is a rank vauation dominating a oca domain R. Let Γ be the vaue group of the composed vauation ν of the quotient fied of R, and et p be the center of ν on R, t = ν(m R ). Theorem.3 gives us a famiy of growth conditions for ϕ Γ on S R (ν) [ ϕ, nt [ for n sufficienty arge. To be precise, Theorem.3 tes us that for each ϕ Γ, there exist functions d(ϕ ) and s(ϕ ) N such that (8) #(S R (ν) [ ϕ, ϕ + nt [ ) < d(ϕ )n dim R/p for n > s(ϕ ). Exampe.. For every natura number s 3, there exists a rank, we ordered subsemigroup T of the positive part of (Z Q) ex, which is of ordina type ω and satisfies the restrictions (8) for a ϕ N, but is not the semigroup of a vauation dominating an s dimensiona oca domain. i= Proof. Let r = s. Define a subsemigroup of Q 0 by S = {( m + j) + α (m+)r m, j, α N, 0 j < m, 0 α < (m+)r }. Suppose that n is a positive integer. Then there exists a unique expression n = m + j with 0 j < m. We have and since m n < m+, #(S [n, n + [ ) = (m+)r, (9) n r < #(S [n, n + [ ) r n r. For y a positive integer, et f(y) = y n= nr. The first difference function f(y + ) f(y) = y r is a poynomia of degree r in y. Thus f(y) is a poynomia of degree r + in y, with positive eading coefficient. From y #(S [0, y ) = #(S [n, n + [ ) n= 9

10 and (9), we deduce that (0) f(y) < #(S [0, y[ ) r f(y). Suppose that c N. Then #(( c S) [0, y[ ) = #(S [0, cy[ ). Thus () f(cy) < #(( c S) [0, y[ ) r f(cy). For i N, et () c(i) = Let { if i = 0 i if i. T = {m} ( c(m) S) (Z Q) ex. m N T is a we ordered subsemigroup of (Z Q) ex, of ordina type ω. Suppose that T is the semigroup S R (ν) of a vauation ν dominating an s = r + dimensiona oca domain R. Then ν has rank. Let ν be the composed vauation ν (f) = π (ν(f)) for f R, where π : Z Q Z is the first projection. By assumption, the center of ν on R is the maxima idea m R of R. Let p be the center of ν on R. We see from an inspection of T that t = ν(m R ) = (0, ) and t = ν (p ) =. Further, ϕ = (ϕ, c(ϕ ) ) for a ϕ Z +. Observe that for a ϕ Z +, and y N, #(T [ ϕ, ϕ + y t [ ) = #(T {ϕ } [0, y [ ) = #( c(ϕ ) S [0, y [ ). From (), we see that (3) f(c(ϕ )y ) < #(T {ϕ } [0, y [ ) r f(c(ϕ )y ). Thus T satisfies the growth conditions (8) on a oca domain R with dim R/p r +. Since T has rank, we must have that dim R dim R/p +. Since we are assuming that R has dimension s = r +, we have that dim R/p = r + and dim R p =. Since ν is a discrete rank vauation dominating R p, this is consistent. Theorem.7 tes us that there exists a function s(y ) and d Z + such that #([0, y [ [0, y [ T ) dy a y b for y s(y ), where a = dim R/p = r + and b = dim R p =. From (3) and (), we see that y f(y ) + f(iy ) #([0, y [ [0, y [ T ). i= There exists a positive constant e such that f(y ) e y r+ for a y N, and thus there exists a positive constant e such that for y, y N. Thus we have y f(y ) + f(iy ) ey r+ y r+ dyy a b = dy r+ y > ey r+ y r+ for a arge y, which is impossibe. i= 0

11 . Wid behavior of the tide function The tide function ϕ, defined in Definition., gives critica information about the behavior of vauations of rank arger than one. This is iustrated by its roe in the statement of Theorem.7, which shows that there is some order in the behavior of semigroups of higher rank vauations. However, the sums in this theorem are a defined starting from the functions ϕ. This function can be extremey chaotic, as we wi iustrate in this section. We wi give exampes of rank two vauations, showing that ϕ can decrease arbitrariy fast as ϕ increases (Exampe.), ϕ can increase arbitrariy fast as ϕ increases (Exampe.3), and that ϕ can jump back and forth from negative numbers which decrease arbitrariy fast to positive numbers which increase arbitrariy fast as ϕ increases (Exampe.4). These properties are a independent of order preserving isomomorphism of the vaue group. To construct our exampes, we wi make use of the foowing technica emma, and some variants of it. This emma is a generaization of the notion of generating sequences of vauations on reguar oca rings of dimension ([0], [4], [7]). Lemma.. Suppose that σ : Z + N is a function. Let K(x, y, z) be a rationa function fied in three variabes over a fied K. Set P 0 = x, P = y and P i+ = z σ(i) Pi P0 i+ P i for i. Define, by induction on i, η 0 =, and (4) η i+ = η i + i+ for i 0. Define γ 0 = 0 and (5) γ i = ( σ() i + σ() σ(i) + + i ) for i > 0.. Suppose that f(x, y, z) K(z)[x, y]. Then for N such that deg y f <, there is a unique expansion (6) f = α a α (z)x α 0 P α P α where a α (z) K(z) and the sum is over α = (α 0, α,..., α ) N {0, }.. For f K(z)[x, y], define from the expansion (6), (7) ν(f) = min α {(0, ord z (a α (z)) + i=0 α i (η i, γ i )} ( Z Z) ex. Then ν defines a rank vauation on K(x, y, z), which is composite with a rank vauation ν of K(x, y, z). The vaue group of ν is Z = i= Z. i The vauation ν dominates the oca ring R = K[x, y, z] (x,y,z) and the center of ν on R is the prime idea (x, y). Proof. We have that deg y P i = i for i. Suppose that f K(z)[x, y] and is such that deg y f <. By the eucidean agorithm, we have a unique expansion f = g 0 (x, y) + g (x, y)p

12 with g 0, g K(z)[x, y] and deg y g 0 <, deg y g <. Iterating, we have a unique expansion of f of the form of (6). We have (8) ν(p i ) = (η i, γ i ) and (9) ν(z σ(i) P i ) = ν(p i+ 0 P i ) < ν(p i+ ) for a i. Observe that for (0) α = (α 0,..., α ), β = (β 0,..., β ) N {0, }, α i η i = i=0 β i η i impies α = β. The function ν defined by (7) thus has the property that there is a unique term in the expansion (6) for which the minimum (7) is achieved. We wi verify that ν defines a vauation on K(x, y, z). Suppose that f, g K(z)[x, y]. Let i=0 () f = α and () g = β a α (z)x α 0 P α P α a β (z)x β 0 P β P β be the expressions of f and g of the form (6). f + g = α (a α (z) + b α (z))x α 0 P α P is the expansion of f+g of the form (6). Since ord z (a α (z)+b α (z)) min{ord z (a(z)), ord z (b(z))} for a α, we have that ν(f + g) min{ν(f), ν(g)}. We wi now show that ν(fg) = ν(f) + ν(g). Suppose that (3) s = δ d δ (z)x δ 0 P δ P δ is an expansion, with d δ K(z), and δ = (δ 0,..., δ ) N + for a δ. We define Λ(s) = min δ {(0, ord z d δ (z)) + δ i (η i, γ i )}. Observe that if s is an expansion of the form (6); that is, δ N {0, } for a δ, then Λ(s) = ν(s). Let c ε (z) = a α (z)b β (z). α+β=ε i=0

13 Let (4) s 0 = ε c ε (z)x ε 0 P ε P ε. s 0 is an expansion of the form (3), and fg = s 0. To simpify the indexing ater on in the proof, we observe that we can initiay take as arge as we ike. Let α (in the expansion ()) be such that ν(f) = ν(a α (z)x α 0 P α P α ) and et β (in the expansion ()) be such that Let ε = α + β. Then ν(g) = ν(b β (z)x β 0 P β P β ). c ε (z)x ε ε 0 P P ε is the ony term in s 0 which achieves the minimum Λ(s 0 ). We have that c ε (z) = a α (z)b β (z) and (5) Λ(s 0 ) = ν(f) + ν(g). If ε N {0, } whenever c ε 0, then we can can compute ν(fg) = Λ(s 0 ) and we are done. Otherwise, there exists an i such that there exists an ε with ε i and c ε (z) 0. We then substitute the identity: (6) Pi = z σ(i) P i+ + xi+ P zσ(i) i into s 0 to obtain an expansion of the form (3), where a terms c ε (z)x ε 0 P ε P γ with ε i are modified to the sum of two terms c ε (z) z σ(i) xε 0+ i+ P ε P ε i + i P ε i i P ε i+ i+ Pε +c ε(z) z σ(i) xε 0 P ε P ε i i P ε i i Coecting terms with ike monomias in x, P,..., P, we obtain a new expansion s = d δ (z)x δ 0 P δ P δ P ε i++ i+ P ε. of fg. From the identities (9), we see that the minimum Λ(s ) is ony obtained by the term d δ (z)x δ δ 0 P P δ {, where d δ (z)x δ δ 0 P P δ c ε (z)x ε ε 0 P = P ε if ε i < c ε (z) x ε z σ(i) 0 +i+ P ε P ε i + i P ε i i P ε i+ i+ Pε if ε i. We have Λ(s ) = Λ(s 0 ). By descending induction on the invariants n = max{δ + + δ some δ i and d δ (z) 0} and m = #{(δ,..., δ ) N δ + + δ = n, some δ i and d δ (z) 0}, making substitutions of the form (6), we eventuay obtain an expression s of fg of the form (3), with (δ 0,..., δ ) N {0, } for a. We then compute ν(fg) = Λ(s) = Λ(s 0 ) = ν(f) + ν(g). We have thus competed the verification that ν is a vauation. 3

14 Exampe.. Suppose that f : N Z is a decreasing function, and K is a fied. Then there exists a vauation ν of the three dimensiona rationa function fied K(x, y, z) with vaue group ( Z Z) ex, which dominates the reguar oca ring R = K[x, y, z] (x,y,z), such that for any vauation ω equivaent to ν with vaue group ( Z Z) ex, for a sufficienty arge n N, there exists λ Z [0, n[ such that π ( λ) < f(n), where π : Z Z Z is the second projection. Proof. We choose positive integers σ(i) so that γ i = ( σ() i + σ() σ(i) + + i ) < f(ii+3 ) for a positive integers i, where η i are defined by (4), and so that γ i Z. Let ν be the vauation dominating R defined by Lemma., with this choice of σ. The vaue group of ν is ( Z Z) ex. Let ω be a vauation equivaent to ν with vaue group ( Z Z) ex. We have ω(x) = (a, b) for some a Z + and b Z, and ω(z) = (0, c) for some c Z + since convex subgroups have to be preserved under an automorphism of ordered groups. From the reations (9) we see that ω(p i ) = η i ω(x) + γ i ω(z) for i, which impies b = 0, since π (ω(p i )) Z. We aso have ω(a(z)) = ord z (a)ω(z) for a(z) K(z). Let ω be the vauation on K(x, y, z) defined by ω (f) = π (ω(f)), where π : Z Z Z is the first projection. Let e = a, and n 0 = a e+. Suppose that n n 0. We wi find λ Z [0, n[ such that π ( λ) < f(n). There exists i e such that a i+ n < a i+3. Let λ = aη i. From η i = 3 (i+ i ), we obtain λ = aη i < a i+ n. Since c Z +, i e and f is decreasing, we have Thus cγ i γ i < f(i i+3 ) f( a i+3 ) < f(n). π ( λ) π (ω(p i )) = cγ i < f(n). Exampe.3. Suppose that g : N Z is an increasing function, and K is a fied. Then there exists a vauation ν of the three dimensiona rationa function fied K(x, y, z) with vaue group ( Z Z) ex, which dominates the reguar oca ring R = K[x, y, z] (x,y,z), such that for any vauation ω equivaent to ν with vaue group ( Z Z) ex, for a sufficienty arge n N, there exists λ Z [0, n[ such that π ( λ) > g(n), where π : Z Z Z is the second projection. Proof. The proof is a variation of the proof of Exampe.. We outine it here. We first must estabish a modification of Lemma.. Suppose that τ : Z + N is a function. Set Q 0 = x, Q = y 4

15 and Q i+ = Q i z τ(i) Q0 i+ Q i for i. As in Lemma., define by induction on i, η 0 =, and (7) η i+ = η i + i+ for i 0. Define δ 0 = 0 and (8) δ i = τ() i + τ() τ(i) + + i for i > 0. Suppose that f(x, y, z) K[x, y, z]. Then for N such that deg y f <, there is a unique expansion (9) f = α a α (z)x α 0 P α P α where a α (z) K[z] and the sum is over α = (α 0, α,..., α ) N {0, }. This is estabished as (6) in the statement of Lemma.. We have a stronger statement which is vaid in the poynomia ring K[x, y, z], since the eading coefficients of the Q i, as poynomias in y, have as their eading coefficient. For f K[x, y, z], define from the expansion (9), (30) ν(f) = min α {(0, ord z (a α (z)) + i=0 α i (η i, δ i )} ( Z Z) ex. Then ν defines a rank vauation on K(x, y, z), which is composite with a rank vauation ν of K(x, y, z). The vaue group of ν is Z = i= Z. i ν dominates the oca ring R = K[x, y, z] (x,y,z) and the center of ν on R is the prime idea (x, y). We have ν(q i ) = (η i, δ i ) and (3) ν(q i ) = ν(z τ(i) Q i+ 0 Q i ) < ν(q i+ ) for a i. We now construct the exampe. We choose positive integers τ(i) so that δ i = τ() i + τ() τ(i) + + i > g(ii+3 ) for a positive integers i, and δ i Z for a i. Let ν be the vauation constructed above, which dominates R. The vaue group of ν is ( Z Z) ex. Let ω be a vauation equivaent to ν with vaue group ( Z Z) ex. We have ω(x) = (a, b) for some a Z + and b Z, and ω(z) = (0, c) for some c Z +. From the reations (3) we see that ω(q i ) = η i ω(x) + δ i ω(z) for i, which impies b = 0, since π (ω(q i )) Z. We aso have ω(a(z)) = ord z (a)ω(z) for a(z) K(z). Let ω be the vauation on K(x, y, z) defined by ω (f) = π (ω(f)), where π : Z Z Z is the first projection. Let e = a and n 0 = a e+. Suppose that n n 0. We wi find λ Z [0, n[ such that π ( λ) > g(n). 5

16 There exists i e such that a i+ n < a i+3. Let λ = aη i. From η i = 3 (i+ i ), we obtain λ = aη i < a i+ n. Since c Z +, i e and g is increasing, we have Thus cδ i δ i > g(i i+3 ) g( a i+3 ) > g(n). π ( λ) = π (ω(q i )) = cδ i > g(n). Exampe.4. Suppose that f : N Z is a decreasing function, g : N Z is an increasing function, and K is a fied. Then there exists a rank vauation ν of the five dimensiona rationa function fied K(x, y, u, v, z) with vaue group (H Z) ex, where H = ( Z + Z ) R, which dominates the reguar oca ring R = K[x, y, u, v, z] (x,y,u,v,z), such that for any vauation ω equivaent to ν with vaue group (H Z) ex, for a sufficienty arge n N, there exists λ H [0, n[ such that π ( λ ) < f(n)and there exists λ H [0, n[ such that π ( λ ) > g(n), where π : Z Z Z is the second projection. Proof. We need an extension of the method of Lemma. for constructing vauations which we first outine. Suppose that σ : Z + N and τ : Z + N are functions. Define P 0 = x, P = y and P i+ = z σ(i) Pi P0 i+ P i for i. Define Q 0 = u, Q = v and Q i+ = Q i zτ(i) Q i+ 0 Q i for i. Suppose that f K[z, x, y, u, v] and deg f <. Then there is a unique expansion f = β g β (z, x, y)u β 0 Q β Qβ where the sum is over β = (β 0, β,..., β ) N {0, } and g β K[z, x, y] for a β (since the eading coefficient of each Q i with respect to y is ). Each g β (z, x, y) has a unique expansion g β = a α,β (z)x α 0 P α P α α where the sum is over α = (α 0, α,..., α ) N {0, } and a α,β (z) K(z) for a α, β. Thus f has a unique expansion (3) f = α,β a α,β (z)x α 0 P α P α u β 0 Q β Qβ with α, β N {0, }, and a α,β (z) K(z). Set η 0 = and η i+ = η i + i+ for i 0. Set γ 0 = 0 and for i > 0. Set δ 0 = 0 and γ i = ( σ() i + σ() σ(i) + + i ) δ i = τ() i + τ() τ(i) + + i 6

17 for i > 0. Let H = ( Z + Z ) R. We define from the expansion (3), (33) ν(f) = min α,β {(0, ord z (a α,β (z)) + α i (η i, γ i ) + β i (η i, δi )} (H Z) ex. We have ν(z σ(i) Pi ) = ν(p0 i+ P i ) < ν(p i+ ) for a i, ν(p i ) = (η i, γ i ) for a i, ν(q i ) = ν(z τ(i) Q i+ 0 Q i ) < ν(q i+ ) for a i, ν(q i ) = (η i, δi ) for a i. Further, there is a unique term in the expansion (3) which achieves the minimum (33). We have if and ony if f has the form and if and ony if f has the form ν(f) Z Z f = a α,0 (z)x α 0 P α P α + higher vaue terms, ν(f) Z Z f = a 0,β (z)u β 0 Q β Qβ + higher vaue terms. Observe that for a β, we have a 0,β (z) K[z] in the expansion (3). We now construct the vauation of the exampe. For i Z +, choose σ(i) N such that γ i < f(i i+3 ) and choose τ(i) Z + so that g(i i+3 ) < δ i for a i Z +, and γ i, δ i Z for a i. Let ν be the vauation constructed above, which dominates R. The vaue group of ν is (H Z) ex. Let ω be a vauation equivaent to ν with vaue group (H Z) ex. There exist α, β, γ, δ Z, b, b Z, c Z +, such that where ω(x) = (a, b ), ω(u) = (a, b ), ω(z) = (0, c) a = α + β > 0, a = γ + δ > 0 and αδ βγ 0. We have ω(p i ) = η i ω(x) + γ i ω(z) and ω(q i ) = η i ω(u) + δ i ω(z) for i, which impies b = b = 0, since π (ω(p i )), π (ω(q i )) Z. Set e = max{ a, a }. Let n 0 = e e+. Suppose that n n 0. We wi show that there exists λ H [0, n[ such that π ( λ ) < f(n) and there exists λ H [0, n[ such that π ( λ ) > g(n). There exists i e such that e i+ n < e i+3. Let λ = a η i. From η i = 3 (i+ i ) we obtain λ = a η i < a i+ e i+ n. 7

18 Since c Z +, i e and f is decreasing, we have cγ i γ i < f(i i+3 ) f(e i+3 ) < f(n). Thus π ( λ ) π (ω(p i )) = cγ i < f(n). Let λ = a η i. We have λ = a η i < a i+ e i+ n. Since c Z +, i e and g is increasing, we have cδ i δ i > g(i i+3 ) g(e i+3 ) > g(n). Thus π ( λ ) = π (ω(q i )) = cδ i > g(n). References [] S. Abhyankar, On the vauations centered in a oca domain, Amer. J. Math. 78 (956), [] N. Bourbaki, Agèbra Commutative, Chapitres 8 et 9, Eéments de Math., Hermann, Paris (965) [3] S.D. Cutkosky, Loca factorization and monomiaization of morphisms, Astérisque 60, 999. [4] S.D. Cutkosky, Semigroups of vauations dominating oca domains, arxiv: [5] S.D. Cutkosky and L. Ghezzi, Competions of vauation rings, Contemp. Math. 386 (005), [6] S.D. Cutkosky and B. Teissier, Semigroups of vauations on oca rings, to appear in Mich. Math. J. [7] C. Favre and M. Jonsson, The vauative tree, Lecture Notes in Math 853, Springr Verag, Berin, Heideberg, New York, 004. [8] W. Heinzer and J. Say, Extensions of vauations to the competion of a oca domain, Journa of Pure and Appied Agebra 7 (99), [9] F.-V. Kuhmann, Vaue groups, residue fieds, and bad paces of agebraic function fieds, Trans. Amer. Math. Soc. 40 (936), [0] S. MacLane, A construction for absoute vaues in poynomia rings, Trans. Amer. Math. Soc. 40 (936), [] S. MacLane and O. Schiing, Zero-dimensiona branches of rank on agebraic varieties, Annas of Math. 40 (939), [] M. Moghaddam, A construction for a cass of vauations of the fied K(X,..., X d, Y ) with arge vaue group, Journa of Agebra, 39, 7 (008), [3] B.H. Neumann, On ordered division rings, Trans. Amer. Math. Soc., 66 (949), 0-5. [4] M. Spivakovsky, Vauations in function fieds of surfaces, Amer. J. Math. (990), [5] B. Teissier, Vauations, deformations, and toric geometry, Proceedings of the Saskatoon Conference and Workshop on vauation theory (second voume), F.-V. Kuhmann, S. Kuhmann, M. Marsha, editors, Fieds Institute Communications, 33, 003, [6] O. Zariski and P. Samue, Commutative Agebra Voume II, D. Van Nostrand, Princeton, 960. [7] B.H. Neumann, On ordered division rings, Trans. Amer. Math. Soc., 66 (949), 0-5. Dae Cutkosky, Bernard Teissier, 0 Mathematica Sciences Bdg, Institut Mathématique de Jussieu University of Missouri, UMR 7586 du CNRS, Coumbia, MO 65 USA 75 Rue du Chevaeret, 7503 Paris,France cutkoskys@missouri.edu teissier@math.jussieu.fr 8