SEMIGROUPS OF VALUATIONS DOMINATING LOCAL DOMAINS STEVEN DALE CUTKOSKY Let (R, m R ) be a equicharacteristic local domai, with quotiet field K. Suppose that ν is a valuatio of K with valuatio rig (V, m V ). Suppose that ν domiates R; that is, R V ad m V R = m R. The possible value groups Γ of ν have bee extesively studied ad classified, icludig i the papers MacLae [8], MacLae ad Schillig [9], Zariski ad Samuel [12], ad Kuhlma [7]. The most basic fact is that there is a order preservig embeddig of Γ ito R with the lex order, where is the dimesio of R. The semigroup S R (ν) = {ν(f) f m R {0} is however ot well uderstood, although it is kow to ecode importat iformatio about the topology, resolutio of sigularities ad ideal theory of R. I Zariski ad Samuel s classic book o Commutative Algebra [12], two geeral facts about the semigroup S R (ν) are prove (Appedix 3 to Volume II). 1. S R (ν) is a well ordered subset of the positive part of the value group Γ of ν of ordial type at most ω h, where ω is the ordial type of the well ordered set N, ad h is the rak of the valuatio. 2. The ratioal rak of ν plus the trascedece degree of V/m V over R/m R is less tha or equal to the dimesio of R. The secod coditio is the Abhyakar iequality [1]. Prior to this paper, o other geeral costraits were kow o value semigroups S R (ν). I fact, it was eve ukow if the above coditios 1 ad 2 characterize value semigroups. I this paper, we costruct a example of a well ordered subsemigroup of Q + of ordial type ω, which is ot a value semigroup of a equicharacteristic oetheria local domai. This shows that the above coditios 1 ad 2 do ot characterize value semigroups o equicharacteristic oetheria local domais. We costuct this i Example 2.3 by fidig a ew costrait, Theorem 2.1, o a semigroup beig a value semigroup of a equidimesioal oetheria local domai of dimesio. I Corollary 2.2, we give a stroger costrait o regular local rigs. I [4], Teissier ad the author give some examples showig that some surprisig semigroups of rak > 1 ca occur as semigroups of valuatios o oetheria domais, ad raise the geeral questio of fidig ew costraits o value semigroups ad classifyig semigroups which occur as value semigroups. The oly semigroups which are realized by a valuatio o a oe dimesioal regular local rig are isomorphic to the atural umbers. The semigroups which are realized by a valuatio o a regular local rig of dimesio 2 with algebraically closed residue field are much more complicated, but are completely classified by Spivakovsky i [10]. A differet proof is give by Favre ad Josso i [5], ad the theorem is formulated i the cotext of semigroups by Cutkosky ad Teissier [4]. However, very little is kow i higher dimesios. The classificatio of semigroups of valuatios o regular local rigs of The author was partially supported by NSF. 1
dimesio two does suggest that there may be costraits o the rate of growth of the umber of ew geerators o semigroups of valuatios domiatig a oetheria domai. We prove that there is such a costrait, givig a ew ecessary coditio for a semigroup to be a value semigroup. This is accomplished i Theorem 2.1 ad Corollary 2.2. The costrait is far from sharp, eve for oe dimesioal regular local rigs. The criterio of Theorem 2.1 is however sufficietly strog to allow us to give a very simple example, Example 2.3, of a well ordered subsemigroup S of Q + of ordial type ω which is ot the semigroup of a valuatio domiatig a equidimesioal oetheria local domai. 1. Semigroups of valuatios domiatig polyomial rigs Give a set T, T will deote the cardiality of T. If S is a totally ordered abelia semigroup, S + will deote the set of positive elemets of S. Let k be a field, be a positive iteger, ad A = k[x 1,..., x ] be a polyomial rig i variables. Let m be the maximal ideal m = (x 1,..., x ) of A. A is aturally a k vector space. Let F,d be the k subspace of m of polyomials of degree d with 0 as the costat term. Suppose that ν is a valuatio of the quotiet field of A with valuatio rig V. Suppose A V ad m V A = m. Let Γ be the value group of ν. Let Ψ be the covex subgroup of real rak 1 of Γ, so that Ψ is order isomorphic to a subgroup of R. Let S(ν) = ν(m {0}) Ψ. If S(ν), the S(ν) is a well ordered semigroup of ordial type ω, by the first coditio stated i the itroductio, sice S(ν) is a value semigroup o the quotiet A/{f A ν(f) Ψ}. Let the smallest elemet of S(ν) be s 0. Let S d (ν) = ν(f d, {0}). Lemma 1.1. The upper boud holds for all d N. ( ) + d S d (ν) < Proof. For λ Γ, the set C λ = {f F d, ν(f) > λ} is a k subspace of F d,. Sice S d (ν) is well ordered ad dim k F d, = ( ) +d d 1, the lemma follows. For a, b Ψ, let [a, b) = {c Ψ a c < b}. Lemma 1.2. For all d N, ( ) + d S(ν) [0, (d + 1)s 0 ) <. Proof. Sice every elemet of m has value s 0, every elemet of m d+1 has value (d + 1)s 0. For f m, we have a expressio f = g + h with g F d, ad h m d+1. If ν(f) < (d+1)s 0, we must have ν(f) = ν(g). Thus S(ν) [0, (d+1)s 0 ) = S d (ν) [0, (d+1)s 0 ). Example 1.3. There exists a well ordered subsemigroup U of Q + such that U has ordial type ω ad U S(ν) for ay valuatio ν as above. Proof. Let T be ay subset of Q + such that 1 is the smallest elemet of T ad ( ) 2 T [, + 1) = 2
for all 1. For all positive itegers r, let rt = {a 1 + + a r a 1,..., a r T }. Let U = ωt = r=1rt be the semigroup geerated by T. By our costructio, U [0, r) < for all r N. Thus U is well ordered ad has ordial type ω. Suppose that there exists a field k, a positive iteger, ad a valuatio ν such that S(ν) = U. We have ( ) ( ) 2 2 = T [, + 1) U [, + 1) U [0, + 1) 1, where the last iequality is by Lemma 1.2, which is a cotradictio. 2. Semigroups of valuatios domiatig local domais Let ν be a valuatio of a field K. Let V be the valuatio rig of ν, ad let m V be the maximal ideal of V. Suppose that R is a oetheria local domai with quotiet field K such that ν domiates R; that is, R V ad m V R = m R is the maximal ideal of R. Let Γ be the value group of ν. Γ has fiite rak, sice ν domiates the oetheria local rig R, ad by the secod coditio stated i the itroductio. Let Ψ be the covex subgroup of real rak 1 of Γ. Defie a semigroup of real rak 1 by S R (ν) = ν(m R {0}) Ψ. For a, b Ψ, let [a, b) = {c Ψ a c < b}. Let ˆR be the m R -adic completio of R, which aturally cotais R. The there exists a prime ideal Q of ˆR such that the valuatio ν iduces uiquely a valuatio ν of the quotiet field of ˆR/Q which domiates ˆR/Q, ν has rak 1, S ˆR/Q (ν) = S R (ν), ad s 0 = mi{ν(f) f m ˆR/Q }. We will call Q the prime ideal of elemets of  of ifiite value. The costructio of Q ad related cosideratios are discussed i Spivakovsky [10], Heizer ad Sally [6], Cutkosky [2], Cutkosky ad Ghezzi [3] ad Teissier [11]. We will outlie the costructio of Q i the followig paragraph. Let α = mi{ν(g) g m R }. Suppose that f ˆR. The for ay Cauchy sequece {g i } i R which coverges to f, exactly oe of the followig two coditios must hold. 1) There exists N such that ν(g i ) α for all i 0 or 2) For all N, there exists i N such that ν(g i ) > α. Moreover, either coditio 1) holds for all Cauchy sequeces {g i } i R which coverge to f or coditio 2) hold for all Cauchy sequeces {g i } i R which coverge to f. If coditio 1) holds for f, the there exists λ S R (ν) such that for ay Cauchy sequece {g i } i R which coverges to f, there exists N (which depeds o the Cauchy sequece) such that ν(g i ) = λ for all i. I this, way may exted ν to a fuctio o ˆR, defiig for f ˆR, ν(f) = λ if 1) holds, ad ν(f) = if 2) holds. Let Q = {f ˆR ν(f) = }. Q is a prime ideal i ˆR. We exted ν to a valuatio ν of the quotiet field of ˆR/Q with the desired properties by defiig ν(f + Q) = ν(f) for f ˆR. If Γ has rak 1, we have Q R = (0), ad we have a iclusio R ˆR/Q. Theorem 2.1. Suppose that A is a equicharacteristic local domai, ad ν is a valuatio which domiates A. Let s 0 = mi{ν(f) f m A } ad = dim A/mA m A /m 2 A. The ( ) + d (1) S A (ν) [0, (d + 1)s 0 ) < 3
for all d N. Proof. Let  be the m A-adic completio of A, with atural iclusio A Â. Let Q be the prime ideal of elemets of  of ifiite value, ad let ν be the valuatio of the quotiet field of Â/Q iduced by ν which domiates Â/Q, so that SÂ/Q (ν) = S A (ν), ad s 0 = mi{ν(f) f mâ/q }. Let k be a coefficiet field of Â/Q. There exists a power series rig R = k[[x 1,..., x ]], ad a surjective k-algebra homomorphism ϕ : R Â/Q. Let P be the kerel of ϕ, so that R/P = Â/Q. We may thus idetify ν with a rak 1 valuatio of the quotiet field of R/P domiatig R/P such that S R/P (ν) = SÂ/Q (ν), ad s 0 = mi{ν(f) f m R/P }. Let ν 1 be a valuatio of the quotiet field of R which domiates R P, ad let µ be the composite valuatio of ν 1 ad ν. For f R, we have that ν 1 (f) = 0 if ad oly if f P, ad S R (µ) = µ(m R P ) = ν(m R/P {0}) = S R/P (ν). We also have that s 0 = mi{µ(f) f m R }. R cotais the polyomial rig A = k[x 1,..., x ]. Suppose that f m R P. There exists a Cauchy sequece {g i } i A (for the m R -adic topology) that coverges to f. Sice ν has rak 1, ad ν 1 (f) = 0, there exists l N such that µ(h) > µ(f) wheever h m l R. Further, sice {g i } is a Cauchy sequece, there exists e N such that f g i m l R wheever i e. Thus µ(f) = µ(g e ). It follows that S (A)m (µ ) = S R (µ) = S A (ν) where µ is the restrictio of µ to A, ad s 0 = mi{µ (f) f m }. The coclusios of the theorem ow follow from Lemma 1.2. Whe ν has rak 1, S A (ν) = ν(m A {0}) is the semigroup of ν o A. Thus (1) gives us a ecessary coditio for a subsemigroup S of R + to be the semigroup of a valuatio o a equicharacteristic local domai R. Whe A is a regular local rig, the i the theorem is the dimesio of A. Thus we have the followig Corollary, which gives a ecessary coditio for a subsemigroup of R + to be the subsemigroup of a valutio o a equidimesioal regular local rig. Corollary 2.2. Suppose that A is a equicharacteristic regular local rig of dimesio, ad ν is a rak 1 valuatio which domiates A. Let s 0 = mi{ν(f) f m A }. The ( ) + d (2) ν(m A {0}) [0, (d + 1)s 0 ) < for all d N. The ecessary coditio (2) is far from givig a sufficiet coditio for a subsemigroup of R + to be the semigroup of a valuatio o a equidimesioal regular local rig A, eve i the case whe A has dimesio = 1. Suppose that ν is a valuatio domiatig a equidimesioal regular local rig A of dimesio = 1. ν must the be equivalet to the m A -adic valuatio of A, ad we thus have that ν(m A {0}) = s 0 N. Thus ν(m A {0}) [0, (d + 1)s 0 ) = d for all d 1, which is sigificatly less tha the boud ( ) 1+d 1 = d 2 + d of the corollary. The criterio of the theorem is strog eough to give us a simple method of costructig well ordered semigroups of ordial type ω which are ot value semigroups. The existece of such examples was ot previously kow. 4
Example 2.3. There exists a well ordered subsemigroup U of Q + such that U has ordial type ω ad U ν(m A {0}) for ay valuatio ν domiatig a equicharacteristic oetheria local domai A. Proof. Let U be the semigroup costructed i Example 1.3. By Theorem 2.1 ad the proof of Example 1.3, U caot be a semigroup of a valuatio domiatig a equicharacteristic local domai. Refereces [1] Abhyakar, S., O the valuatios cetered i a local domai, Amer. J. Math. 78 (1956), 321-348. [2] S.D. Cutkosky, Local factorizatio ad moomializatio of morphisms, Astérisque 260, 1999. [3] S.D. Cutkosky ad L. Ghezzi, Completios of valuatio rigs, Cotemp. math. 386 (2005), 13-34. [4] S.D Cutkosky ad B. Teissier, Semigroups of valuatios o local rigs, to appear i Mich. Math. J. [5] C. Favre ad M. Josso The valuative tree, Lecture Notes i Mathematics 1853, Spriger Verlag, Berli, Heidelberg, New York, 2004. [6] W. Heizer, W. ad J. Sally, Extesios of valuatios to the completio of a local domai, Joural of Pure ad Applied Algebra 71 (1991), 175-185. [7] F.-V. Kuhlma, Value groups, residue fields, ad bad places of algebraic fuctio fields, Tras. Amer. Math. Soc. 356 (2004), 4559-4600. [8] S. MacLae, A costructio for absolute values i polyomial rigs, Tras. Amer. Math. Soc. 40 (1936), 363-395. [9] S. MacLae ad O, Schillig, Zero-dimesioal braces of rak 1 o algebraic varieties, Aals of Math. 40 (1939), 507-520. [10] M. Spivakovsky, Valuatios i fuctio fields of surfaces, Amer. J. Math. 112 (1990), 107-156. [11] B. Teissier, Valuatios, deformatios, ad toric geometry, Proceedigs of the Saskatoo Coferece ad Workshop o valuatio theory (secod volume), F-V. Kuhlma, S. Kuhlma, M. Marshall, editors, Fields Istitute Commuicatios, 33, 2003, 361-459. [12] O. Zariski ad P. Samuel, Commutative Algebra Volume II, D. Va Nostrad, Priceto, 1960. 5