RAMIFICATION OF VALUATIONS

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1 RAMIFICATION OF VALUATIONS STEVEN DALE CUTKOSKY AND OLIVIER PILTANT 1. Introducton The ramfcaton theory of valuatons was developed classcally n fnte extensons of rngs of algebrac ntegers, and for mappngs of algebrac curves. In these cases, the correspondng homomorphsms of local rngs of ponts are ramfed maps R S of dscrete (rank 1) valuaton rngs, R = V and S = V. These valuaton rngs are local Dedeknd domans. Suppose that (y) s the maxmal deal of R, (x) s the maxmal deal of S. We have an expresson y = x e δ (1) where δ S s a unt. If the value groups are Γ = Z and Γ = Z we have a natural somorphsm Γ /Γ = Z e. Ths theme s developed nto the beautful theory of ramfcaton of dscrete (rank 1) valuaton rngs. If R contans a feld k, we observe that ˆR Ŝ s the fnte extenson R/m R[[y]] S/m S [[x]]. Suppose that k has characterstc zero, and let k be an algebrac closure of S/m S. Then we have an acton of the fnte Abelan group Hom(Γ /Γ, (k ) ) = Γ /Γ on Ŝ S/m S k, and the nvarant rng s (Ŝ S/mS k ) Γ /Γ = ˆR R/mR k (2) The theory of valuaton rngs n arbtrary felds, and the ramfcaton theory of valuatons was ntated by Krull. Suppose that R s a local doman. A monodal transform R R 1 s a bratonal extenson of local domans such that R 1 = R[ P x ] m where P s a regular prme deal of R, 0 x P and m s a prme deal of R[ P x ] such that m R = m R. If P = m R, R R 1 s called a quadratc transform. If R s regular, and R R 1 s a monodal transform, then there exsts a regular system of parameters (x 1,..., x n ) of R and r n such that R 1 = R [ x2 x 1,..., x r x 1 Suppose that ν s a valuaton of the quotent feld R whch domnates R. Then R R 1 s a monodal transform along ν f ν domnates R 1. Suppose that K s an algebrac functon feld. If K has dmenson 2, then K has non Noetheran valuaton rngs. However, whenever there s a suffcently strong theory of resoluton of sngulartes, the valuaton rngs of K can be wrtten as unons of algebrac regular local rngs wth quotent feld K. When k has characterstc zero, ths follows from Zarsk s Theorem on local unformzaton along a valuaton [21]. A stronger verson of ths theorem s proven n Theorem 6.2. ]. m The frst author was partally supported by NSF. The second author was supported by CNRS and Ecole Polytechnque. 1

2 2 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT In Chapters 2 to 6, we assume K has characterstc zero. Suppose that K s a fnte extenson of K, V s a valuaton rng of K. We show n Theorem 6.1 and Theorem 6.3 that we can express V V as V = R, V = S so that R and S are algebrac normal local rngs wth quotent felds K and K respectvely, S s regular and obtaned from a fxed S 0 by a product of monodal transforms along ν, R has torc sngulartes, and S les above R. If k s an algebrac closure of V /m V, then there s an acton of the fnte Abelan group Γ /Γ on Ŝ S/m S k such that (Ŝ S/m S k ) Γ /Γ = ˆR R/m R k and these actons are compatble wth ncluson of the S. We thus obtan the strongest possble generalzaton of the classcal theory of equatons (1) and (2) to algebrac functon felds of arbtrary dmenson and characterstc zero. Such a statement was unantcpated by prevous work. It can be vewed as a relatve local unformzaton theorem. We also nterpret the nvarants of ramfcaton of valuatons and of the Galos theory of ramfcaton of valuatons, to show that they also generalze from the classcal case of local Dedeknd domans n the best possble way (Theorem 5.2 and Remark 6.4). The frst author s proof of the Weak smultaneous resoluton conjecture s the man step n ths constructon. Abhyankar s Weak smultaneous resoluton local conjecture (page 144 [6]), asserts that f we start wth an algebrac regular local rng S wth quotent feld K whch s domnated by V, then there exsts a sequence of monodal transforms S S along V (blowups of regular prmes, localzed at the center of V ) such that there exsts an algebrac normal local rng R wth quotent feld K such that S les above R. Abhyankar has proven ths theorem for two dmensonal functon felds n all characterstcs. It s a key step n hs proof of resoluton of sngulartes of surface sngulartes n char 0. We have proven n [11] that the Weak smultaneous resoluton local conjecture s true n functon felds of arbtrary dmenson and characterstc 0. We prove a stronger verson of the Weak smultaneous resoluton local conjecture n Theorem 4.2. Ths theorem s a corollary of the local monomalzaton theorem of [10]. The subtlety of the conjecture can be understood by the fact that the Global weak smultaneous local conjecture (page 144 [6]) s false, even n characterstc zero [12]. Suppose that k s a feld of characterstc zero, S s an algebrac regular local rng wth quotent feld K whch s domnated by V and R s an algebrac regular local rng wth quotent feld K whch s domnated by S. The local monomalzaton theorem [10] proves that there then exst sequences of monodal transforms R R 0 and S S such that V domnates S, S domnates R 0 and there are regular parameters (x 1,..., x n ) n R 0, (y 1,..., y n ) n S, unts δ 1,..., δ n S and a matrx A = (a j ) of nonnegatve ntegers such that det(a) 0 and x 1 x n = y a11 1 y a1n n δ 1. = y an1 1 yn ann δ n. The dffculty n obtanng ths result s to acheve the condton det(a) 0. A refnement of ths Theorem s possble, gvng a descrpton of A dependng on nvarants of V, whch we call Strong Monomalzaton. Ths s proven n Theorem 4.8. A further refnement s obtaned n Theorem 4.9. Theorem 4.9 s necessary to

3 RAMIFICATION OF VALUATIONS 3 prove Theorems 6.1 and 6.3. Several hard facts make t dffcult to extend the classcal theory of equatons (1) and (2) to algebrac functon felds of postve characterstc. Frst, the local unformzaton theorem has been proven sofar n characterstc p > 0 only when K has dmenson two ([1]), and when K has dmenson three and p 2, 3, 5 ([5]). Also, equaton (2) mples that the nduced ncluson of functon felds ( QF ˆR R/mR k ) QF (Ŝ S/mS k ) s cyclc Galos, whereas t need not even be Galos n postve characterstc (examples and [8]). Moreover, even n the Galos case, the local nerta group of (2) s not n general Abelan (example [8]). In partcular, t s not n general somorphc to Γ /Γ. In Chapter 7 we study the ramfcaton n surfaces over a feld of postve characterstc. Most of ths chapter s devoted to gettng a rght formulaton of (1) n dmenson two. Suppose that K /K s a fnte, separable extenson of two dmensonal algebrac functon felds, over an algebracally closed feld k of characterstc p > 0. Suppose that V s a valuaton rng of K and V = V K. Let Γ be the value group of V, Γ be the value group of V. We further consder a bratonal extenson of algebrac regular local rngs R S where R has quotent feld K, S has quotent feld K, and V domnates R and S. In Theorem 7.3, we prove that Strong Monomalzaton holds whenever Γ s fntely generated. Snce K s a two dmensonal algebrac functon feld, ths ncludes all valuatons of K except those whch are nondscrete and ratonal. We now restrct to the case where Γ s nondscrete and ratonal. Smultaneous Resoluton s the statement that there exsts a commutatve dagram of algebrac regular local rngs R S R S such that the vertcal arrows are products of monodal transforms along V, and R S s the localzaton of a fnte map. Provng Smultaneous Resoluton s extremely useful for applcatons to local unformzaton snce t mples that local unformzaton goes up n a feld extenson ([1]). In the case where k had characterstc zero, Smultaneous Resoluton held along V. Ths s proven for ratonal valuatons n algebrac functon felds of dmenson 2 and characterstc zero by Abhyankar (Theorem 2 [3]) and follows for ratonal valuatons n algebrac functon felds of arbtrary dmenson and characterstc 0 from Strong Monomalzaton n characterstc 0 (Theorem 4.8). An example of Abhyankar (Theorem 12 [3]) shows that Smultaneous Resoluton s n general false for valuatons of ratonal rank larger than 1. A drect consequence of (2) of Theorem 7.33 s that Smultaneous Resoluton s true n dmenson two and characterstc p > 0 whenever the nondscrete ratonal group Γ s not p-dvsble. One essental new nvarant to be consdered n characterstc p > 0 s the defect of V over V whch s a power of p (cf. Defnton 7.1). V /V s defectless f k has characterstc zero. We prove that V /V s defectless (n characterstc p > 0) f Γ s fntely generated ((3) of Theorem 7.3, see also [15]). In Theorem 7.33 we obtan stable forms for mappngs R S where S s a product of quadratc transforms along V and R R s the maxmal factorzaton by quadratc transforms of R S. The ramfcaton ndex and defect of V over V can be computed from the equatons

4 4 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT defnng these mappngs. In Theorem 7.35 we prove that Strong Monomalzaton holds whenever V /V s defectless. In Theorem 7.38 we gve an example of an extenson of two dmensonal algebrac functon felds wth valuatons V /V such that V /V has postve defect, and Strong Monomalzaton does not hold. 2. notatons We wll denote the maxmal deal of a local rng R by m R. We wll denote the quotent feld of a doman R by QF (R). Suppose that R S s an ncluson of local rngs. We wll say that R domnates S f m S R = m R. Suppose that K s an algebrac functon feld over a feld k. We wll say that a subrng R of K s algebrac f R s essentally of fnte type over k. Suppose that K s a fnte extenson of an algebrac functon feld K, R s a local rng wth QF (K) and S s a local rng wth QF (K ). We wll say that S les over R and R les below S f S s a localzaton at a maxmal deal of the ntegral closure of R n K. If R s a local rng, ˆR wll denote the completon of R at ts maxmal deal. If M s a fnte feld extenson of a feld L, we wll denote the group of L-automorphsms of M by Gal(M/L). Good ntroductons to the valuaton theory whch we requre n ths paper can be found n Chapter VI of [22] and n [4]. A valuaton ν of K wll be called a k-valuaton f ν(k) = 0. We wll denote by V ν the assocated valuaton rng, whch necessarly contans k. A valuaton rng V of K wll be called a k-valuaton rng f k V. The resdue feld V/m V of a valuaton rng V wll be denoted by k(ν). The value group of a valuaton ν wll be denoted by Γ ν. If X s an ntegral k-scheme wth functon feld K, then a pont p X s called a center of the valuaton ν (or the valuaton rng V ν ) f V ν domnates O X,p. If R s a subrng of V ν then the center of ν (the center of V ν ) on R s the prme deal R m Vν. Suppose that R s a local doman. A monodal transform R R 1 s a bratonal extenson of local domans such that R 1 = R[ P x ] m where P s a regular prme deal of R, 0 x P and m s a prme deal of R[ P x ] such that m R = m R. R R 1 s called a quadratc transform f P = m R. If R s regular, and R R 1 s a monodal transform, then there exsts a regular system of parameters (x 1,..., x n ) n R and r n such that R 1 = R [ x2 x 1,..., x r x 1 Suppose that ν s a valuaton of the quotent feld R wth valuaton rng V ν whch domnates R. Then R R 1 s a monodal transform along ν (along V ν ) f ν domnates R 1. ]. m 3. Conventons on valuatons We recall some classcal nvarants of valuatons (Chapter VI, [22], [4]), and establsh some notatons whch we wll follow. Suppose that K s a feld of algebrac functons over a feld k, and ν s a k-valuaton of K wth valuaton rng V and value group Γ. The prmes of V are a fnte chan 0 = p 0 p r = m V V. The rank of V s the length r of ths chan. r trdeg k K < by Corollary, page 50, Secton 11, Chapter VI, [22]. The solated subgroups of the value group Γ of V are 0 = r 0 = Γ.

5 RAMIFICATION OF VALUATIONS 5 The are defned as follows. Set U = {ν(a) a p }. s the complement of U and U n Γ. For < j, (V/p ) pj s a rank j valuaton rng wth value group / j and wth quotent feld (V/p ) p. V s sad to be composte wth the valuatons (V/p ) pj. Set λ = trdeg k (V/p ) p for 0 r. The ratonal rank of (V/p 1 ) p s s = ratrank(v/p 1 ) p := dm Q ( 1 / ) Q for 1 r. s and λ are < by Theorem 1 [2] or by Proposton 2, Appendx 2 [22]. Now suppose that K s a fnte extenson of K, and ν s an extenson of ν to K. Let V be the valuaton rng of ν, and let Γ be the value group. The prmes of V are a fnte chan 0 = p 0 p r V wth p V = p, 0 r, and wth solated subgroups 0 = r 0 = Γ whch have the property that Γ = for 0 r and / s a fnte (Abelan) group for 0 r (Secton 11, Chapter VI [22]). We further have that for 0 r and trdeg k (V /p ) p = trdeg k (V/p ) p = λ ratrank(v /p 1) p = ratrank(v/p 1 ) p = s for 1 r. Set t = λ 1 λ for 1 r. The ramfcaton ndex of ν relatve to ν s defned to be (page 53, Secton 11, Chapter VI, [22]) e = [Γ : Γ]. (3) The resdue degree of ν wth respect to ν s defned to be (page 53, Secton 11, Chapter VI) f = [V /m V : V/m V ]. (4) 4. Ramfcaton of valuatons n algebrac functon felds Theorem 4.1. (Local Monomalzaton)(Theorem 1.1 [10]) Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte algebrac extenson of K, ν a k-valuaton of K. Suppose that S s an algebrac regular local rng wth quotent feld K whch s domnated by ν and R s an algebrac regular local rng wth quotent feld K whch s domnated by S. Then there exst sequences of monodal transforms R R 0 and S S such that ν domnates S, S domnates R 0 and there are regular parameters (x 1,..., x n ) n R 0, (y 1,..., y n ) n S, unts δ 1,..., δ n S and a matrx A = (a j ) of nonnegatve ntegers such that det(a) 0 and x 1 = y a11 1 y a1n n δ 1. x n = y an1 1 yn ann δ n. The standard theorems on resoluton of sngulartes allow one to easly fnd R 0 and S such that (5) holds, but, n general, we wll not have the essental condton det(a j ) 0. The dffculty n the proof of ths Theorem s to acheve the condton det(a j ) 0. (5)

6 6 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT Let α be the mages of δ n S/m S for 1 n. Let C = (a j ) 1, a matrx wth ratonal coeffcents. Defne regular parameters (y 1,..., y n ) n ( ) Ŝ by c1 ( ) cn δ1 δn y = y α 1 for 1 n. We thus have relatons wth α S/m S for 1 n n α n x = α y a1 1 y an n (6) ˆR 0 = R 0 /m R0 [[x 1,..., x n ]] Ŝ = S/m S[[y 1,..., y n ]]. Theorem 4.2. Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte algebrac extenson of K, ν a k-valuaton of K. Suppose that S s an algebrac local rng wth quotent feld K whch s domnated by ν and R s an algebrac local rng wth quotent feld K whch s domnated by S. Then there exsts a commutatve dagram (7) R 0 R S V ν R S where S S and R R 0 are sequences of monodal transforms along ν such that R 0 S have regular parameters of the form of the conclusons of Theorem 4.1, R s an algebrac normal local rng wth torc sngulartes whch s the localzaton of the blowup of an deal n R 0, and the regular local rng S s the localzaton at a maxmal deal of the ntegral closure of R n K. Proof. By resoluton of sngulartes [14] (c.f. Theorem 2.6, Theorem 2.9 [10]), we frst reduce to the case where R and S are regular, and then construct, by the local monomalzaton theorem, Theorem 4.1 a sequence of monodal transforms along ν (8) R 0 S V ν R S so that R 0 s a regular local rng wth regular parameters (x 1,..., x n ), S s a regular local rng wth regular parameters (y 1,..., y n ), there are unts δ 1,..., δ n n S, and a matrx of natural numbers A = (a j ) wth nonzero determnant d such that x = δ y a1 1 yn an for 1 n. After possbly rendexng the y we may assume that d > 0. Let (b j ) be the adjont matrx of A. Set n n f = x bj j = ( δ bj j )y d j=1 for 1 n. Let R be the ntegral closure of R 0 [f 1,..., f n ] n K, localzed at the center of ν. Snce m R S = m S, Zarsk s Man Theorem (10.9 [5]) shows that R s an algebrac normal local rng wth quotent feld K such that S les above R. We thus have a sequence of the form (7). As an mmedate consequence, we obtan a proof n characterstc zero of the weak smultaneous resoluton local conjecture. whch s stated explctly on page 144 of [6], and s mplct n [3]. Abhyankar proves ths for algebrac functon felds of dmenson two and any characterstc n [1] and [4]. In the paper [11], we have gven a drect proof of ths result, also as a consequence of Theorem 4.1. j=1

7 RAMIFICATION OF VALUATIONS 7 Corollary 4.3. (Theorem 1.1 [11]) Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte algebrac extenson of K, ν a k-valuaton of K, and S an algebrac regular local rng wth quotent feld K whch s domnated by ν. Then for some sequence of monodal transforms S S along ν, there exsts a normal algebrac local rng R wth quotent feld K, such that the regular local rng S s the localzaton at a maxmal deal of the ntegral closure of R n K. Proof. There exsts a normal algebrac local rng R wth quotent feld K such that ν domnates R (take R to be the local rng of the center of ν on a normal projectve model of K). There exsts a fnte type k-algebra T such that the ntegral closure of R n K s a localzaton of T, and T s generated over k by g 1,..., g m V such that ν (g ) 0 for all. There exsts a sequence of monodal transforms S S 1 along ν such that T S 1 (Theorem 2.7 [10]). S 1 domnates R. After replacng S wth S 1, we can assume that S domnates R. Theorem 4.2 apples to ths stuaton, so we can construct a dagram of the form (7). The sequence of monodal transforms S S s necessary n Theorem 4.2 and Corollary 4.3 as can be seen by the followng smple example whch was communcated to us by Wllam Henzer. Let x, y be algebracally ndependent over a feld k, and let S = k[x 3, x 2 y] (x3,x 2 y). Consder the automorphsm of K = k(x, y) over k that maps x to y and y to x. The mage of S s the 2 dmensonal regular local rng S where S = k[y 3, y 2 x] localzed at (y 3, y 2 x). Regardng S and S as subrngs of the formal power seres k[[x, y]], we see that the ntersecton of S and S s k. Hence f K s the fxed feld of the above automorphsm, so that K = k(x + y, xy), we have S K = S K = k. When K s Galos over K, t s not dffcult to construct usng Galos theory and resoluton of sngulartes a regular local rng S wth quotent feld K and a normal local rng R wth quotent feld K such that S les over R (Theorem 7 [3], Theorem 6.2), although even n the Galos case the full statements of Theorem 4.2 and Corollary 4.3 do not follow from these results (Theorem 7 [3], Theorem 6.2). The general case of non Galos extensons s much more subtle, and not as well behaved, as can be seen from Theorem 3.1 of [12]. Ths Theorem shows that a genercally fnte morphsm of projectve surfaces cannot n general be bratonally modfed to produce a proper morphsm from a nonsngular surface to a normal surface. Theorem 3.1 [12] s thus a counterexample to Abhyankar s weak smultaneous resoluton global conjecture, whch s stated explctely on page 144 of [6] and s mplct n [3]. The followng lemma prepares the proof of our hgher dmensonal verson of (2) n the ntroducton. Lemma 4.4. Suppose that k 1 k 2 s a fnte extenson of felds of characterstc zero, A = (a j ) s an n n matrx of natural numbers wth det(a) 0 and L = k 1 (x 1,..., x n ) L 1 = k 2 (x 1,..., x n ) L = k 2 (y 1,..., y n ) are nclusons of ratonal functon felds, gven by wth α k 2 for 1 n. Then x = α y a1 1 y an n (1) [L : L] = det(a) [k 2 : k 1 ]. (2) L s Galos over L f and only f the followng condtons hold: k 2 s Galos over k 1 and there s a prmtve e th root of unty n k 2 where e = lcm{ord(b) b Z n /AZ n }.

8 8 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT (3) If L s Galos over L, then there s a natural exact sequence 0 Gal(L /L 1 ) = Z n /AZ n Gal(L /L) Gal(k 2 /k 1 ) 0. (9) Proof. Let 0 d = det(a). Set x = x α L 1 for 1 n. If v = (v 1,..., v n ) Z n, we wll wrte y v = y v1 1 yvn n. Let e 1,..., e d Z n be representatves of dstnct cosets of Z n /A t Z n. We wll show that {y e1,..., y e d } s a bass of L over L 1. Suppose that there s a relaton m f (x)y e = 0 wth f (x) L 1. k 2 [x 1,..., x n ], =1 After clearng denomnators, we may assume that each f (x) f (x) = I α,i x I wth α,i k 2. 0 =,I α,i y At I+e A t I + e = A t J + e j mples e = e j, and thus I = J so α,i y e1,..., y e d are lnearly ndependent over L. Set B = (b j ) = da 1 = ±adj(a). = 0 for all, I, and for 1 n so that the monomals y d = x b1 1 x bn n L 1 y 1 1 yn n 0 j d 1 generate L over L 1 whch thus mples y e1,..., y e d generate L over L 1. Thus [L : L 1 ] = Z n /A t Z n = d and 1. of the Lemma follows. Now suppose that k 1 = k 2 contans a prmtve d th root of unty ω. Then x L, and y d L for 1 n. σ Gal(L /L) mples σ(y d ) = y d for 1 n so that σ(y ) = ω c y for some c Z. Gven c = (c 1,..., c n ) Z n, defne a k 2 algebra automorphsm σ c : L L by σ c (y ) = ω c y. σ c s an L automorphsm f and only f Ac dz n. Thus Defne a group homomorphsm Gal(L /L) = {c Z n Ac dz n }/dz n. Ψ : Z n /AZ n Gal(L /L) by Ψ(c) = Bc, where B = da 1 = ±adj(a). Ψ s well defned and an somorphsm. Thus Gal(L /L) = d. By 1. of ths Lemma, L s Galos over L. Now consder the general case, wth no restrctons on k 1 and k 2. Let ω be a prmtve d th root of unty n some extenson feld of k 2. Set k = k 2 (ω). Set L 0 = k (x 1,..., x n ), L = k (y 1,..., y n ). By the argument for k 1 = k 2, we know that L /L 0 s Galos wth Gal(L /L 0 ) = Z n /AZ n. Suppose that L /L s Galos. Then there s a natural homomorphsm Gal(L /L 0 ) Gal(L /L)

9 RAMIFICATION OF VALUATIONS 9 whch s an ncluson snce L s the jon of L 0 and L. If σ Gal(L /L 0 ) then for 1 n, σ(y ) y = ω λ k 2 for some λ Z. σ e = Id and ω λe = 1 and d e λ for 1 n. There exsts σ Gal(L /L 0 ) of order e. If σ(y) y = ω λ for 1 n, we then have gcd(λ 1,..., λ n ) = d e and thus ω d e k 2 whch s a prmtve e th root of unty. L /L Galos mples the fxed feld of the mage of Gal(L /L) n Gal(k 2 /k 1 ) by the natural morphsm s k 1. Thus k 2 /k 1 s Galos and the dagram (9) s short exact. Now suppose that k 2 /k 1 s Galos and k 2 contans a prmtve e th root of unty. Then there s a natural ncluson Z n /AZ n = Gal(L /L 1 ) Gal(L /L). To show that L /L s Galos, s thus suffces to show that any σ Gal(k 2 /k 1 ) extends to an L automorphsm of L. σ extends to an L automorphsm of L 1 such that σ(x ) = x for all. Snce L s Galos over L 1, σ extends to an L automorphsm of L. Remark 4.5. Wth the assumptons of Lemma 4.4, assume that L s Galos over L. Then Gal(L /L) acts fathfully on T = k 2 [y 1,..., y n ] by k 1 [x 1,..., x n ] automorphsms, and we have natural nclusons of nvarant rngs k 1 [x 1,..., x n ] T Gal(L /L) T Zn /AZ n T. (10) Suppose that τ k 2 s a prmtve e th root of unty, d = det(a). To c Z n /AZ n the correspondng k 2 -algebra automorphsm σ c of T s defned by σ c (y ) = τ <B,c> e d y for 1 n, where B s the th row of da 1 = ±adj(a). Theorem 4.6. Suppose that R 0 R S V ν s a sequence of the form of (7) of Theorem 4.2. Let k be an algebrac closure of S/m S. Then ˆR R/mR k = ( Ŝ S/MS k ) Zn /AZ n by the fathful acton of Z n /AZ n on Ŝ S/m S k of Remark 4.5. If then ˆR S/mS k = k [[x e1,..., x er ]], R = R 0 [x e1,..., x er ] P, where P = (x e1,..., x er ). In partcular, R has normal torc sngulartes. Proof. Set k 1 = R 0 /m R0, k 2 = S/m S. From (6) we see that there are regular parameters y 1,..., y n n Ŝ and α k 2 such that x = α y a1 1 y an n for 1 n. Let d = det(a) > 0. Set F = Z n /AZ n. Set x = x α, 1 n. By Lemma 4.4, k (y 1,..., y n ) s Galos over k (x 1,..., x n ) wth Galos group F. By Remark 4.5, we have an expanson k [y 1,..., y n ] F = k [y c1,..., y cr ] where c N r and y c = x e wth e Z n, and these nvarants nclude x 1,..., x n. k [y c1,..., y cr ] = k [x e1,..., x er ]

10 10 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT s normal, and thus k 1 [x e1,..., x er ] s normal. We have x e = ɛ y c n K where ɛ S are unts. R 0 [x e1,..., x er ] has a maxmal deal m = (x e1,..., x er ). Set R 1 = R 0 [x e1,..., x er ] m. R 1 /m R1 = R 0 /m R0. We have R 0 R 1 S. ˆR1 = k 1 [[x e1,..., x er ]] s normal (Theorem 32, Secton 13, Chapter VIII [22]), so R 1 s normal snce ˆR 1 K = R 1 (by Lemma 2, [1]). Snce m R1 S = m S, R 1 les below S by Zarsk s Man Theorem (10.9 [5]). Thus R 1 = S K = R by Proposton 1 (v) [1]. Theorem 4.7. Suppose that assumptons are as n Theorem 4.2. Let k be an algebrac closure of V /m V. Then there exsts a sequence R 0 R S V ν of the form of (7) of Theorem 4.2, wth the followng property. Suppose that there s a commutatve dagram R 0 (1) R(1) S(1) V ν R 0 R S such that the top row s also a sequence of the form of (7) of Theorem 4.2, so that there are regular parameters (x 1 (1),..., x n (1)) n R 0 (1), (y 1 (1),..., y n (1)) n S(1), unts δ (1) S(1) and a matrx A(1) of natural numbers (wth nontrval determnant) such that x (1) = y 1 (1) a1(1) y n (1) an(1) δ (1) for 1 n. Then Z n /AZ n = Z n /A(1)Z n. Let L be the quotent feld of ˆR R/mR k, M be the quotent feld of Ŝ S/M S k, L 1 be the quotent feld of ˆR(1) R(1)/mR(1) k, M 1 be the quotent feld of Ŝ(1) S(1)/M S(1) k. Then there are natural restrcton maps of Galos groups whch are somorphsms. Proof. Let Gal(M 1 /L 1 ) Gal(M/L) (11) R 0 R S V ν be a sequence of the form of (7) of Theorem 4.2. Let R 0 (1) R(1) S(1) V ν R 0 R S be a commutatve dagram such that the top row s also a sequence of the form of (7) of Theorem 4.2. There exst deals I R and J S, f I, g J, maxmal deals m 1 n R[ I f ] and n 1 n S[ J g ] such that R(1) = R[ I f ] m 1, S(1) = S[ J g ] n 1. Let R = ˆR R/mR k, S = Ŝ S/M S k, R 1 = ˆR(1) R(1)/mR(1) k, S 1 = Ŝ(1) S(1)/M S(1) k. We have natural nclusons R[ I f ] R 1 and S[ J g ] S 1. Let m = R[ I f ] m R 1, n = S[ J g ] m S 1. Let R = R[ I f ] m, S = S[ J g ] n. We have a commutatve dagram of

11 RAMIFICATION OF VALUATIONS 11 nclusons of local rngs R 1 S 1 R S R S. By constructon, R 1 = ˆ R and S1 = ˆ S. Let L = QF(R), M = QF(S), L1 = QF(R 1 ), M 1 = QF(S 1 ). R and S are normal local rngs, snce R = R1 L and S = S 1 M, by Lemma 2 [1]. By Theorem 4.6, M s Galos over L and M 1 s Galos over L 1. There s a natural somorphsm (the notaton G s s defned n Secton 4) Gal(M 1 /L 1 ) = G s ( S/ R) by Lemma 7 [1]. (The proof of ths Lemma generalzes wthout dffculty to felds of the form of M and L). In partcular, there s a 1-1 restrcton homomorphsm Gal(M 1 /L 1 ) Gal(M/L). If ths map s not an somorphsm, we can replace R 0 S wth R 0 (1) S(1). After repeatng ths process a fnte number of tmes, we wll fnd an extenson R 0 S such that the conclusons of the Theorem hold. Theorem 4.8. (Strong Monomalzaton) Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte algebrac extenson of K, ν a k-valuaton of K. Suppose that S s an algebrac local rng wth quotent feld K whch s domnated by ν and R s an algebrac local rng wth quotent feld K whch s domnated by S. Let notaton be as n Secton 3 for V = V ν, V = V ν. Then there exsts a commutatve dagram (12) R 0 S V R S such that R R 0 and S S are sequences of monodal transforms such that V domnates S, S domnates R 0 and there are regular parameters (x 1,..., x n ) n R 0, (y 1,..., y n ) n S such that p R 0 = (x 1,..., x t1+ +t ) p S = (y 1,..., y t1+ +t )

12 12 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT for 1 r and there are relatons x 1 = y g11(1) 1 y g1s 1 (1) s 1 y h1,t 1 +1(1) t 1+1. x s1 = y gs 1 1(1) 1 y gs 1 s 1 (1) x s1+1 = y s1+1y hs 1 +1,t 1 +1(1) t 1+1. x t1 = y t1 y ht 1,t 1 +1(1) t 1+1 x t1+1 = y g11(2) t 1+1 yg1s 2 (2). x t1+s 2 = y gs 2 1(2) t 1+1 y gs 2 s 2 (2) s 1 y hs 1,t 1 +1(1) t 1+1 y hs 1 +1,m(1) m y ht 1 n n(1) δ 1t1 t 1+s 2 y h1,t 1 +t 2 +1(2) t 1+t 2+1 t 1+s 2 y hs 2,t 1 +t 2 +1(2) t 1+t 2+1 x t1+s 2+1 = y t1+s 2+1y hs 2 +1,t 1 +t 2 +1(2) t 1+t 2+1. x t1+t 2 = y t1+t 2 y ht 2,t 1 +t 2 +1(2) t 1+t 2+1 x t1+ +t r 1+1 x t1+ +t r 1+s r x t1+ +t r 1+s r+1 x t1+ +t r y h1m(1) m δ 11 y hs 1 m m(1) δ 1s1 δ 1,s1+1 y h1m(2) m δ 21 w hs 2 +1,m(2) m y ht 2 m m(2) δ 2t2. = y g11(r) (r) t 1+ +t r 1+1 yg1sr t 1+ +t r 1+s r δ r1. gsr 1(r) sr (r) = yt 1+ +t r 1+1 ygsr t 1+ +t r 1+s r δ rsr = x t1+ +t r 1+s r+1δ r,sr+1. = y t1+ +t r δ rtr where m = t t r 1 + s r and for 1 r, g 11 () g 1s () det.. 0, g s1() g ss () y hs 2 n n(2) δ 2s2 δ 2,s2+1 δ j are unts n S, h jk () are natural numbers such that for 1 l k r 1, Let h,j (l) = 0 f 1 t l and t t k + s k+1 < j t t k+1. T = {j t t k < j t t k + s k+1 for some 0 k r 1}. Then {ν (y j ) j T } s a ratonal bass of Γ Q, {ν (x j ) j T } s a ratonal bass of Γ Q. Theorem 4.8 can be vsualzed as follows. For 1 r there are t t matrces ( ) (gjk ()) 0 M = 0 I t s correspondng to the composte valuaton rngs (V/p 1 ) p, such that M M A = M (13) M r

13 RAMIFICATION OF VALUATIONS 13 The smbols 0 n (13) denote t t j matrces whose last t j s j columns are dentcally zero. Proof. For valuatons V of rank 1 ths s mmedate from Theorem 5.1 [10]. Suppose that V has rank r > 1 and that the Theorem s true for valuatons of rank less than r. To reach the conclusons of the Theorem, we need only modfy the proof of Theorem 5.3 [10] by observng that we can assume by nducton that the upper λ λ matrx of exponents of (131) n the proof of Theorem 5.1 [10] has the desred form, and notce that we actually have e j = 0 f j > λ + s r n (131). We can then construct a sequence of monodal transforms S(m ) S(m +1) along ν by choosng t > max{a j, g j (r)} and defnng { y (m yλ+1 (m ) = + 1) t y λ+sr (m + 1) t y (m + 1) 1 λ y (m + 1) λ + 1 n We then obtan the conclusons of Theorem 4.8. Theorem 4.9. Wth the assumptons of Theorem 4.8, further suppose that u j V, 1 j l, and v j V, 1 j m. Then there exsts a commutatve dagram R 0 S V R S such that the conclusons of Theorem 4.8 hold and (1) v j = x bj1 1 x bjn n δ j R 0 for 1 j m, where δ j R 0 s a unt, and b j = 0 f t t l + s l < t l+1 for some l. (2) u j = y dj1 1 yn djn ɛ j S for 1 j l, where ɛ j S s a unt, and d j = 0 f t t l + s l < t t l+1 for some l. Proof. Frst assume that ν has rank 1. As n the begnnng of the proof of Theorem 5.1 [10], frst construct a sequence of monodal transforms R R 1 along ν such that R 1 /m 1 V/m V s algebrac, and there are regular parameters (x 1 (1),..., x n (1)) n R 1 such that ν(x 1 (1)),..., ν(x s (1)) are a bass of Γ Q. For 1 m, wrte v = f g wth f, g R 1. By Theorem 4.8 [10] (wth S = R, l = n, m = n) appled to f = f or f = g n (60) and by (2) of Theorem 4.10 [10] (wth S = R, l = n) we can perform a sequence of monodal transforms along ν R 1 R 2 where R 2 has regular parameters (x 1 (2),..., x n (2)) such that ν(y 1 (2)),..., ν(y s (2)) are a bass of Γ Q and f = x 1 (2) c1 x s (2) cs α g = x 1 (2) d1 x s (2) ds β where α, β are unts n R 2. We remark that (A3) on page 83 [10] mples ν(m(u(t)) of (64) n Theorem 4.8 [10] s a constant whch does not depend on N for N N 0. Now by (25) of Lemma 4.2 [10], appled to the par f, g we can perform a further sequence of monodal transforms R 2 R 3 along ν to acheve that v R 3 for 1 m, and there exst regular parameters (x 1 (3),..., x n (3)) n R 3 such that ν(x 1 (3)),..., ν(x s (3)) are a bass of Γ Q, v = x 1 (3) d 1 x s (3) d s ɛ for 1 m where d 1,..., d s are natural numbers, ɛ are unts n R 3.

14 14 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT Let S S 2 be a sequence of monodal transforms along ν so that S 2 domnates R 3. As n the proof of Theorem 5.1 [10], we can perform a sequence of monodal transforms S 2 S 3 along ν so that u S 3 for all, S 2 has a regular system of parameters (y 1 (3),..., y n (3)) such that x = y 1 (3) c1 y s (3) cs φ 1 s, φ are unts n S 2, (y 1 (3),..., y s (3)) s a bass of Γ Q and det(c j ) 0. By Theorem 5.1 [10], we can perform a sequence of monodal transforms R 4 S 4 R 3 S 3 so that R 4 has regular parameters (x 1 (4),..., x n (4)), S 4 has regular parameters (y 1 (4),..., y n (4)) such that x 1 (4) = y 1 (4) e11 y s (4) e1s δ 1. x s (4) = y 1 (4) es1 y s (4) ess δ s x s+1 (4) = y s+1 (4). x n (4) = y n (4) where det(e j ) 0, δ 1,..., δ s are unts n S 4, (ν(x 1 (4)),..., ν(x s (4)) s a ratonal bass of Γ Q. We further have v = x 1 (4) d1 x s (4) ds ɛ 1 m, ɛ R 4 unts (snce the monodal transforms used n the proof of Theorem 5.1 [10] preserve ths form) and u S 4, 1 l. Now by (60) of Theorem 4.8 and (2) of Theorem 4.10 [10], wth l = n and m = n, appled to R 4 S 4 and f = u 1 u l, we acheve a commutatve dagram R S R 4 S 4 where the vertcal arrows are sequences of monodal transforms along ν such that the conclusons of the Theorem hold n R S. Now assume that V has rank r > 1. For the general case of rank r > 1 we must modfy the proof of Theorem 5.3 [10]. We assume (by nducton) that the Theorem s true for valuatons of rank < r. We frst construct (as n the proof of Theorem 5.3 [10]) sequences of monodal transforms R(1) S(1) R S along ν such that f p (1) = p R(1), q (1) = q S(1) then trdeg (R(1)/p(1)) p (1) (V /p ) p = 0 for 1 r and (usng the nducton assumpton) that the conclusons of the Theorem hold for R(1) pr 1(1) S(1) qr 1(1). Set λ = t t r 1.

15 RAMIFICATION OF VALUATIONS 15 As n the proof of Theorem 5.3 [10], we can construct sequences of monodal transforms R S R(1) S(1) along ν such that f q r 1 = p r 1 S, p r 1 = p r 1 R, then R(1) pr 1(1) = R p, S(1) qr 1(1) = S q, there exst regular parameters (x 1,..., x n) n R, (y r 1 r 1 1,..., y n) n S such that, as n (129) of the proof of Theorem 5.3 [10], x 1 = ψ 1 (y 1 ) g11(1) (y s 1 ) g1,s 1 (1) (y t 1+1) h1,t 1 +1(1) (y λ )h 1λ(1). x λ = ψ λ y λ wth ψ 1,, ψ λ S q r 1, and 1 j l wth γ j S q r 1 and u j = γ j (y 1 ) ej1 (y λ) e jλ (14) v j = γ j (x 1) ej1 (x λ) e jλ (15) 1 j m wth γ j R p r 1. We further have that the exponents of y 1,..., y λ (respectvely x 1,..., x λ ) n (14) (respectvely (15)) are of the form of the conclusons of the Theorem (by our nducton assumpton on the rank), or u j = γ j, v j = γ j. We now construct a sequence of monodal transforms along ν (as n the proof of Theorem 5.3 [10]). R(m ) S(m ) R S so that there exst regular parameters (y 1 (m ),..., y n (m )) n S(m ), (x 1 (m ),..., x n (m )) n R(m ) such that (as n (131) of page 121 of the proof of Theorem 5.3 [10]) x 1 (m ) = y 1 (m ) g11(1) y s1 (m ) g1s 1 (1) y t1+1(m ) h1,t 1 +1(1) y λ (m ) h 1λ(1) y λ+1 (m ) e 1,λ+1 y λ+sr (m ) e 1,λ+sr ψ1. x λ (m ) = y λ (m )y λ+1 (m ) e λ,λ+1 y λ+sr (m ) e λ,λ+sr ψλ x λ+1 (m ) = y λ+1 (m ) g11(r) y λ+sr (m ) gsr sr (r) δ λ+1 + f1 λ+1 y 1 (m ) + + f λ+1 λ y λ (m ). x n (m ) = y n (m )δ n + f n 1 y 1 (m ) + + f n λ y λ(m ) where δ are unts n S(m ), f j S(m ) for 1 λ, ψ = u y λ+1 (m ) a,λ+1 y λ+sr (m ) a,λ+sr + h 1 y 1 (m ) + + h λy λ (m ) where u are unts n S(m ), and we further have v j = α j x 1 (m ) ej1 x λ (m ) e jλ 1 j m, where e j = 0 f t t l + s l < t t l+1 for some l. u j = β j y 1 (m ) fj1 y λ (m ) f jλ 1 j l, where f j = 0 f t t l + s l < t t l+1 for some l,

16 16 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT α j = α j x λ+1 (m ) a j,λ+1 x λ+sr (m ) a j,λ+sr + h j 1x 1 (m ) + + h j λx λ (m ), (16) where α j are unts n R(m ), h j R(m ) and β j = β j y λ+1 (m )ãj,λ+1 y λ+sr (m )ãj,λ+sr + hj 1 y 1(m ) + + h j λ y 1(m ) + + h j λ y λ(m ) (17) where β j are unts n S(m ), h j S(m ). Set t 1 = max{a j } n (16). Defne a sequence of monodal transforms along V R(m ) R(m +1) where R(m +1) has regular parameters (x 1 (m +1),..., x n (m + 1)) defned by by { x (m xλ+1 (m ) = + 1) t1 x λ+sr (m + 1) t1 x (m + 1) 1 λ x (m + 1) λ + 1 n Set t 2 = max{ã j, g j (r), a j, t 1 }. Now perform a sequence of monodal transforms along ν, S(m ) S(m + 1) where S(m + 1) has regular parameters (y 1 (m + 1),..., y n (m + 1)) defned by { y (m yλ+1 (m ) = + 1) t2 y λ+sr (m + 1) t2 y (m + 1) 1 λ y (m + 1) λ + 1 n Then S(m + 1) domnates R(m + 1) and the conclusons of the Theorem hold n R(m + 1) S(m + 1). Theorem Suppose that assumptons are as n Theorem 4.2. Let k be an algebrac closure of V /m V. Then there exsts a sequence R 0 R S V ν of the form of (7) of Theorem 4.2, whch satsfes the conclusons of Theorem 4.7, and wth the followng property. Suppose that there s a commutatve dagram R 0 (1) R(1) S(1) V ν R 0 R S such that the top row s also a sequence of the form of (7) of Theorem 4.2, so that there are regular parameters (x 1 (1),..., x n (1)) n R 0 (1), (y 1 (1),..., y n (1)) n S(1), unts δ (1) S(1) and a matrx A(1) of natural numbers (wth nontrval determnant) such that x (1) = y 1 (1) a1(1) y n (1) an(1) δ (1) for 1 n. Then there s an somorphsm of Abelan groups Z n /A(1)Z n = Γ /Γ. Γ /Γ acts fathfully on Ŝ(1) S(1)/m S(1) k by k -algebra automorphsms, and there s an somorphsm (Ŝ(1) S(1)/m S(1) k ) Γ /Γ = ˆR(1) R(1)/mR(1) k. Proof. There exst u 1,..., u l V such that Γ /Γ s generated by ν (u 1, ),..., ν (u l ). By Theorem 4.9 and Theorem 4.7, there exsts a sequence R 0 R S of the form of (7) of Theorem 4.2 such that the conclusons of Theorem 4.7 hold, and there are

17 RAMIFICATION OF VALUATIONS 17 unts β S and natural numbers e j such that wth the notaton of Theorem 4.9 (and Theorem 4.8), u = β for 1 l. Observe that Z n /AZ n = Z n /A t Z n snce A and A t have the same nvarant factors. We wll prove that Z n /A t Z n = Γ /Γ. Then the conclusons of the Theorem wll follow from Theorem 4.7 and 4.6. We have a group homomorphsm defned by j T y ej j Ψ : Z n Γ /Γ (b 1,..., b n ) b 1 ν (y 1 ) + + b n ν (y n ). Ψ s onto snce {ν (u ) 1 l} generate Γ /Γ. By defnton of A, Ψ(A t Z n ) Γ. Let {e } be the standard bass of Z n. Suppose that Ψ( λ m e m ) = 0. Then λm ν (y m ) = ν(f) for some f K. By Theorem 4.9, there exsts a dagram such that f R 0 (1) satsfes b l natural numbers, δ R 0 (1) a unt, and R 0 (1) S(1) R 0 S f = l T x l (1) b l δ, y = l T y l (1) c l ɛ for 1 n, c l natural numbers, ɛ S(1) unts. ν ( y λm m ) = ν ( l T y l(1) λmc ml ) = ν (f) = ν ( x m (1) bm ) = ν ( m ( l T y l(1) a ml(1) ) bm ) mples λ m c ml = a ml b m m m for all l T snce {ν (y l (1)) l T } are lnearly ndependent. Thus y λ m m = f δ where δ S(1) s a unt. Set R(0) = R, S(0) = S. For j = 0, 1, let K(j) = QF( ˆR(j) R(j)/mR(j) k ), L(j) = QF(Ŝ(j) S(j)/m S(j) k ). Wth notatons as n (6), set and where R(j) := ˆR(j) R(j)/mR(j) k = k [[x 1 (j),..., x n (j)]] S(j) := Ŝ(j) S(j)/m S(j) k = k [[y 1 (j),..., y n (j)]] x (j) = y 1 (j) a1(j) y n (j) an(j)

18 18 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT for 1 n. Let A T (j) be the T T submatrx of A(j) whch s the matrx of exponents of x (j) = for λ k = t 1 + t k, λ k < λ k + s k+1. We have a commutatve dagram l T,l λ k y l (j) al(j) K(1) L(1) K(0) L(0) where the horzontal arrows are fnte Galos, wth respectve Galos groups and G 1 = Z T /A T (1)Z T G = Z T /A T (0)Z T. We have ym (0) λm = fδ where δ S(1) s a unt. Suppose that σ G = G 1. Then σ( y m (0) λm ) ym (0) λm =. σ(δ) δ We can wrte δ = c + h wth c k, h m S(1). Thus σ(δ) δ mod m S(1). σ( y m (0) λm ) = ω y m (0) λm for some d th root of unty ω k, wth d = G. Thus and mples y m (0) λm σ(δ) = 1 δ σ( y m (0) λm ) = y m (0) λm K(0), and ym (0) λm = x m (0) dm for some d m Z. Thus λ m e m A t Z n. In partcular, Z n /A t Z n = Γ /Γ. 5. Ramfcaton n Galos extensons Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte Galos extenson of K wth Galos group G = Gal(K /K), ν a k-valuaton of K, wth value group Γ. Let ν be the restrcton of ν to K, and let Γ be the value group of ν. Let V be the valuaton rng of ν and V be the valuaton rng of ν. Suppose that R s a normal local rng wth quotent feld K and S s a normal local rng wth quotent feld K whch les above R. We can then defne the splttng groups and nerta groups G s (S/R) = {g G g(s) = S}, G (S/R) = {g G s (S/R) g(u) u mod m S for all u S}.

19 RAMIFICATION OF VALUATIONS 19 G (S/R) G s (S/R) G. G (S/R) s a normal subgroup of G s (S/R) (Theorem 1.48 [4]). The splttng feld K s of S over R s the fxed feld of G s (S/R). The nerta feld K of S over R s the fxed feld of G (S/R). We have a correspondng sequence of felds K K s K K. K s s the smallest subfeld of K such that S s the only local rng lyng above S K s (Proposton 1.46 [4]). We have a sequence R R s R S (18) where R s = S K s s the localzaton of the ntegral closure of R n K s at the center of ν. R = S K s the localzaton of the ntegral closure of R n K at the center of ν. R R s s unramfed, wth R/m R = R s /m R s, R s R s unramfed, R /m R s Galos over R s /m R s wth Galos group G s (S/R)/G (S/R) (by Theorem 1.48 [4]). We wll wrte G s (ν /ν) = G s (V /V ), G (ν /ν) = G (V /V ). Snce we are n characterstc zero, by Theorem 3 [16] or Corollary, Secton 12, Chapter VI [22], there s an somorphsm G (ν /ν) = Γ /Γ. (19) Lemma 5.1. Let assumptons be as n Theorem 4.2. Suppose that K s Galos over K. Let t 1,..., t f k(ν ) be a k(ν) bass. Then there exsts an algebrac regular local rng R wth quotent feld K whch s domnated by ν such that f R s an algebrac normal local rng domnated by ν, R R, S s the localzaton of the ntegral closure of R n K at the center of ν, then R/m R k(ν) s algebrac, G s (S/R) = G s (ν /ν) and G (S/R) = G (ν /ν). Further, [S/m S : R/m R ] = f, and {t 1,..., t f } s a bass of S/m S over R/m R, where f = [k(ν ) : k(ν)] s the resdue degree of ν wth respect to ν. Proof. Let V = V 1, V 2,..., V n be the dstnct valuaton rngs of K lyng over V. Then T = n =1 V s the ntegral closure of V n K (by Propostons 2.36 and 2.38 [4]). Let m = m V T be the maxmal deals of T. By the Chnese remander theorem, there exsts u T such that u m 1 and u m for = 2,..., n. Let u m + a 1 u m a m = 0 be the equaton of ntegral dependence of u over V. Let R 0 K be an algebrac regular local rng wth quotent feld K whch s domnated by ν. As a consequence of resoluton of sngulartes (cf. Theorem 2.7 [10]) there exsts a sequence of monodal transforms R 0 R 1 along ν such that a R 1 for 1 n. Let {w 1,..., w r } be a transcendence bass of k(ν) = V/m V over k. r < by Theorem 1 [7] or Appendx 2 [22]. Let w 1,..., w r be lfts of the w to V. w V mples there exsts a sequence of monodal transforms R 1 R 2 along ν such that w R 2 for all (Theorem 2.7 [10]). Suppose that R s an algebrac normal local rng wth quotent feld K such that R 2 R and R s domnated by ν. Let W be the ntegral closure of R n K. u W m V and u W m V for 2 n. Let S be the localzaton of W at the center of ν. g G s (S/R) mples u g(m V ) and consequently g(v ) = V, so that g G s (ν /ν). Thus G s (S/R) G s (ν /ν). By Proposton 1.50 [4], G s (S/R) = G s (ν /ν). R 2 R mples k(w 1,..., w r ) R/m R so that k(ν) s algebrac over R/m R. k(ν ) s fnte over k(ν) by Corollary 2, Secton 6, Chapter VI [22] (although k(ν) need not be fnte over k(w 1,..., w r )). Let t 1,..., t f k(ν ) be a k(ν) bass. Let t

20 20 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT be lfts of the t to T. Let t m + a 1 t m a m = 0, 1 f be equatons of ntegral dependence of t over V. There exsts a sequence of monodal transforms R 2 R along ν such that a j R for all, j. Now suppose that R s an algebrac normal local rng such that R R and R s domnated by ν. Let S be the localzaton at the center of ν of the ntegral closure of R n K. R 2 R mples G s (S/R) = G s (ν /ν). G (ν /ν) G (S/R) by Proposton 1.50 [4]. By Theorem 1.48 [4] S/m S s Galos over R/m R and k(ν ) s Galos over k(ν). We have an exact dagram 0 0 G (S/R) G (ν /ν) 0 G s (S/R) = G s (ν /ν) 0 Gal(S/m S /R/m R ) Gal(k(ν )/k(ν)) 0 0 By constructon, t S for all whch mples that t 1,..., t f S/m S. t 1,..., t f are necessarly lnearly ndependent over R/m R. Thus f [S/m S : R/m R ] = Gal(S/m S /R/m R ). f = Gal(k(ν )/k(ν)) = Gal(S/m S /R/m R ) [G (S/R) : G (ν /ν)] mples G (S/R) = G (ν /ν), and t 1,..., t f s a R/m R bass of S/m S. Theorem 5.2. Let assumptons be as n Theorem 4.2. Suppose that K s Galos over K. Let t 1,..., t f k(ν ) be a k(ν) bass. Suppose that R 0 R S s a sequence of the form of (7) of Theorem 1 such that the R of Lemma 5.1 s contaned n R 0. Set k 1 = R 0 /m R0, k 2 = S/m S. Then QF (Ŝ) s Galos over QF ( ˆR), wth Galos group G s (ν /ν), k 2 s Galos over k 1 wth Galos group G s (ν /ν)/g (ν /ν). Wth the notaton of (6) and Theorem 4.6, the completon of the sequence of (18) s ˆR 0 ˆR = ˆR s ˆR Ŝ ˆR 0 = k 1 [[x 1,..., x n ]] ˆR = (Ŝ)Gs (ν /ν) = k 1 [[x e1,..., x er ]] (Ŝ)G (ν /ν) = k 2 [[x e1,..., x er ]] Ŝ = k 2[[y 1,..., y n ]] wth x = α y a1 1 y an n α k 2 for 1 n. There s an exact sequence 0 G (ν /ν) = Z n /AZ n G s (ν /ν) = Gal(QF (Ŝ)/QF ( ˆR) Gal(k 2 /k 1 ) 0. f = [S/m S : R/m R ], e = det(a), [QF (Ŝ) : QF ( ˆR)] = ef and {t 1,..., t f } s a bass of k 2 = S/m S over k 1 = R/m R where e = [Γ /Γ] s the ramfcaton ndex of ν wth respect to ν, f = [k(ν ) : k(ν)] s the resdue degree of ν wth respect to ν..

21 RAMIFICATION OF VALUATIONS 21 Proof. QF (Ŝ) s Galos over QF ( ˆR) = QF ( ˆR s ) wth Galos group G s (S/R) by Lemma 7 a), b) [1] and QF (Ŝ) s Galos over QF ( ˆR ) wth Galos group G (S/R) by Proposton 1.49 [4] and Lemma 7 a), b) [1] appled to K K K and R R S. R/m R = R 0 /m R0 by Theorem 4.6. Wth the notatons of (6) and Theorem 4.6, ˆR0 = k 1 [[x 1,..., x n ]], ˆRs = k 1 [[x e1,..., x er ]], ˆR = k 2 [[x e1,..., x er ]], Ŝ = k 2 [[y 1,..., y n ]]. S/m S s Galos over R/m R = R s /m R s mples k 2 s Galos over k 1. Wth the notatons of Lemma 4.4, set L = k 1 (x 1,..., x n ) L 1 = k 2 (x 1,..., x n ) L = k 2 (y 1,..., y n ). We have that L /L s Galos wth Gal(L /L) = Hom ˆR(Ŝ, Ŝ) = Gs (S/R). We further have that L /L 1 s Galos wth Gal(L /L 1 ) = Hom ˆR(Ŝ, Ŝ) = G (S/R). By Lemma 5.1, G s (S/R) = G s (ν /ν), G (S/R) = G (ν /ν) and [S/m S : R/m R ] = f. By (19) and Lemma 4.4, e = det(a) and [QF (Ŝ) : QF ( ˆR)] = ef. Lemma 5.3. Let k be a feld of characterstc zero, K an algebrac functon feld over k, K a fnte algebrac extenson of K, ν a k-valuaton of K, ν = ν K. Suppose that S 0 s an algebrac normal local rng wth quotent feld K such that S 0 s domnated by ν. Then there exsts an algebrac regular local rng R wth quotent feld K whch s domnated by ν such that f R s an algebrac normal local rng wth quotent feld K, S s an algebrac regular local rng wth quotent feld K such that R contans R and S s domnated by V, R s domnated by S, then S contans S 0. Proof. There exsts a sequence of monodal transforms S 0 S 1 along ν such that S 1 domnates an algebrac regular local rng R 1 wth quotent feld K. Let U 1 be the ntegral closure of R 1 n K. There exsts f 1,..., f m K (wth ν (f ) 0 for all ) such that S 1 s a localzaton of U 1 [f 1,..., f m ]. Let T be the ntegral closure of V n K, so that V s the localzaton of T at T m V. Thus for 1 m, f = b c wth b, c T, ν (b ) 0, ν (c ) = 0. Let b m + d 1 b m d,m = 0 c n + e 1 c n e n = 0, 1 m, be equatons of ntegral dependence of b, c over V, so that all d j, e j V. There exsts a sequence of monodal transforms R 1 R along ν such that all d j, e j R. Suppose that R, S are as n the statement of the Lemma, so that R contans R. Then U 1 [b 1,..., b m, c 1,..., c m ] s contaned n S. c S and ν (c ) = 0 mples c m V S = m S and c s a unt n S. Thus f S for 1 m, and S 1 S. 6. Extensons of valuaton rngs are torc Theorem 6.1. Suppose that the assumptons of Theorem 4.2 hold. Let g 1,..., g f be a bass of V /m V over V/m V. Then there exsts an algebrac regular local rng R wth quotent feld K whch s domnated by V, such that f R 0 R S s of the form of (7) of Theorem 4.2 and R R 0, then the conclusons of Theorem 4.10 hold for R 0 R S. Further, wth the notatons of (3) and (4), [S/m S : R/m R ] = f, det(a) = e, [QF (Ŝ) : QF ( ˆR)] = ef and {g 1,..., g f } s a bass of S/m S over R/m R = R 0 /m R0. Proof. As n the proof of Lemma 5.1, there exsts an algebrac regular local rng R 1 wth quotent feld K whch s domnated by ν such that f R s an algebrac regular local rng wth quotent feld K such that R 1 R and R s domnated by ν, then

22 22 STEVEN DALE CUTKOSKY AND OLIVIER PILTANT g 1,..., g f S/m S, where S s the localzaton of the ntegral closure of R n K at the center of ν. Ths argument does not requre that K /K be Galos. Let K be a Galos closure of K over K, ν be an extenson of ν to K. By Lemma 5.1, there exsts an algebrac regular local rng S 0 wth quotent feld K whch s domnated by ν such that f S s a normal algebrac local rng of K whch contans S 0 and S s the localzaton of the ntegral closure of S n K at the center of ν, then G s (S /S) = G s (ν /ν) and G (S /S) = G (ν /ν). By Lemma 5.3 and Lemma 5.1, there exsts an algebrac regular local rng R whch s domnated by ν and has quotent feld K, such that f R s an algebrac local rng wth quotent feld K whch s domnated by ν, R R, and S les above R n K and s domnated by ν, S les above R n K and s domnated by ν, then G s (S /R) = G s (ν /ν) G (S /R) = G (ν /ν), G s (S /S) = G s (ν /ν ) G (S /S) = G (ν /ν ). By Theorem 4.10 and Lemma 5.3, there exsts an algebrac regular local rng R wth quotent feld K whch contans R and s domnated by ν such that f R 0 R S s a sequence of the form (7), then the conclusons of Theorem 4.10 hold. Snce R R 0, we further have [S/m S : R/m R ] = [S /m S : R/m R ]/[S /m S : S/m S ] = [G s (S /R) : G (S /R)]/[G s (S /S) : G (S /S)] = [G s (ν /ν) : G (ν /ν)]/[g s (ν /ν ) : G (ν /ν )] = [k(ν ) : k(ν)]/[k(ν ) : k(ν )] = [k(ν ) : k(ν)] = f Snce g 1,..., g f S/m S, we have that g 1,..., g f s an R/m R bass of S/m S. Snce the conclusons of Theorem 4.10 hold (and by (19)), e = det(a). By Theorem 4.6, ˆR Ŝ s the completon of the fnte extenson k 1 [x e1,..., x er ] k 2 [y 1,..., y n ] where k 1 = R/m R, k 2 = S/m S, so that wth the notaton of Lemma 4.4, [QF (Ŝ) : QF ( ˆR)] = [L : L]. By Lemma 4.4, [QF (Ŝ) : QF ( ˆR)] = [S/m S : R/m R ] det(a) = fe. Theorem 6.2 s essentally proven by Zarsk n [21]. Certanly the expresson of V as a unon of algebrac regular local rngs follows easly from the results of [21]. The remanng statements follow easly from Zarsk s results when V has rank 1, and can be deduced wth some effort for hgher rank. Theorem 6.2. Let k be a feld of characterstc zero, K an algebrac functon feld over k, V a k-valuaton rng of K. Then there exsts a partally ordered set I and algebrac regular local rngs {R() I} wth quotent feld K whch are domnated by V such that V = lm R(j) = j I R(j) and such that, wth the notatons on V defned n Secton 3, R(j) has regular parameters (x 1 (j),..., x n (j)) such that (1) p (j) = p R(j) = (x 1 (j),..., x t1+ +t (j)) for 1 r and {ν(x t1+ +t 1+1(j)),..., ν(x t1+...+t 1+s (j)} s a ratonal bass of ( 1 / ) Q for 1 r.

23 RAMIFICATION OF VALUATIONS 23 (2) Let T = {j t t k < j t t k + s k+1 for some 0 k r 1}. If j < k I then there are relatons x (j) = x c (k) dc δ (20) c T,c>t 1+ +t a where d c are natural numbers and δ R(k) s a unt for = t 1 + +t a +b T wth 1 b s a+1. If D(j, k) s the T T matrx (d c ) of (20) then det(d(j, k)) 0. (3) For j I, let Λ j be the free Z-module Λ j = T ν(x (j))z. Then Γ = lm Λ j = j I Λ j. Proof. Let R be an algebrac regular local rng such that V domnates R. By Theorem 4.8 (wth S = R ) there exsts a sequence of monodal transforms R R(0) along V such that 1. of ths theorem holds on R(0). Suppose that m s a postve nteger and f = (f 1,..., f m ) V m. We wll construct a sequence of monodal transforms R 0 R(f) along V such that f 1,..., f m R(f), 1. of ths theorem holds for R(f) and 2. of ths theorem holds for R 0 R(f). We wll further have ν(f) Λ f. By Theorem 4.9, wth the R, S of the statement of Theorem 4.9 set as R = S = R(0), and v = x (0) f T, and v T +1 = f 1,..., v T +m = f m, there exsts a sequence of monodal transforms R(0) R(f) along V such that 1. of ths theorem holds for R(f), 2. of ths theorem holds for R(0) R(f), f R(f) and ν(f) Λ f. DetD(0, f) 0 snce {ν(x (0) T } and {ν(x (f)) T } are two bases of Γ Q. Let I = m N+ V m be the dsjont unon. For f I we construct R(f) as above. If f = 0 we let R(0) be the R(0) constructed above. Defne a partal order on I by f g f R(f) R(g). Suppose that R(α) R(β). We have R(0) R(α) R(β). x (0) = j T x j (α) cj δ for T wth δ a unt n R(α) and x (0) = j T x j (β) dj ɛ for T wth ɛ a unt n R(β). Thus n R(β) there are factorzatons x (α) = j T x j (β) ej λ for T and λ a unt n R(β). We have det(d(α, β)) 0 snce 1. holds for R(α) and R(β). Thus 2. holds for R(α) R(β). To show that V = lm R(j), we must verfy that I s a drected set. That s, for α, β I, there exsts γ I such that R(α) R(γ) and R(β) R(γ). There exsts f 1,..., f m V such that f A = k[f 1,..., f m ], m = A m V1 then R(α) = A m. There exsts g 1,..., g n V such that f B = k[g 1,..., g n ], n = B m V1 then R(β) = B n. Set γ = (f 1,..., f m, g 1,..., g n ). By constructon, A, B R(γ). Snce m V R(γ) = m γ s the maxmal deal of R(γ), we have R(α), R(β) R(γ). 3. holds by our constructon,snce ν(f) Λ f f f V.