Algebraic series and valuation rings over nonclosed fields

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1 Joural of Pure ad Aled Algebra 008) Algebrac seres ad valuato rgs over oclosed felds Steve Dale Cutkosky a, Olga Kashcheyeva b, a Deartmet of Mathematcs, Uversty of Mssour, Columba, MO 65, USA b Deartmet of Mathematcs, Statstcs ad Comuter Scece, Uversty of Illos, Chcago, IL 60607, USA Receved 5 October 007; receved revsed form 8 November 007 Avalable ole 4 March 008 Commucated by A.V. Geramta Abstract Suose that k s a arbtrary feld. Let k[[,..., be the rg of formal ower seres varables wth coeffcets k. Let k be the algebrac closure of k ad σ k[[,...,. We gve a smle ecessary ad suffcet codto for σ to be algebrac over the quotet feld of k[[,...,. We also characterze valuato rgs V domatg a ecellet Noethera local doma R of dmeso, ad such that the rak creases after assg to the comleto of a bratoal eteso of R. Ths s a geeralzato of the characterzato gve by M. Svakovsky [Valuatos fucto felds of surfaces, Amer. J. Math. 990) the case whe the resdue feld of R s algebracally closed. c 008 Elsever B.V. All rghts reserved. MSC: 3F5; 3F30. Itroducto Suose that k s a arbtrary feld. Cosder the feld k,..., )), whch s the quotet feld of the rg k[[,..., of formal ower seres the varables,...,, wth coeffcets k. Suose that k s a algebrac closure of k, ad σ k[[,..., s a formal ower seres. I ths aer, we gve a very smle ecessary ad suffcet codto for σ to be algebrac over k,..., )). We rove the followg theorem, whch s restated a equvalet formulato Theorem 3.. Theorem.. Suose that k s a feld of characterstc 0, wth algebrac closure k. Suose that σ,..., ) = α,..., k[[,...,,..., N where α,..., k for all. Let L = k{α,...,,..., N}) Corresodg author. E-mal addresses: cutkoskys@mssour.edu S.D. Cutkosky), olga@math.uc.edu O. Kashcheyeva) /$ - see frot matter c 008 Elsever B.V. All rghts reserved. do:0.06/j.jaa

2 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) be the eteso feld of k geerated by the coeffcets of σ,..., ). The σ,..., ) s algebrac over k,..., )) f ad oly f there ests r N such that [kl r : k <, where kl r s the comostum of k ad L r k. I the case that L s searable over k Corollary 3.4), or that k s a ftely geerated eteso feld of a erfect feld Corollary 3.3), we have a stroger codto. I these cases, σ s algebrac f ad oly f [L : k <. The fteess codto [L : k < does ot characterze algebrac seres over arbtrary base felds k of ostve characterstc. To llustrate ths, we gve a smle eamle, Eamle.3, of a algebrac seres oe varable for whch [L : k =. I Secto, we rove Theorem. the case =. The most dffcult art of the roof arses whe k s ot erfect. Our roof uses the theorem of resoluto of sgulartes of a germ of a lae curve sgularty over a arbtrary feld cf. [,0,8). I Secto 3, we rove Theorem. for ay umber of varables. The roof volves ducto o the umber of varables, ad uses the result for oe varable rove Secto. I the case whe k has characterstc zero ad =, the coclusos of Theorem. are classcal. We recall the very strog kow results, uder the assumto that k has characterstc zero, ad there s oly oe varable = ). The algebrac closure of the feld of formal meromorhc ower seres k)) the varable s k)) = F F )) = where F s ay fte feld eteso of k cotaed the algebrac closure k of k. The equalty ) s stated ad rove Rbebom ad Va de Dres artcle [4. A roof ca also be deduced from Abhyakar s Theorem [3 or Secto.3 of [). The equalty ) already follows for a algebracally closed feld k of characterstc zero from a classcal algorthm of Newto [6,8. If k has characterstc > 0, the the algebrac closure of k)) s much more comlcated, eve whe k s algebracally closed, because of the estece of Art Schreer etesos, as s show Chevalley s book [7. I fact, the seres σ ) =, = cosdered by Abhyakar [4, s algebrac over k)), as t satsfes the relato σ σ = 0. Whe k s a algebracally closed feld of arbtrary characterstc, the geeralzed ower seres feld k Q )) s algebracally closed, as s show by Rbebom [3. The aroach of studyg the algebrac closure of k)) through geeralzed ower seres s develoed by Behess [5, Hah [, Huag, [5, Pooe [, Rayer [, Stefaescu [6 ad Vadya [8. A comlete soluto whe k s a erfect feld s gve by Kedlaya [6. He shows that the algebrac closure of k)) cossts of all twst recurret seres u = α k Q )) such that all α le a commo fte eteso of k. Whe >, the algebrac closure of k,..., )) s kow to be etremely comlcated, eve whe k s algebracally closed of characterstc 0. I ths case, dffcultes occur whe the ramfcato locus of a fte eteso s very sgular. There s a good uderstadg some mortat cases, such as whe the ramfcato locus s a smle ormal crossgs dvsor ad the characterstc of k s 0 or the ramfcato s tame Abhyakar [3, Grothedeck ad Murre [) ad for quas-ordary sgulartes Lma [8, Gozález-Pérez [0). More geerally, subrgs of a ower seres rg ca be very comle, ad are a source of may etraordary eamles, such as [9,5,4. As a alcato of our methods, we gve a characterzato of valuato rgs V whch domate a ecellet, Noethera local doma R of dmeso two, ad such that the rak creases after assg to the comleto of a bratoal eteso of R. The characterzato s kow whe the resdue feld of R s algebracally closed Svakovsky [7). I ths case R/m R algebracally closed) the rak creases uder comleto f ad oly f dm R V ) = 0 V/m V s algebrac over R/m R ) ad V s dscrete of rak. However, the characterzato s more subtle over oclosed felds. I Theorem 4., we show that the codto that the rak creases uder comleto s characterzed by the two codtos that the resdue feld of V s fte ) )

3 998 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) over the resdue feld of R, ad that V s dscrete of rak. The case whe the resdue feld of V s fte algebrac over the resdue feld of R ad the value grou s dscrete of rak ca occur, ad the rak of such a valuato does ot crease whe assg to comleto. I Corollary 4.3, we show that there ests a valuato rg V domatg R whose value grou s dscrete of rak wth dm R V ) = 0 such that the rak of V does ot crease uder comleto f ad oly f the algebrac closure of R/m R has fte degree over R/m R. We ot out the cotrast of the coclusos of Theorem. wth the results of Secto 4. The fteess codto [L : k < of the coeffcet feld of a seres over a base feld k does ot characterze algebracty of a seres a ostve characterstc, whle the corresodg fteess codto o resdue feld etesos does characterze algebracty the crease of rak) the case of valuatos domatg a local rg of Theorem 4.. We llustrate ths dstcto Eamle 4.4 by costructg the valuato rg determed by the seres of Eamle.3. We coclude by showg a smle stadard ower seres reresetato of the valuato assocated to the algebrac seres of ), whose eoets do ot have bouded deomators. The cocet of the rak creasg whe assg to the comleto already aears mlctly Zarsk s aer [9. Some aers where the cocet s develoed are [7,3,9. If R s a local or quas local) rg, we wll deote ts mamal deal by m R.. Seres oe varable Lemma.. Suose that R s a two-dmesoal regular local rg, ad m R s art of a regular system of arameters. Suose that k 0 s a coeffcet feld of ˆR ad y ˆR s such that, y are regular arameters ˆR. Ths determes a somorhsm ˆR λ 0 k 0 [[, y of ˆR wth a ower seres rg. Suose that α s searably algebrac over k 0. Let y = y mamal deal R[ m R ad a somorhsm [ m R λ R k 0 α)[[, y whch makes the dagram α. The there ests a λ 0 ˆR k 0 [[, y [ m R λ R k 0 α)[[, y commute, where the vertcal arrows are the atural mas. Proof. There ests ỹ R such that ỹ = y + h where h m 3 R ˆR. We have ỹ α = y + h, y + α)) = y + h, y ) where h k 0 α)[[, y s a seres of order. Thus we have atural chage of varables k 0 [[, y = k 0 [[, ỹ ad k 0 α)[[, y = k 0 α)[[, ỹ α. We may thus assume that y R. We have a atural cluso duced by λ 0, [ y [ y R ˆR k 0 α)[[, y. Let =, y ) R[ y. Let ht) be the mmal olyomal of α over k 0, ad f R[ y be a lft of y ) [ y [ h k 0 = y [ y R / R. The =, f ) ad we see that R[ y / = k 0 α). Now the coclusos of the lemma follow from Hesel s Lemma cf. Lemma 3.5 [8).

4 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Theorem.. Suose that k s a feld, wth algebrac closure k. Let k)) be the feld of formal Lauret seres a varable wth coeffcets k. Suose that σ ) = α k)) =d where d Z ad α k for all. Let L = k{α N}), ad suose that L s searable over k. The σ ) s algebrac over k)) f ad oly f [L : k <. Proof. We reduce to the case where d, by observg that σ s algebrac over k)) f ad oly f d σ s. Frst suose that [L : k <. Let M be a fte Galos eteso of k whch cotas L. Let G be the Galos grou of M over k. G acts aturally by k algebra somorhsms o M[[, ad the varat rg of the acto s k[[. Let f y) = τ G y τσ )) M[[[y. Sce f s varat uder the acto of G, f y) k[[[y. Sce f σ ) = 0, we have that σ s algebrac over k)). Now suose that σ ) = = α s algebrac over k)). The there ests g, y) = a 0 )y + a )y + + a ) k[[[y such that a 0 ) 0,, g s rreducble ad g, σ )) = 0. Let y 0 = y, y = y α, y = y α,..., y = y α,... ad defe S 0 = k[[, y, S = kα )[[, y,... S = kα,..., α )[[, y,.... We have atural clusos S 0 S S. By Lemma., there ests a sequece of clusos R 0 R R 3) where R 0 = k[[[y,y) ad each R s a localzato at a mamal deal of the blow u of the mamal deal m R of R, ad we have a commutatve dagram of homomorhsms S 0 S S R 0 R R where the vertcal arrows duce somorhsms of the m R -adc comletos ˆR of R wth S. We further have that s art of a regular system of arameters R for all, ad m R R = R for all. By our costructo, we have that R /m R = kα,..., α ) for all. For all, wrte g = b g where g R ad does ot dvde g R. I k[[, y, we have a factorzato ) y σ = y α j j. j=+ Sce y σ dvdes g k[[, y, we have that y j=+ α j j dvdes g k[[, y. Thus g s ot a ut k[[, y, ad s thus ot a ut R. 4)

5 000 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Let C be the curve germ g = 0 the germ SecR 0 ) of a osgular surface. The sequece 3) s obtaed by blowg u the closed ot SecR ), ad localzg at a ot whch s o the strct trasform of C. g = 0 s a local equato of the strct trasform of C SecR ). By embedded resoluto of lae curve sgulartes [,0 or a smle geeralzato of Theorem 3.5 ad Eercse 3.3 of [8) we obta that there ests 0 such that the total trasform of C SecR ) s a smle ormal crossgs dvsor for all 0. Sce b g = g = 0 s a local equato of the total trasform of C SecR ), we have that, g are regular arameters R for all 0. Thus g 0 = 0g for all 0, ad R = R [ g,g ) for all 0 +. We thus have that R /m R = R0 /m R0 for all 0, ad we see that L = 0 R /m R = R 0 /m R0 = kα,..., α 0 ). Thus [L : k <. Eamle.3. The coclusos of Theorem. may fal f L s ot searable over k. Proof. Let be a rme ad {t N} be algebracally deedet over the fte feld Z. Let k = Z {t N}). Defe σ ) = t k[[. Let = f y) = y t k[[[y. = σ ) s algebrac over k[[ sce f σ )) = σ )) t = 0. However, = [k{t N}) : k =. Suose that k s a feld of characterstc > 0 ad L s a eteso feld of k. For N, let L = { f f L}. If k has characterstc = 0, we take L = L for all. Theorem.4. Suose that k s a feld of characterstc > 0, wth algebrac closure k. Let k)) be the feld of formal Lauret seres the varable wth coeffcets k. Suose that σ ) = α k)) =d where d Z ad α k for all. Let L = k{α N}), ad assume that L s urely searable over k. The σ ) s algebrac over k)) f ad oly f there ests N such that L k. Proof. As the roof of Theorem., we may assume that d. Frst suose that L k for some. The τ) = σ ) k[[, ad σ ) s the root of y τ) = 0. Thus σ s algebrac over k)). Now suose that σ ) = = α k[[ s algebrac over k)). The there ests g, y) = a 0 )y + a )y + + a ) k[[[y such that a 0 ) 0,, g s rreducble ad g, σ )) = 0.

6 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Let K be the quotet feld of k[[[y, ad let R 0 := S 0 := k[[[y,y). We wll frst costruct a seres of subrgs S of K. Defe a local k-algebra homomorhsm π 0 : S 0 k[[ by rescrbg that π 0 ) = ad π 0 y) = σ ). The kerel of π 0 s the rme deal gs 0. y = α + k[[ =0 defes a k-algebra homomorhsm S 0 [ y k[[ whch eteds π 0. Let λ) N be the smallest atural umber such that α λ) k. The the mamal deal k[[ of k[[ cotracts to [ y y ) λ) ) k[[ S 0 =, α λ). Set y = y ) λ) α λ). Let [ y S = S 0.,y ) Let π : S k[[ be the local k-algebra homomorhsm duced by π 0. We have that, y s a regular system of arameters S, wth y = = S /m S = kα ) ad α λ) + λ). [S /m S : S 0 /m S0 = [kα ) : k = λ). Let λ) N be the smallest atural umber such that α λ)+λ) kα ). Let y = y λ) ) λ) α λ)+λ). The there s a easo k[[ y = = α λ)+λ) + λ)+λ). y Let S = S [, α λ),y ) K. We have a local k-algebra homomorhsm π : S k[[ whch eteds π. We have S /m S = kα, α λ) ), so that [S /m S : S /m S = [kα, α λ) ) : kα ) = λ). We terate the above costructo, defg for, y = = ) y λ) λ)+ +λ ) j= α λ)+ +λ) j+ jλ)+ +λ) α λ)+ +λ) where λ) N s the smallest atural umber such that α λ)+ +λ) kα, α λ),..., α λ)+ +λ ) ).

7 00 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Defe S = S [ y λ)+ +λ ), α λ)+ +λ ),,y ) to costruct a fte commutatve dagram of regular local rgs, whch are cotaed K, We have ad S 0 S S π 0 π π k[[ = k[[ = = k[[ = S /m S = S /m S [α λ)+ +λ ) [S /m S : S /m S = λ). 5) ad For all, the feld k := kα, α λ),..., α λ)+ +λ ) ) S, S /m S = k [α λ)+ +λ ). We ow costruct a sequece R 0 R R of bratoally equvalet regular local rgs such that there s a commutatve dagram of local k-algebra homomorhsms satsfyg R 0 R R S 0 S S m R S = m S ad S /m S = R /m R for all. The vertcal arrows are clusos. Ths s certaly the case for R 0 = S 0, so we suose that we have costructed the sequece out to R S, ad show that we may eted t to R + S +. We have α λ)+ +λ+) + kα, α λ),..., α λ)+ +λ ) ) = R /m R. Thus there ests ϕ R such that the class of ϕ R /m R s [ϕ = α λ)+ +λ+) +. Our assumtos m R S = m S ad S /m S = R /m R mly that m R /m + R = m S /m + S as R /m R vector saces for all N. By 6), there ests z R such that z = y + h 6)

8 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) wth h m +λ)+ +λ) S. We the have that m R =, z ), sce m R /m R = m S /m S as R /m R vector saces, ad by Nakayama s Lemma. Now z λ)+ +λ) ) λ+) = = for some h S [. λ)+ +λ) y ) y λ+) λ)+ +λ) ) λ+) y λ)+ +λ) + + h h λ)+ +λ) ) λ+) ) z λ+) [ [ z y ϕ R λ)+ +λ) S λ)+ +λ) λ)+ +λ) has resdue y λ)+ +λ) ) λ+) S + / S + = S /m S [ Let m S+ R [ R + = R [ z λ)+ +λ) z λ)+ +λ) α λ)+ +λ+) + y λ)+ +λ). Thus z =, λ)+ +λ), z λ)+ +λ) ) λ+) ) λ+). ϕ) We have m R+ S + = m S+ by Nakayama s Lemma) ad R + /m R+ = S+ /m S+. We have factorzatos g, y) = β g where β N ad g R s ether rreducble or a ut. g s a strct π trasform of g R. Sce π ) 0, we have that g s cotaed the kerel of the ma R S k[[, ad thus the deal g ) s the otrval) kerel of R k[[. I artcular, g M R for all. Each eteso R R + ca be factored as a sequece of λ)+ +λ) bratoally equvalet regular local rgs, each of whch s a quadratc trasform the blow u of the mamal deal followed by localzato). The j-th local rg wth j < λ)+ +λ), has the mamal deal, z ). j By embedded resoluto of lae curve sgulartes [,8,0, we obta that there ests 0 such that g = 0 s a smle ormal crossgs dvsor SecR ) for all 0, so that, g s a regular system of arameters R for all 0. Thus [ z R + = R for all 0, ad λ)+ +λ) g, λ)+ +λ) ) S + /m S+ = R+ /m R+ = R /m R = S /m S for all 0. Thus λ) = 0 for all 0 +. Let M = kα, α λ),..., α λ)+ +λ 0 ) 0 ) = S 0 /m S0. From 5), we see that L λ)+ +λ 0 ) M. Sce M s a ftely geerated urely searable eteso of k, there ests r N such that M r k. Thus L λ)+ +λ 0 )+r k. ϕ).

9 004 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Theorem.5. Suose that k s a feld of characterstc 0, wth algebrac closure k. Let k)) be the feld of formal Lauret seres wth coeffcets k. Suose that σ ) = α k)) =d where d Z ad α k for all. Let L = k{α N}). The σ ) s algebrac over k)) f ad oly f there ests N such that [kl : k <, where kl s the comostum of k ad L k. Proof. Frst suose that [kl : k < for some. After ossbly relacg wth a larger value of, we may assume that kl s searable over k. The σ ) s algebrac over k)) by Theorem., ad thus σ ) s algebrac over k)). Now suose that σ ) s algebrac over k)). Let M be the searable closure of k L. The σ ) s algebrac over M)). Sce L s a urely searable eteso of M, t follows from Theorem.4 that τ) = σ ) M[[ for some N. Sce τ) s algebrac over k)), we have that [kl : k < by Theorem.. Corollary.6. Suose that k s a feld of characterstc 0 such that k s a ftely geerated eteso of a erfect feld, wth algebrac closure k. Let k)) be the feld of formal Lauret seres wth coeffcets k. Suose that σ ) = α k)) =d where d Z ad α k for all. Let L = k{α N}). The σ ) s algebrac over k)) f ad oly f [L : k <. Proof. If [L : k <, the σ ) s algebrac over k)) by Theorem.5. Suose that σ ) s algebrac over k)). By assumto, there ests a erfect feld F ad s,..., s r k such that k = Fs,..., s r ). By Theorem.5, there ests such that [kl : k <. Thus kl = Fs,..., s r, β,..., β s ) where β,..., β s kl are algebrac over k. Thus Now L Fs,..., s r, β,..., β s ). [Fs,..., s r ) : Fs,..., s r ) < ad sce β,..., β s are algebrac over Fs,..., s r ), Thus [Fs,..., s r, β,..., β s ) : Fs,..., s r ) <. [L : k [Fs,..., s r, β,..., β s ) : k <. 3. Seres several varables We wll ow geeralze Theorem.5 to hgher dmesos. Deote by X a -dmesoal determate vector,,..., ) ad by I a -dmesoal eoet vector,,..., ) N. The for l wrte X l =,,..., l ), I l =,,..., l ) ad Xl I = X I l l = l l. If E s a feld deote by E[[X the formal ower seres rg varables wth coeffcets E ad by EX)) the quotet feld of E[[X. Also deote by E c the erfect closure of E ad by E the algebrac closure of E.

10 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) Lemma 3.. Suose that E s a feld ad F s a feld eteso of E. Let σ = I N α I X I F[[X, wth α I F, be a formal ower seres varables wth coeffcets F. For ay l ad I N defe the followg ower seres varable wth coeffcets F a I,l = j=0 α J j l, where J =,,..., l, j, l+,..., ). The σ s algebrac over EX)) mles a I,l s algebrac over E l )). Proof. We use ducto o the umber of varables. If = the statemet s trval. Suose that >. After ossbly ermutg the varables we may assume that l =. Wrte X =,..., ) ad for all m N cosder the ower seres varables δ m = R N, r =m α R X R = R N, r =m α R r r r. If δ s algebrac over EX )) t wll follow from the ductve hyothess that a I, s algebrac over E )). We wll show that δ m s algebrac over EX )) for all m N. Cosder the algebrac deedecy relato for σ over EX)) c t X)σ t + c t X)σ t + + c X)σ + c 0 X) = 0. By clearg the deomators we may assume that c j E[[X for all 0 j t. Let g be the hghest ower of that dvdes c j for all j. Set c j = g c j ),,...,, 0). The c j E[[X ad the followg equato holds: c t X )δ t 0 + c t X )δ t c X )δ 0 + c 0 X ) = 0, where c j 0 for some 0 j t. Thus δ 0 s algebrac over EX )). Set σ = σ δ 0). The σ F[[X ad t s algebrac over EX)). Argug as above we get that δ s algebrac over EX )). I geeral we defe σ m = σ m δ m ) recursvely for all m N ad use σ m to rove that δ m s algebrac over EX )). Theorem 3.. Suose that k s a feld of characterstc 0. Suose that σ = α I X I k[[x, wth α I k I N s a formal ower seres varables wth coeffcets k. Let L = k{α I I N }) be the eteso feld of k geerated by the coeffcets of σ. The σ s algebrac over kx)) f ad oly f there ests r N such that [kl r : k <, where kl r s the comostum of k ad L r k. Proof. Frst suose that there ests r N such that [kl r : k <. After ossbly creasg r we may assume that kl r s a searable eteso of k. Let M be a fte Galos eteso of k whch cotas kl r. Notce that kl r = k{α r I I N }) ad, therefore σ r M[[X. Let G be the Galos grou of M over k. G acts aturally by k algebra somorhsms o M[[X, ad the varat rg of the acto s k[[x. Let f y) = τ G y τσ r )) M[[X[y. Sce f s varat uder the acto of G, f y) k[[x[y. Sce f σ r ) = 0, we have that σ s algebrac over k[[x. To rove the other mlcato we use ducto o the umber of varables. Whe = the statemet follows from Theorem.5. Assume that >. For all I N let a I = α J j, wth J =,,...,, j), j=0

11 006 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) be a ower seres varable wth coeffcets k. If K = k )) the by Lemma 3. a I s algebrac over K for all I N. The σ = a I X I j=0 {I N =0} s a seres varables wth coeffcets K. By the ductve hyothess there ests N N ad r N such that K {a r I I N }) = K a r I, a r I,..., a r I N ). Thus, for all I N we have a r I s a olyomal a r I, a r I,..., a r I N wth coeffcets K. F I N, f j N set J =,,...,, j) ad wrte ) α r J jr = a r I = γ S,m m a r I ) s a r I ) s a r I N ) s N, m= M S S {0,,...,T } N where T N, S = s, s,..., s N ) s a de vector, M S N ad γ S,m k for all S ad m. Ths mles that for all I N ad j N, α r J s a olyomal the coeffcets of ower seres a r I, a r I,..., a r I N over k. Moreover, for all r r we also have α r J s a olyomal the coeffcets of ower seres a r I the feld eteso of k geerated by the coeffcets of ower seres a r I, a r I,..., a r I N over k. Thus kl r s, a r I,..., a r I N. Alyg Theorem.5 to each of the seres a I, a I,..., a IN we see that there ests R N such that kl R s ftely geerated over k. Smlarly to the case of oe varable we deduce the followg corollary. Corollary 3.3. Suose that k s a feld of characterstc 0 such that k s a ftely geerated eteso of a erfect feld. Suose that σ = α I X I k[[x, wth α I k I N s a formal ower seres varables wth coeffcet k. Let L = k{α I I N }) be the eteso feld of k geerated by the coeffcets of σ. The σ s algebrac over kx)) f ad oly f [L : k <. Also otce that f E s a feld of characterstc 0 ad a s searable algebrac over E; the for all r N we have E[a r = E[a. Thus f F s a searable eteso of E, E F r = F for all r N. So we have the followg statemet case of searable etesos. Corollary 3.4. Suose that k s a feld of characterstc 0. Suose that σ = α I X I k[[x, wth α I k I N s a formal ower seres varables wth coeffcet k. Let L = k{α I I N }) be the eteso feld of k geerated by the coeffcets of σ. Suose that L s searable over k. The σ s algebrac over kx)) f ad oly f [L : k <. 4. Valuatos whose rak creases uder comleto Suose that K s a feld ad V s a valuato rg of K. We wll say that the rak of V creases uder comleto f there ests a aalytcally ormal local doma T wth quotet feld K such that V domates T ad there ests a eteso of V to a valuato rg of the quotet feld of ˆT whch domates ˆT whch has hgher rak tha the rak of V. Suose that V domates a ecellet local rg R of dmeso. The by resoluto of surface sgulartes [7, there ests a regular local rg R 0 ad a bratoal eteso R R 0 such that V domates R 0. Let R 0 R R 7)

12 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) be the fte sequece of regular local rgs obtaed by blowg u the mamal deal of R ad localzg at the ceter of V. Sce R has dmeso, we have that V = =0 R as s show [), ad thus V/m V = =0 R /m R. We see that V/m V s coutably geerated over R/m R. Suose that the rak of V creases uder comleto. The there ests such that for all, there ests a valuato rg V of the quotet feld of the regular local rg ˆR whch eteds V, domates ˆR, ad has rak larger tha. By the Abhyakar equalty [ or Proosto 3 of Aed [30), we have that R has dmeso, V s dscrete of rak, ad V /m V s algebrac over ˆR /m ˆR. Thus V/m V s algebrac over R/m R ad V s dscrete of rak. It was show by Svakovsky [7 the case that R/m s algebracally closed that the coverse holds, gvg the followg smle characterzato. Theorem 4. Svakovsky [7). Suose that V domates a ecellet two-dmesoal local rg R who resdue feld R/m R s algebracally closed. The the rak of V creases uder comleto f ad oly f dm R V ) = 0 ad V s dscrete of rak. The codto that the trascedece degree dm R V ) of V/m V over R/m R s zero s just the statemet that V/m V s algebrac over R/m R. I the case that R/m R s algebracally closed, dm R V ) = 0 f ad oly f V/m V = R/m R. Usg a smlar method to that used the roof of our algebracty theorem o ower seres, Theorem., we rove the followg eteso of Theorem 4.. Theorem 4.. Suose that V s a valuato rg of a feld K, ad V domates a ecellet two-dmesoal local doma R whose quotet feld s K. The the rak of V creases uder comleto f ad oly f V/m V s fte over R/m R ad V s dscrete of rak. Proof. Frst assume that the rak of V creases uder comleto. Cosder the sequece 7). We observed above after 7) that V/m V s algebrac over R/m R ad V s dscrete of rak. Further, there ests R ad a valuato V of the quotet feld of ˆR whch domates ˆR whose tersecto wth the quotet feld K of R s V, ad the rak of V s. Wthout loss of geeralty, we may assume that R = R 0. For 0, let R ) be the otrval) rme deal ˆR of Cauchy sequeces whose value s greater tha for ay N Secto 5 of [9). Sce ˆR s a two-dmesoal regular local rg, R ) s geerated by a rreducble elemet ˆR for all. Let f be a geerator of R 0 ). By resoluto of lae curve sgulartes [,8,0, there ests the sequece 7) such that f = h f, where h R s such that h = 0 s suorted o the ecetoal locus of SecR ) SecR), ad f ˆR s such that ˆR /f ˆR s a regular local rg. We ecessarly have that R ) = f ˆR. Aga, wthout loss of geeralty, we may assume that = 0. Let T 0 = ˆR 0, ad let T 0 T T be the fte sequece of regular local rgs obtaed by blowg u the mamal deal of the regular local rg T ad localzg at the ceter of V. We the have a commutatve dagram R 0 R R T 0 T T ˆR 0 ˆR ˆR. 8) There ests R 0 such that, f 0 s a regular system of arameters T 0. Thus T = T 0 [ f 0 f, 0 ). Defe f = f 0 for. The T = T 0 [ f, f ) ad R ) = f ˆR for all 0. Thus R /m R = T /m T = T0 /, f 0 ) = R 0 /m R0 for all. Sce V/m V = 0 R /m R = R 0 /m R0 ad R 0 /m R0 s fte over R/m R, we have the coclusos of the theorem. Now assume that V/m V s fte over R/m R ad V s dscrete of rak. Cosder the sequece 7). There ests such that R /m R = V/m V. Wthout loss of geeralty, we may assume that R = R. Let ν be a valuato of K such that V s the valuato rg of ν. We may also assume that there are regular arameters, y R such that ν) = geerates the value grou Z of ν. Let π : R R/m R = V/m V be the resdue ma. Let y 0 = y. There ests 0 N

13 008 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) such that νy) = 0. Let α 0 R be such that πα 0 ) = [ y V/m 0 V. Let y = y α 0 0, ad let = νy ). We have > 0. Iterate, to costruct y R ad N wth νy ) = for N by choosg α R such that y + = y α satsfes + >. Thus {y } s a Cauchy sequece R. Let σ be the lmt of {y } ˆR. Let ˆν be a eteso of ν to the quotet feld of ˆR whch domates ˆR. The ˆνσ ) > for all N, so that ˆν has rak >, ad we see that the rak of V creases uder comleto. We see that the codto that V/m V s fte over R/m R thus dvdes the class of dscrete rak valuato rgs wth dm R V ) = 0 to two subclasses, those whose rak creases uder comleto [V/m V : R/m R < ), ad those whose rak does ot crease [V/m V : R/m R = ). We have the followg recse characterzato of whe ths dvso to subclasses s otrval. Corollary 4.3. Suose that R s a ecellet two-dmesoal local rg. The there ests a rak dscrete valuato rg V of the quotet feld of R whch domates R such that dm R V ) = 0 ad the rak of V does ot crease uder comleto f ad oly f [k : k =, where k s the algebrac closure of k = R/m R. Proof. Suose that [k : k <, ad V s a rak dscrete valuato rg of the quotet feld of R whch domates R such that dm R V ) = 0. The [V/m V : k [k : k <. Thus the rak of V must crease uder comleto by Theorem 4.. Now suose that [k : k =. We wll costruct a rak dscrete valuato rg V of the quotet feld of R whch domates R such that dm R V ) = 0 ad the rak of V does ot crease uder comleto. There ests a two-dmesoal regular local rg R 0 whch bratoally domates R. We have [k : R 0 /m R0 =. Let, y 0 be a regular system of arameters R 0. We wll ductvely costruct a fte bratoal sequece of regular local rgs R 0 R R such that R has a regular system of arameters, y ad [R /m R : R /m R > for all. Suose that we have defed the sequece out to R. Choose α + k R /m R. Let h + t) be the mmal olyomal of α + the olyomal rg R /m R [t. We have a somorhsm R [ m R / R [ m R Let y + be a lft of h + y ) to R R + = R [ m R,y + ) = R /m R [ y. [ m R.. Let We have that R + /m R+ = R /m R α + ). Let V = =0 R. V s a valuato rg whch domates R as s show [). V/m V = =0 R /m R so that dm R V ) = 0 ad [V/m V : k =. V must have rak sce [V/m V : k = for stace by the Abhyakar equalty, [ or Proosto 3 [30). By our costructo, ν) ν f ) for ay f m V = = m R. Thus the value grou of V s dscrete. Sce [V/m V : k =, by Theorem 4. the rak of V does ot crease uder comleto. Whe a valuato rg V wth quotet feld K s equcharacterstc ad dscrete of rak, t ca be elctly descrbed by a reresetato a ower seres rg oe varable over the resdue feld of V. I fact, sce V s dscrete of rak, t s Noethera Theorem 6, Secto 0, Chater VI [30). As V s equcharacterstc, the m V -adc comleto ˆV of V has a coeffcet feld L by Cohe s theorem, ad thus ˆV = L[[t s a ower seres rg oe varable over L = V/m V. We have V = K ˆV. The subtlety of ths statemet s that f k s a subfeld of K cotaed V such that V/m V s ot searably geerated over k, the there may ot est a coeffcet feld L of ˆV whch cotas k. Although the comleto of a rak valuato rg s a ower seres rg, ostve characterstc, the valuato determed by assocatg to a system of arameters secfc ower seres may ot be easly recogzable from a seres reresetato of the valuato rg. Ths ca be see from the cotrast of the coclusos of Theorem. wth

14 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) the results of ths secto. The fteess codto [L : k < of the coeffcet feld of a seres over a base feld k does ot characterze algebracty of a seres ostve characterstc, whle the corresodg fteess codto o resdue feld etesos does characterze algebracty the case of valuatos domatg a local rg of Theorem 4.. We llustrate ths dstcto the followg eamle. Eamle 4.4. The valuato duced by the seres of Eamle.3, whose coeffcet feld s ftely algebrac over the base feld k, has a resdue feld whch s fte over k. Proof. Wth otato of Eamle.3, we have a k-algebra homomorhsm R = k[u, v u,v) π k[[ defed by the substtutos u =, v = σ ) = t. = π s sce, y ad the t are algebracally deedet over k. The order valuato o k[[ duces a rak valuato ν o the quotet feld of R. Let v = u v ) t. R = R[ u v u,v ) s domated by ν. From the easo v = t +, = we ductvely defe ad v j+ = v j u t j+ = t + j = R j+ = R [ v j u u,v j+ ) for j. The R j are domated by ν for all j, so that V = j R j s the valuato rg of ν. We have that the resdue feld of V s V/m V = R /m R = kt ). Ths s a fte eteso of k, cotrast to the fact that the feld of coeffcets L = k{t N}) of σ ) has fte degree over k. A esecally strage reresetato of a rak dscrete valuato s gve by the eamle ) of a ower seres whose eoets have ubouded deomators. Let k be a feld of characterstc > 0, ad cosder the seres σ = = of ). σ s algebrac over k), wth rreducble relato σ σ = 0. Cosder the two-dmesoal regular local rg R 0 = k[, y,y). ad y are regular arameters R 0. Let y = σ ). We see from 9) that y does ot have a fractoal ower seres reresetato terms of. However, by eadg terms of y, we have a easo = y + y) 0) whch reresets as a fractoal ower seres y wth bouded deomators. Let g = y y R 0. g = 0 has a sgularty of order R. Let [ R = R y, y y,y). 9)

15 00 S.D. Cutkosky, O. Kashcheyeva / Joural of Pure ad Aled Algebra 008) = y ad y are regular arameters R. g = y g, where g = y y s a strct trasform of g R. g = 0 s osgular. From the equato g = 0 we deduce that y = ) = + + ) + ) = ), = obtag a stadard ower seres easo of y terms of. We obta a fractoal ower seres of terms of y wth bouded deomators ether from the equato g = 0, or by substtuto 0). Ackowledgemet The frst author was artally suorted by NSF. Refereces [ S. Abhyakar, Desgularzato of lae curves, AMS Proc. Sym. Pure Math. I) ) 45. [ S. Abhyakar, O the valuatos cetered a local doma, Amer. J. Math ) [3 S. Abhyakar, O the ramfcato of algebrac fuctos, Amer. J. Math ) [4 S. Abhyakar, Two otes o formal ower seres, Proc. Amer. Math. Soc ) [5 A. Behess, La Clôture algébrque du cors des séres formelles, A. Math. Blase Pascal 995) 4. [6 E. Breskor, H. Körrer, Plae Algebrac Curves, Brkhauser, Basel, 986. [7 C. Chevalley, Itroducto to the theory of algebrac fuctos of oe varable, : AMS, Mathematcal Surveys, vol. 6, Amer. Math. Soc., NY, 95. [8 S.D. Cutkosky, Resoluto of Sgulartes, : Graduate Studes Math., vol. 63, AMS, Provdece, Rhode Islad, 004. [9 S.D. Cutkosky, L. Ghezz, Comletos of valuato rgs, Cotem. Math ) [0 P.D. Gozález-Pérez, Decomosto buches of the crtcal locus of a quas-ordary ma, Comos. Math ) [ A. Grothedeck, J. Murre, The tame fudametal grou of a formal eghbourhood of a dvsor wth ormal crossgs o a scheme, : Lect. Notes Math., vol. 08, Srger Verlag, Hedelberg, Berl, New York, 97. [ H. Hah, Uber de chtarchmedsche Groössesysteme, : Gesammelte Abhadluge, vol. I, Srger, Vea, 995. [3 W. Hezer, J. Sally, Etesos of valuatos to the comleto of a local doma, J. Pure Al. Algebra 7 99) [4 W. Hezer, C. Rotthaus, S. Wegad, Geerc fber rgs of med ower seres/olyomal rgs, J. Algebra ) [5 M.F. Huag, Ph.D. Thess, Purdue Uversty, 968. [6 K. Kedlaya, The algebrac closure of the ower seres feld ostve characterstc, Proc. Amer. Math. Soc. 9 00) [7 J. Lma, Desgularzato of -dmesoal schemes, A. of Math ) [8 J. Lma, Quas-ordary sgulartes of surfaces C 3, : Sgulartes Part, Arcata, Calf., 98, : Proc. Symos. Pure Math., vol. 40, Amer. Math. Soc., Provdece, RI, 983, [9 M. Nagata, Local Rgs, Iterscece, 96. [0 U. Orbaz, Embedded resoluto of algebrac surfaces after Abhyakar, : V. Cossart, J. Graud, U. Orbaz Eds.), Resoluto of Surface Sgulartes, : Lect. Notes Math., vol. 0, Srger Verlag, Hedelberg, Berl, New York, 984. [ B. Pooe, Mamally comlete felds, Eseg. Math ) [ F. Rayer, A algebracally closed feld, Glasgow J. Math ) [3 P. Rbebom, Felds: Algebracally closed ad others, Mauscrta Math ) [4 P. Rbebom, L. Va de Dres, The absolute Galos grou of a ratoal fucto feld characterstc zero s a sem-drect roduct, Caad. Math. Bull ) [5 C. Rotthaus, Ncht Ausgezechete, Uversell Jaosche Rge, Math. Z ) [6 D. Stefaescu, O meromorhc formal ower seres, Bull. Math. Soc. Sc. Math. R.S. Roumae 7 983) [7 M. Svakovsky, Valuatos fucto felds of surfaces, Amer. J. Math. 990) [8 S. Vadya, Geeralzed Puseu easos ad ther Galos grous, Illos J. Math 4 997) 9 4. [9 O. Zarsk, The reducto of the sgulartes of a algebrac surface, A. of Math ) [30 O. Zarsk, P. Samuel, Commutatve Algebra Volume II, D. Va Nostrad, Prceto, 960.