UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) Jilong Tong

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1 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) Jlong Tong Contents Introducton Notatons and revews Generators of the R-algebra R[G] Geometry of unpotent groups Some explct models of G 2 a,k References Introducton The theory of unpotent algebrac groups, and n partcular that of commutatve unpotent algebrac groups, over a feld represents a very beautful theory (see for example [3] [10] [14] [16]), whch plays also an mportant role of n the study of algebrac groups over a feld. Hence we can also expect that over a general base scheme, a study of unpotent group scheme can gve applcatons n the study of famly of algebrac groups. On the other hand, famly of unpotent group arses also naturally n practce, and leads to nterestng questons of affne schemes. In Exposé XVII of [5], a general theory of unpotent groups over a feld or over a general base scheme s gven. Besdes of ths, one can stll fnd n Exposé XXVI of [6] some studes of famly of unpotent groups, but only n a very specal case, namely, the unpotent radcal of a parabolc subgroup of some reductve group. In [20], Dolgachev and Wesfeler proposed a theory of Key words and phrases. unpotent group scheme, p-polynomal scheme, models. c 1997, Socété MathéMatque de France

2 2 JILONG TONG unpotent groups n a more general settng. More precsely, the authors of loc. ct. consdered affne unpotent group schemes G flat over an affne ntegral scheme S = Spec(R) such that the generc fber s somorphc to an affne space as scheme, and they got many nterestng propertes about such group schemes. For example, f R s a dscrete valuaton rng, the authors were able to fnd a famly of good generators of the affne rng R[G] of G/S and determned all the relatons between these generators. When R s of equal characterstc p > 0 and f G K G n a,k, they proved that the group scheme G/S s a socalled p-polynomal S-scheme. Wth ths result and suppose moreover that G/S has smooth connected fbers, together wth some computatons wth the p-polynomals, the authors proved that the groups scheme G/S s somorphc to G n a,s after an eventual extenson of dscrete valuaton rngs of R. Besdes of these, one fnds n loc. ct. also many results concernng deformatons and cohomology of such group schemes. The present report s then an attempt to understand the papers [20] of Dolgachev-Wesfeler. Snce the majorty of the results n loc. ct. are based on the assumpton that the base scheme S s the spectrum of a dscrete valuaton rng, n ths report, we wll work manly over such a base. Ths report contans no orgnal result, and every statement n ths report s contaned n [20] (or [3], [11]), though n some places, the treatments gven here are slghtly dfferent from the orgnal ones n [20]. But of course, I am responsble for any error n ths report. Ths s the expanded verson of a talk gven n the summer school n Lumny organzed n the occason of the reprnt of SGA3. The author wants to thank P. Glle for the knd nvtaton. Durng the preparaton of ths notes, the author benefts from the communcatons wth M. Raynaud, Q. Lu and D. Tossc, he thanks them sncerely. 1. Notatons and revews 1.1. Notatons and conventons Unless mentoned explctly, n ths report, the letter R denotes always a dscrete valuaton rng, wth K ts fracton feld and k t resdue feld. Moreover, we note by S the spectrum of R Let X K be a scheme of fnte type over K. In ths report, a model of X K over S wll be a flat S-morphsm of fnte type X/S whose generc fber s X K For m, n Z tow ntegers such that m n, we wll denote by [m, n] the set {m, m + 1,, n} Z.

3 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) Let [1, n], and for any r Z 0, we wll denote by m(, r) = (0,, 0, r, 0,, 0) Z n 0 where the nteger r s located n the -th component For G an affne group scheme over an affne scheme Spec(A), we denote by A[G] the functon rng of G, and by µ : A[G] A[G] A A[G] ts map of comultplcaton. Moreover, we denote by η : A[G] A[G] A A[G] the morphsm obtaned by x µ(x) x 1 1 x Unpotent groups over a feld: defntons and examples Let k be a algebracally closed feld, and G be a group scheme over k. Recall that the group scheme G s unpotent f t verfes one of the followng two equvalent condtons: G has a central composton seres of the form 0 = H 0 H 1 H r 1 H r = G whose successve quotents H /H 1 for = 1, 2,, r are somorphc to an algebrac subgroup of G a,k ([5] Exposé XVII, Défnton 1.1). G s affne, and n ts functon rng k[g], there exst generators t 1,, t n of k[g] as k-algebra such that the comultplcaton map µ verfes µ(t ) = t t + j a j b j, = 1,, n wth a j, b j k[t 1,, t 1 ] ([15] Chap. VII 1.6, Remarque 2) More generally, untl the end of 1.2, let k be an arbtrary feld, wth k an algebrac closure of k. A group scheme G/k s called unpotent f the k-group scheme G k := G Spec(k) Spec( k) s unpotent n the sense of When the group scheme G/k s smooth and connected group scheme, then the condton that G/k s unpotent s also equvalent to the followng condton ([5] Exposé XVII, Proposton 4.1.1): G has a composton seres whose successve quotents are forms of G a,k. In partcular, once the base feld k s perfect, snce there s no non trval form of G a,k over k (loc. ct., Lemme 2.3 bs), G has a composton seres whose successve quotents are somorphc to G a,k. As a result, ts underlyng scheme s somorphc to some affne space A n k. More generally, wthout the perfectness of the feld k, we have the followng result due to Lazard.

4 4 JILONG TONG Proposton 1.1 (Lazard [3] IV 4 n 4, Théorème 4.1) Let G/k be an affne k-group scheme. The followng three assertons are equvalents: There s an somorphsm of k-schemes G A n k wth n = dm(g); G has a composton seres wth successve quotents somorphc to G a,k ; G s reduced and solvable. Moreover, there exsts nteger N 1 and a domnant morphsm of k-schemes A N k G. Defnton 1.2 ([17] Chapter IV Defnton 4.1.2) A connected k-unpotent group scheme G/k s call k-splt (or splt over k), f G/k verfes one of the three equvalent condtons n Proposton 1.1. Hence, for G/k splt over k of dmenson n, one can fnd generators x 1,, x n of k[g] such that the comultplcaton map satsfes µ(x ) = x x + j a j b j wth a j, b j k[x 1,, x 1 ]. In the followng, such a famly of generators of k[g] wll be called prmtve. In ths report, we are manly nterested n such unpotent groups and ther affne models over a dscrete valuaton rng Some examples of unpotent groups. Let G be a smooth connected unpotent group over a feld k of characterstc p > If G s of one dmensonal, then G s a form of the addtve group G a,k. Hence f k s perfect, G G a,k. But over an mperfect feld, there exsts non trval form of G a,k. For example, let a k k p, and consder the followng closed subgroup scheme of G 2 k = Spec(k[x, y]) defned by the followng equaton x + x p + ay p = 0 whch can be trvalzed by the nseparable extenson k k(a 1/p ). 2. For r Z 1, let ( p r Φ r (X) = 1 p p r 1 =1 ) X X pr Z[X] Z[X]. We consder the k-algebra of polynomals n two varables k[x, y], and defne the followng map µ : k[x, y] k[x, y] k[x, y] by: µ(x) = x x, µ(y) = y y + r 1 a r Φ r (x) for a r k such that a r = 0 for almost all r. We verfy that ths gves a structure of Hopf algebra on k[x, y], and the group scheme obtaned n ths way s an extenson of Spec(k[x]) = G a,k by Spec(k[x, y]/(x)) = G a,k, whch s also commutatve. Conversely, any k-splt two dmensonal

5 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER)5 (connected) commutatve unpotent group s gven by such formulas ([3] II théorème). Fnally, we refer to [10] for a detal dscusson of unpotent groups over general felds. See also [3] [14] and [16] Characterstc zero case. Let S be an arbtrary noetheran base scheme, and G/S be a flat S-group scheme of fnte type. Recall that G/S s unpotent f the geometrc fbers of G/S are unpotent groups n the sense of 1.2. In ths secton, we wll brefly revew the results of unpotent group schemes when the base S s of characterstc zero,.e., s a scheme over Q. Under ths assumpton, accordng to a result of Carter, the group scheme G/S has smooth fbers, and hence G/S s smooth by the flatness of G/S. Moreover, over an algebracally closed feld of characterstc zero, the only subgroup scheme of G a s (0) and G a tself ([5] Exposé XVII Proposton 1.5), the group scheme G/S has hence connected fbers. As a result, the group scheme G/S s separated ([4] Exposé VI Corollare 5.5) Exponental maps. Let S be a noetheran scheme of characterstc zero, and G be a group scheme flat of fnte type over S. Let Spec(R) be an affne scheme over S, we wll denote by an element of G(R[[T ]]) (resp. of G(R[T ])) by the functonal symbol lke f(t ). For any complete lnear topology R-algebra A (resp. any R-algebra A), and any element t A whch s topologcally nlpotent (resp. any element t A), we denote by f(t) G(A) the mage of f(t ) G(R[[T ]]) n G(A) (resp. of f(t ) G(R[T ]) n G(A)) by the canoncal map G(R[[T ]]) G(A), whch s nduced by R[[T ]] A sendng T to t A (resp. by R[T ] R sendng T to t A). Proposton 1.3 ([3] II 6 n 3). 1. Pour each x Le(G R ) = ker(g(r[ε]) G(R)), there exsts a unque element e T x G(R[[T ]]) such that e εx = x G(R[ε]); e (T +T )x = e T x e T x G(R[[T, T ]]). Moreover, let x, y Le(G R ) be two elements verfyng [x, y] = 0, then e T (x+y) = e T x e T y. 2. Let V be a vector bundle on S, and G = GL(V ). Then for any element x Le(G R ) = End R (V OS R), we have e T x = 0 T x! GL(V OS R[[T ]]) In partcular, f x End R (V OS R) s nlpotent, we have e T x G(R[T ]).

6 6 JILONG TONG Corollary 1.4 ([3] II 6 n 3 Corollare 3.5). Let ρ: G GL n (V ) be a fathful representaton of G on a vector bundle V of fnte rank over S. If x Le(G R ) End(V R ) s nlpotent. Then e T x G(R[T ]) G(R[[T ]]). Let G/S be a unpotent group scheme whch can be realzed as subgroup scheme of some GL(V ) for some vector bundle V on S by the morphsm ρ: G GL(V ). Ths latter morphsm nduces a morphsm between ther Le-algebras: Le(ρ): Le(G) Le(GL(V )) = End(V ) Snce G/S has unpotent fbers and the base S s noetheran, the mage of the prevous map Le(ρ) s contaned n the set of nlpotent endomorphsms of V. In partcular, we can apply the constructon that we sketched here. Hence, for any affne scheme Spec(R) over S, and for any x Le(G R ), we have e T x G(R[T ]). Now, we consder the morphsm of R-algebras R[T ] R sendng T to 1 R, and the nducng morphsm of groups G(R[T ]) G(R) we get hence an element e x := e x 1 G(R), and hence a map Le(G R ) G(R). More generally, we get n such a way a morphsm of S-schemes (called the exponental map of the unpotent group scheme G/S): (1) exp: W(Le(G)) G, x e x. whose geometrc fbers are somorphsms of schemes ([3] IV 2 n 4 Proposton 4.1). Snce the two S-schemes are smooth of fnte type, ths mples then the exponental map (1) s an somorphsm of S-schemes Exponental map over a normal base. Wth the help of [11], t s possble to extend the constructon of exponental map n the prevous to unpotent group schemes over a general normal base scheme. Recall that S of characterstc zero, and for G/S a (flat) unpotent group scheme, ts Le-algebra Le(G) s nlpotent. Hence, by usng the Baker-Campbell-Hausdorff formula, W(Le(G)) becomes an S-group scheme whch s n general not commutatve (see for example [2] Chapter II 6.5 Remark (3)). Proposton 1.5. Let S be a noetheran normal scheme of characterstc zero, and G/S be a flat unpotent group. Then there s a unque morphsm of S-schemes exp: W(Le(G)) G whch can be characterzed by the followng condtons: (a) exp sends the zero secton of W(Le(G)) to the neutral element e of G/S; (b) exp(n x) = exp(x) n G for any local secton x of W(Le(G));

7 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER)7 (c) The nduced map of exp between the tangent spaces s dentty. T 0 (exp): Le(G) T 0 (W(Le(G))) T e (G) = Le(G) Moreover, f we mpose the group scheme structure on W(Le(G)) gven by the Baker-Campbell-Hausdorff formula as n [2] (Chapter II 6.5), the exponental map above s an somorphsm of S-group schemes. Proof. Snce G/S s separated and W(Le(G)) s flat over S, to prove the unqueness, we only need to verfy the correspondng statement over the generc pont of S. Hence, we are reduced to the case where S = Spec(K) s the spectrum of a feld of characterstc zero. By Galos descent, we can even assume that K s algebracally closed. Let f, g two morphsm of S- schemes verfyng the three propertes (a)-(c) of the proposton. Snce K s algebracally closed, we only need to show that f(x) = g(x) for for any x Le(G) = W(Le(G))(K). Let g = K x Le(G) be the lnear subspace generated by such a x Le(G). Ths gves a Le-subalgebra of Le(G). We consder then the compostons f, g of f, g wth the followng canoncal map W(g) W(Le(G)). Accordng to our assumpton on f and g, the two maps f and g nduces the same morphsms between the tangent spaces on 0, and also verfy f (ny) = f (y) n, g (ny) = g (y) n y W(g). As a result, both maps are morphsms of groups schemes over S. Hence, the kernel H of the double morphsm W(g) f g G s an algebrac subgroup of W(g). On the other hand, snce f, g have the same nduced map bwtween the tangent spaces, ths mples that Le(H) = Le(W(g)) = g. Snce K s of characterstc zero, we get H = W(g). Hence f = g. In partcular, f(x) = g(x), as s requred. It remans to prove the exstence of such a morphsm under our assumpton on the base. Recall that by the Baker-Campbell-Hausdorff formula, W(Le(G)) becomes a smooth group scheme over S wth connected fbers. We wll frst construct the exponental map when the base S s local normal of dmenson one. In ths case, on applyng the Lemma IX 2.2 of [11], G/S can be realzed as a closed subgroup scheme of GL n for some nteger n. Hence one can construct the exponental map as n exp: W(Le(G)) G.

8 8 JILONG TONG By the Proposton 1.3, the morphsm of schemes exp verfes the propertes (a)-(c) of the proposton. To see that exp s a morphsm of groups, snce our group scheme G/S s separated and W(Le(G)) s flat over S, we are reduced to show the smlar result between the generc fbers, whch s then well-known ([3] IV 2, 4.3). Fnally, we proceed to the general case. Accordng to the Corollary IX 1.4 of [11] and the fact that W(Le(G))/S s smooth wth connected fbers, we only need to construct the morphsm exp over an open subset W S contan every pont of S of depth 1. Hence, we are reduced to prove that for any pont s S of depth 1, there exsts some open subset V S contanng s, and a morphsm W(Le(G W )) G W nducng the usual exponental map on the generc fbers. Hence, up to replace S by ts localzaton at s S, we are reduced to the prevous case. Ths fnshes the proof. 2. Generators of the R-algebra R[G] In ths part, S = Spec(R) s the spectrum of a dscrete valuaton rng. Let G be a flat affne group scheme of fnte type over S such that ts generc fber G K s K-splt n the sense of Defnton 1.2. The am of ths secton s to explan the exstence of good generators of R[G] as an R-algebra. Let us begn wth some preparatons on the lexcographc order defned on the rng of polynomals Lexcographc orders. Let n > 0 be an nteger, we consder the rng K[x 1, x 2,, x n ] of polynomals n n varables wth coeffcents n K. To denote a monomal x α 1 1 xα 2 2 xαn n, we wll use the usual abbrevaton x n, where x = (x 1,, x n ) and α = (α 1,, α n ) Z n 0. For any α, α Z n 0, we ntroduce the so-called lexcographc order: α > α f there s some nteger [1, n] such that α > α and that α j = α j for any j [ + 1, n]. In partcular, ths gves a total order on Z n 0, and (1, 0,, 0) s the mnmal element dfferent from 0 = (0, 0,, 0) n Z n 0. In ths report, the maxmum max s always taken relatve to ths order. Lemma 2.1. The lexcographc order of Z n 0 satsfes the decreasng chan condton: for α Z n 0 such that (2) α 0 α 1 α α +1, there exsts some 0 Z 0 suffcently large such that α = α 0 for any 0. Proof. Let α = (a,1, a,2,, a,n ). Accordng to the defnton of lexcographc order, we must have the followng descendng sequence of non negatve ntegers: a 1,n a 2,n a,n

9 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER)9 As a result, for some 0 0, and for any 0, we must have a,n = a 0,n. Up to remove all the frst 0 terms of the sequence (2), we may assume that a,n = a 1,n for all 0. Hence, one must have a 1,n 1 a 2,n 1 a,n 1 The same argument shows that up to remove agan fntely many frst terms n the sequence (2), we may assume further more that a,n 1 = a 1,n 1 for all 0. After at most n repettons of ths argument, and up to remove fntely many terms of the sequence (2), we fnd that α = α 0 for any 0. Ths gves the lemma. Corollary 2.2. Any non empty subset of Z n 0 has a mnmal element relatve to the lexcographc order. Defnton 2.3. For any polynomal f = α a αx α K[x 1,, x n ], we defne ts degree, denoted by deg(f), by the element of Z n 0 gven by the followng formula deg(f) = max{α a α 0}. Now, let X be an affne model (1.1.2) of the affne space A n K = Spec(K[x 1, x 2,, x n ]), wth R[X] ts affne rng, whch s an R-algebra of fnte type. From the flatness of R[G] over R, we can vew R[G] naturally as an R subalgebra of K[x 1, x 2,, x n ]. Hence, for any f R[X], we have the well-defned noton deg(f). Followng [20], for any α Z n 0, we defne P α = {f R[X] deg(f) α}, and P α = {f R[X] deg(f) < α}. We have P α P α, ts cokernel wll be denoted by P α. Lemma These three R-modules are torson free. 2. dm K (P α A K) = Suppose that the orgn o K = (0, 0,, 0) A n K can be extended to a S- secton of X/S. Then the R-module P α s also of fnte type. In partcular, the R-module P α s free of rank 1 over R. Proof. Only the thrd statement needs a verfcaton. We wll ntroduce some notatons for ths proof: for any α = (α 1,, α n ) Z n, we put n α = α. =1 and defne d(f) = mn{ α : a α 0} for a polynomal f. In partcular, d(f) 0 wth equalty holds f and only f the polynomal f has non zero constant term.

10 10 JILONG TONG Now, let us begn the proof. Snce R[X] s an R-algebra of fnte type, there exst f 1, f 2,, f m K[x 1,, x n ] such that R[X] = R[f 1,, f m ]. We clam that the constant term of each f s contaned n R. Accordng to our assumpton, the orgn o K of X K = A n K can be extended to an S-secton of X/S, hence there exsts an epmorphsm of R-algebras ρ : R[X] R, such that ts nduced map on the generc fber s the epmorphsm of K-algebras gven by ρ K : K[x 1, x 2,, x n ] K, x 0. If we wrte f = c + f such that c K and f (x 1, x 2,, x n ), we fnd that ρ K (f ) = ρ K (c ) = c. On the other hand, snce ρ K (f ) = ρ(f ) R K, we must have c R. Hence, up to modfy f by f c, we may assume that each f has zero constant term. In partcular, d(f ) > 0. Now as an R-modules, R[X] s generated by the elements f t := f t 1 1 f t 2 2 f m tm for f = (f 1,, f m ) and t = (t 1,, t m ) Z m 0. For each t Zm 0, let λ t be the coeffcent of the term x α of the polynomal f t K[x 1,, x n ], et Λ K be the R-submodule generated by these coeffcents. Then Λ s also the set of coeffcents of the term x α of the elements n R[G]. Snce d(f t ) = m =1 d(f ) t, we fnd that λ t = 0 once the followng nequalty s satsfed: m t d(f ) > α. =1 Snce the complement of the elements t Z m 0 verfyng the nequalty above s fnte (because d(f ) > 0 for all ), there s only fntely many t Z m 0 such that λ t 0. As a result, the R-module Λ s of fnte type over R. Now we consder the subset Λ of K formed by the element λ K such that, there exst an element g P α wth λ as ts coeffcent of the term x α. Ths s a R-submodule of K, whch s also contaned n Λ. Hence Λ s also of fnte type over R, ths means exactly that the R-module P α s of fnte type. Ths fnshes the proof Lnear unpotence of affne group models of a unpotent group Let G/S be an affne flat group scheme of fnte type over S, such that ts generc fber s splt (Defnton 1.2). In partcular, we can fnd x 1,, x n K[G K ] such that K[G K ] = K[x 1, x 2,, x n ] and that (3) µ(x ) = x x + j a j b j wth µ the comultplcaton map, and a j, b j K[x 1,, x 1 ]. In the followng, we wll fx once for all such a prmtve famly of generators of K[G K ].

11 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 11 Snce G/S s a affne group scheme and because of the choce of the prmtve famly {x 1,, x n }, the zero secton of G K = Spec(K[x 1,, x n ]) can be extended to a secton of G over S (.e., the neutral element of G/S), the S- scheme satsfes hence the assumpton of Lemma 2.4 (3). In partcular, for each α Z n 0, the correspondng R-module P α s free of rank 1. For each α Z n 0, let z α P α an element whose mage n P α gves a bass over R. Then R[G] s a free R-module wth a bass gven by the famly {z α : α Z n 0 }. Moreover, as an R-module, R[G] R R[G] s free wth a bass gven by {z α z β : α, β Z n 0 }. Ths famly gves also a bass of K[G K ] K[G K ] over K, and an element a α,β z α z β K[G K ] K[G K ], a α,β K α,β s contaned n R[G] R[G] K[G K ] K[G K ] f and only f the coeffcents a α,β R for any α, β Z n 0. In the followng, to smplfy the presentaton, we wll use the followng notatons: let α Z n 0, we put M := R[G] R R[G], M K = M R K = K[x 1,, x n ] K K[x 1,, x n ]; M K,α M K (resp. M K,α ) the subspace generated over K by the elements a b verfyng the followng propertes: deg(a) < α, deg(b) < α, and deg(a) + deg(b) α (resp. deg(a) + deg(b) < α). M α := M K,α M, and M α := M K,α M. For any a b M K, we defne deg(a b) := deg(a) + deg(b). Lemma 2.5. Let µ : R[G] R[G] R[G] the comultplcaton map of the group scheme G/S. Then for any y R[G], we have µ(y) y y mod M deg(y). Proof. Let α 0 = deg(y). Accordng to the formula (3), we have µ(y) (y y) = l a l b l K[x 1,, x n ] K[x 1,, x n ]. such that the followng two condtons are satsfed: (1) deg(a l ) < α 0, deg(b l ) < α 0 ; (2) deg(a l ) + deg(b l ) α 0. Snce z α P α whch generates also P α R K over K, there exst c α,β K such the followng equalty holds: a l b l = c α,βz α z β l α<α 0,β<α 0 α+β α 0

12 12 JILONG TONG On the other hand, snce y R[G], µ(y) y 1 1 y R[G] R[G]. Moreover, snce the famly {z α z β : α, β Z n 0 } s a bass of the free R- module R[G] R[G], the coeffcents c α,β n the prevous equalty must le n R. From ths, we get the concluson. Corollary 2.6. Let α = (t 1, t 2,, t s, 0,, 0), β = (0,, 0, t s+1, t n ) two elements n Z n 0. Then the mage of z α z β P α+β n P α+β gves also a bass of ths free R-module of rank 1. Proof. By the Lemma 2.4 (), there exsts a R such that az α+β z α z β P α+β R[G]. We only need to prove that a R. Accordng to the prevous Lemma 2.5, we have η(az α+β z α z β ) M α+β. On the other hand, snce z α+β R[G], t follows that η(z α+β ) R[G] R[G] = M, hence Now 1 a η (z αz β ) = η(z α+β ) + 1 a η(z αz β az α+β ) M + M K,α+β. η(z α z β ) = µ(z α )µ(z β ) z α z β 1 1 z α z β = (z α z α + η(z α )) (z β z β + η(z β )) z α z β 1 1 z α z β = z α z β + z β z α + (z α z α ) η(z β ) +(z β z β ) η(α) + η(α) η(β). By the assumptons on α and β, for any α α, and any β < β, we have α + β < β, and snce η(z α ) M α η(z β ) M β, the followng sum (z α z α ) η(z β ) + (z β z β ) η(α) + η(α) η(β). does not contan the term z α z β. As a result, the coeffcent of the term z α z β n η(z α z β ) s equal to 1. Hence, n order that 1 a η(z αz β ) M + M K,α+β, t s necessary that 1 a R. Ths proves that a R s a unt Snce R[G] s fntely generated as an R-algebra, there exst fntely many α 1,, α t such that R[G] s generated by these z α (1 t) as an R-algebra. In fact, we can do better here. Defnton 2.7. We defne by nducton a famly of element {y } R[G] verfyng the followng two propertes: () y 1 = z (1,0,,0) ; () for 1, y +1 s the frst z α such that deg(z α ) > deg(y ) and z α / R[y 1, y 2,, y ] (here f such a z α does not exst, then we stop and get a fnte famly). Lemma 2.8. The constructon n 2.7 stops after fntely many steps.

13 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 13 In order to prove ths lemma, we need some preparatons. Recall that we have chosen {x 1,, x n } s a prmtve famly of generators of K[G K ]/K, hence for any nteger r [1, n], the K-scheme H K = Spec(K[x 1,, x r ]) has a group scheme structure. Moreover, the canoncal njecton K[x 1,, x r ] K[x 1,, x n ] gves us a surjectve morphsm of K-algebrac groups: G K H K. Its kernel s the subgroup scheme F K of G K defned by the deal (x 1, x 2,, x r ) K[x 1,, x n ]. Let F be the schematc closure of F K n G, whch s a flat fnte type subgroup scheme of G K. Lemma 2.9. Wth the notatons as above. 1. The quotent G/F s representable by a group scheme H flat affne of fnte type over S, and ts generc fber s H K = Spec(K[x 1,, x r ]). 2. If we dentfy R[H] as a subalgebra of K[H K ] whch tself s contaned n K[x 1,, x n ], then R[H] = R[G] K[x 1,, x r ]. In partcular, the last ntersecton s of fnte type over R as an R-algebra. Proof. The frst asserton (1) follows from the general result of Artn and Raynaud. More precsely, accordng to 8.4/9 of [1], the fppf quotent H = G/F s representable by an algebrac space over S. Snce F G s a closed subgroup scheme over S, t follows that the quotent H s separated. By by a theorem of Raynaud (Théorème of [12]), ths algebrac space s representable by an S-group scheme, necessarly flat and of fnte type over S. Snce our bass S s normal of dmenson 1, the affneness of H/S follows then from the Lemma IX 2.2 of [11] snce the generc fber H K, beng a quotent of an affne K-group scheme G K, s affne. For (2), snce H = G/F, H s the cokernel of the double morphsm F G pr 2 where pr 2 : F G G s the projecton to the second factor, and m : F G G s the multplcaton map. As a result, a morphsm h : G A 1 S can factor through the surjecton G F f and only f h pr 2 = h m. Snce G/S s separated, the last condton s equvalent to the correspondng equalty n the generc fber: h K pr 2,K = h K m K, whch s agan equvalent to the fact that h K : G K A 1 K can factor through H K. Hence, f we dentfy K[H K ] and R[G] as subset of K[G K ] = K[x 1,, x n ], h K[G K ] les n R[H] f and only f h R[G] K[H K ]. Ths last statement means exactly R[H] = R[G] K[H K ], and hence fnshes the proof of the lemma. m G

14 14 JILONG TONG Proof of Lemma 2.8. Let {y 1, y 2, } be a famly (fnte or nfnte) produced by the constructon n Defnton 2.7. Snce R[G] s fntely generated over R, to prove the lemma, we only need to verfy that R[G] = R[y 1, y 2, ]. Ths follows that there exsts some nteger N such that R[G] = R[y 1,, y N ], hence the constructon n Defnton 2.7 must stop after fntely steps. We wll prove the last statement by nducton on n. The case where n = 1 s clear. Supposons now the lemma has been proven for the nteger n 1 1. We consder the subalgebra R[G] K[x 1,, x n 1 ]. Accordng to Lemma 2.9, ths subalgebra s fntely generated wth generc fber K[x 1,, x n 1 ], whch s also the affne rng of the unpotent group H/S. Moreover, H K A n 1 K. Hence the constructon n Defnton 2.7 applyng to ths subalgebra, together wth nducton hypothess, yeld fntely many elements y 1, y 2,, y s R[G] K[x 1,, x n 1 ] such that R[y 1,, y s ] = R[G] K[x 1,, x n 1 ]. The next element y s+1 {y 1, y 2, } gven by the contructon s then of degree (0,, 0, 1). We wll prove by nducton that the elements y s+1, y s+2, (f exst) are all of the form (0,, 0, ). Suppose that we have shown ths asserton for y wth s + 1. If R[y 1, y 2,, y ] = R[G], then there s nothng to prove. So we suppose n the followng that R[y 1, y 2,, y ] R[G]. Now let r Z 2 be the frst nteger such that z (0,,0,r) / R[y 1,, y s, y ]. We clam that, for α = (t 1,, t n 1, t n ) < (0, 0,, r), we have P α R[y 1,, y ]. Indeed, let β = (t 1,, t n 1, 0), then α β = (0,, 0, t n ) wth t n < r. Hence z β R[G] K[x 1,, x n 1 ] = R[y 1,, y s ], and z α β R[y 1,, y s, y ] (snce t n < r). Moreover, accordng to Corollary 2.6, the mage of z β z α β n P α generates the R-module P α. Hence, for any f P α, there exsts a R such that f az α β z β P α wth az α β z β R[y 1,, y ]. Now by repeatng ths argument and takng account Lemma 2.1, we fnd that f R[y 1,, y ]. Hence z (0,,0,r) s the frst z α / R[y 1,, y ], and we have y +1 = z (0,,0,r). Fnally, because of ths and for the reason of degree, we fnd R[G] = R[y 1,, y s, y s+1, ]. Ths fnshes the proof. Corollary One can fnd a fnte famly {y 1, y 2,, y N } R[G] verfyng the followng propertes: 1. R[G] = R[y 1,, y N ]; 2. deg(y 1 ) < deg(y 2 ) < < deg(y N ); 3. For each 0 P deg(y ) R[y 1,, y 1 ] (here, by conventon, let y 1 = 0). Remark In the followng, a famly of generators n Corollary 2.10 s sad to be a famly of good generators. Clearly, good generators are not unque. For example, let {y : 1 N} a famly of good generators, and let y = λ y + f, for = 1,, N,

15 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 15 wth λ R and f R[y 1,, y 1 ], then {y : 1 N} s stll a famly of good generators. Conversely, any two famles of good generators can be related n the prevous way. Recall that an affne group scheme U/S s called lnearly unpotent f there exst generators u 1,, u s of R[U]/R such that µ(u ) = u u + j λ j a j b j wth a j, b j R[u 1,, u 1 ]. Wth ths termnology, we have Theorem Let G be an affne group model over R of a unpotent group, where the generc fber of G s splt over K (Defnton 1.2). Then G s lnearly unpotent. In partcular, G/S s a unpotent group scheme. Proof. We consder a famly of good generators {y 1,, y N } gven n Corollary For each y, accordng to Lemma 2.5, we have µ(y ) = y y + a j b j R[G] R[G], wth deg(a j ) < deg(y ) and deg(b j ) < deg(y ). Hence by the propertes of the generators {y 1,, y N }, we fnd that a j, b j R[y 1,, y 1 ]. Ths mples exactly that G/S s lnearly unpotent Relatons between the good generators Statement of the man result. We use the notatons as n the prevous. Moreover, for r [1, N], we set Ω r = {j : y j K[x 1,, x r ]}, Ω 0 =, Ω r = Ω r Ω r 1, Ω r = [1, t r+1 1] I = {t 1 = 1, t 2,, t n }, I = [1, N] I, N = t n+1 1. If t Ω, then we put ω(t) =. Proposton Keepng the notatons as above. 1. If t I, then there exst a t K {0}, and an element f t K[y 1,, y t 1 ] such that y t = a t x ω(t) + f t. 2. If t I, then there exsts d t Z 1 such that π dt y t R[y 1,, y t 1 ], π dt 1 y t / R[y 1,, y t 1 ].

16 16 JILONG TONG Proof. If t I, y t s then the frst among the famly {y 1,, y N } whch s contaned n K[x 1,, x ω(t) ]. Hence, ts degree must be m(ω(t), 1). Let a t K be the leadng coeffcent of y t, then y t a t x ω(t) K[x 1,, x ω(t) 1 ]. Now we only need to use the fact that R[y 1,, y t 1 ] B K = K[x 1,, x ω(t) 1 ] to get the asserton (1). Suppose now t Ī, let r I be the bggest nteger such that r < t. Then we have R[y 1,, y r ] R[y 1,, y t 1 ] R[y 1,, y t ]. Moreover, by the frst asserton, the generc fbers of these three R-algebras are all equal to K[x 1,, x ω(t) ]. In partcular, R[y 1,, y t 1 ] R K = R[y 1,, y t ] R K. As a result, for some nteger d 0, we have π d y t R[y 1,, y t 1 ]. The nteger d t that we need n the second statement s then the mnmal non negatve nteger d wth ths property. Moreover, snce R[y 1,, y t 1 ] R[y 1,, y t ], we fnd d t 1. Ths fnshes the proof. The man result of ths secton can be stated as follows: Theorem Keepng the notatons as before, and let p = char(r/π) 0. Then we can fnd a famly of good generators {y 1,, y N } such that π d y = y pr() 1 + a,α y α, Ī wth the followng two propertes: (a) r() Z 1 for all Ī; (b) The sum α<m( 1,p r() ) α<m( 1,p r() ) a,α y α has coeffcents a,α R, and t s reduced wth respect to the ntegers r() ( Ī) n the sense of the Defnton 2.15 below. (c) π d p, where d Z s the nteger n Proposton Moreover, these relatons are the only ones between these good generators Prelmnary of the proof: reduced wrtten forms. The noton of reduced wrtten form s used to produce some nce bass of the free R-modules R[G] and R[G] R R[G]. In ths, let B be an R-algebra of fnte type wth generators u ( [1, t]). We assume that [1, t] = I Ī wth 1 I. For each Ī, we assocate t wth a number r() Z 1.

17 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 17 Defnton Wth the notatons as before. element b B b = a α u α A wrtten form of an α Z t 0 s called reduced (wth respect to the ntegers r() for Ī) f for α = (α 1,, α t ) from a α 0, t follows that for all [1, t 1] such that + 1 Ī. α < p r(+1) Suppose further more that B s the quotent of the polynomal algebra R[U 1,, U t ] modulo the deal generated by the followng elements π d U U pr() 1 + a β U β, for Ī β<m( 1,p r() ) where a β R, and the sum β<m( 1,p r() ) a β U β R[U 1,, U 1 ] s reduced n the sense of Defnton 2.15 wth respect to the ntegers r() (for Ī), and we wll denote by u the mage of U n the quotent B. Lemma Wrte I = {1 = 1 < 2 < < r }, let v j = u j. () The generc fber B K := B R K of B s the polynomal rng n r varables v 1, v 2,, v r wth coeffcents n K. () Let b = u α B be a monomal wth α = (α 1,, α t ). Let b K be ts mage n B K, and deg(a K ) = (δ 1,, δ r ) the degree of b K wth respect to the varables v 1,, v r. Then we have δ q = α p r(j) q < q+1 q<j () Let a = u α and b = u β be two reduced monomals n B = R[u 1,, u t ]. Then f α < β, we have deg(a K ) < deg(b K ). Proof. The frst asserton s clear from the defnton of B. In order to prove (), wthout loss of generalty, we may assume that I = {1} [1, t]. Let u α be a monomal, or more generally, we consder a polynomal (4) a β u β satsfyng the followng two propertes: β

18 18 JILONG TONG (a) α := max{β : a β 0} = (α 1,, α t + p r a) for some a 0 r 0, and (α 1,, α t ) Z t 0 ; (b) For each β = (β 1,, β t ) such that a β 0, there exst ntegers b 1,, b t 0 verfyng b b t = a, and such that β (α 1 + b 1 p r(2),, α t 1 + b t 1 p r(t), α t + b t p r ) (1) We only need to reduce the sum (4) nto a polynomal n u 1, and compute ts degree. To do ths, we wll frst reduce the sum (4) nto an expresson whch nvolves only u 1, u t 1 by replacng n (4) the varable u t by π dt u pr(t) t 1 π dt δ<m(t 1,p r(t) ) a tδ u δ Now we clam that after ths reducton, the new sum n u 1,, u t 1 satsfes agan the smlar propertes (a) and (b). More precsely, let u γ be a monomal appearng after applyng the reducton to u β for β = (β 1, β 2,, β t ), then snce the sum a tδ u δ δ<m(t 1,p r(t) ) s reduced, there exst c 1,, c t 1 such that t 1 =1 c = β t, and that Hence, γ (β 1 + c 1 p r(2), β 2 + c 2 p r(3),, β t 1 + c t 1 p r(t), 0) γ (α 1 + (c 1 + b 1 )p r(2),, α t 1 + (c t 1 + b t 1 )p r(t), 0) The hghest term of the new sum s of degree ( ) α 1,, α t 2, α t 1 + (α t + ap r ) p r(t), 0 Hence, to fnsh the proof, we only need to show that But, snce we fnd t 1 (b + c ) α t + a p r j=1 t b j = a, j=1 and t 1 c j = β t α t + b t p r j=1 t 1 (b j + c j ) a b t + α t + b t p r j=1 (1) For γ = (γ 1,, γ n), δ = (δ 1,, δ n) Z n 0, we defne γ δ f γ δ for all [1, n].

19 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 19 So, we are reduced to show that or equvalently α t + b t p r + a b t α t + ap r a b t (a b t )p r whch s clear. Hence, after repeatng ths reducton t 1 tmes to the monomal u α, we get a polynomal n u 1 wth degree gven by the formula t =1 α 2 j t p r(j) As s requred. Now the proof of () s mmedate from the degree formula. Indeed, let u α and u β be two reduced wrtten form. Let [1, t] such that α > β, and α j = β j for all j. Moreover, suppose q < q+1. Let δ and γ be the degree of u α and u β. We clam that we have δ q > γ q, and δ q = γ q q > q. Accordng to the degree formula, for q > q, the number δ q (respectvely for γ q ) only nvolves α j (respectvely β j ) for j q >, hence accordng to the defnton of the nteger, on must have δ q = γ q. We only need to show δ q > γ q. Snce α > β, and the monomals u α u β are reduced, hence δ q γ q = (α j β j ) p r(j ) q j< q+1 q<j j = (α j β j ) p r(j ) q j q<j = q j = 1 > 0 p r(j) Hence δ q < γ q, ths proves (). q<j j q j< q<j j p r(j ) (p r(j+1) 1) q+1 j q<j j q<j j p r(j ) p r(j ) Proposton Wth the notatons as before. 1. Any element b B has a unque reduced wrtten form n B. 2. B s a flat R-algebra.

20 20 JILONG TONG Proof. We only need to show the exstence of the reduced wrtten form for any element b B. Indeed, let b B wth two reduced wrtten form b = a α u α and b = a αu α α α Defne α 0 = max{α : a α 0}, and α 0 = max{α : a α 0}. Accordng to Lemma 2.16 (), we have deg(b K ) = deg(u α 0 K ), and deg(b K) = deg(u α 0 K ). Applyng agan Lemma 2.16 (), we fnd α 0 = α 0. Now, we consder the element b a α 0 u α 0, t has then the followng two reduced wrtten form b a α 0 u β 0 = a α u α + (a α0 a α 0 )u α0 = a αu α. α<α 0 α<α 0 Hence the same argument applyng to the element b b β0 u β, we fnd that a α0 = b β0. Now we contnue the proof wth the element b a α u α, we fnd fnally that a α = a α, hence the two reduced wrtten form are the same. Now, we proceed to the proof of the exstence of reduced wrtten form. Ths wll be done by nducton on t. Frst we consder the case where t = 1. Recall that 1 I, hence n ths case, B s the polynomal rng n one varable u 1 wth coeffcents n R. Hence, any polynomal r ur 1 s already n ts reduced wrtten form. Now suppose t 2, and the exstence of reduced wrtten form s proven for B = R[u 1,, u t 1 ] wth respect to the relatons π d u = u pr() 1 + a α u α, for all [1, t 1] Ī α<m( 1,p r() ) Let now u α = u α 1 1 u α t 1 t 1 uαt t be a monomal n B, by nducton hypothess, we may assume that the monomal u α 1 1 uα t 1 t 1 s reduced as element n B. We only need to fnd a reduced wrtten for u α. Clearly, f t I, then there s nothng to prove snce u α s already reduced n B n ths case. Hence, from now on, we suppose that t Ī. We wll descrbe n the followng an algorthm actng on u α as follows: 1. If u α s reduced,.e., α t 1 < p r(t), then the algorthm stops, and the monomal u α s reduced. 2. If not,.e., α t 1 p r(t). We replace u α by the followng u α 1 1 uα t 2 t 2 uα t 1 p r(t) t 1 u αt t π dt u t α<m(t 1,p r(t) ) a α u α Then for each term u β = u β 1 1 uβ t 1 t 1 uβt t appearng n the prevous expresson, replace the monomal u β 1 1 uβ t 1 t 1 by ts reduced wrtten form n R[u 1,, u t 1 ] (here, we apply our nducton hypothess). Then for

21 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 21 each monomal appeared n the new expresson after ths reducton, we return to step 1. To fnsh the proof, we must show that ths algorthm stops after fntely many teratons. Indeed, we only need to show that each monomal u γ 1 1 uγ t 1 t 1 uγt t n the new expresson of step 2 verfes the followng nequalty: (5) (γ 1,, γ t 1 ) < (α 1,, α t 1 ). Then, apply Lemma 2.1, we fnd that ths algorthm stops after fntely many steps. To prove our statement, we clam that for any two reduced monomals u α and u β of R[u 1,, u t 1 ] such that α < m(t 1, p r(t) ) and β t 1 p r(t), each monomal of the reduced wrtten form n R[u 1,, u t 1 ] of the product (6) u β m(t 1,pr(t)) u α R[t 1,, u t 1 ] s of degree < β. But note that deg(u β m(t 1,pr(t)) u α ) = deg(u β ) deg(u pr(t) t 1 ) + deg(uα ) < deg(u β ) where the last nequalty follows from the fact that deg(u pr(t) t 1 ) > deg(uα ) snce both the monomals u r(t) t 1 and uα are reduced n B = R[u 1,, u t 1 ] and we have α < m(t 1, p r(t) ). Hence by Lemma 2.16 (), for each monomal u γ n the reduced form of the product (6), one must have γ < β. In ths way, the nequalty (5) s verfed, and ths fnshes the proof. Now, once we show the exstence and the unqueness of the reduced wrtten form, the flatness of B over R follows. In fact, we only need to show that B has no π-torson. Let b B such that πb = 0. Let b = α a α u α be ts reduced wrtten form. Hence πa α u α α s the reduced wrtten form of πb. The fact that πb = 0 mples then πa α = 0 by the unqueness of reduced wrtten form. As a result, we fnd b = 0. Remark Keepng the notatons as before. 1. By usng Proposton 2.17, we fnd that the set of reduced monomals gves a bass of the free R-module B over R. 2. The noton of reduced form stll has a sense n B K. In partcular, the set of reduced monomals gves also a bass of the K-space B K. 3. Let P = d j u u j B K K B K, d j K.,j

22 22 JILONG TONG We say that ths wrtten form s reduced f the for d j 0, the monomals u and u j are reduced. By usng Proposton 2.17, one shows that any element P B K B K has a unque reduced wrtten form. Moreover, f P B R B B K K B K, we may requre that ts reduced wrtten form has coeffcents n R. In partcular, B R B s free over R, wth a bass gven by the famly {u u j : u, u j reduced } A reformulaton of Lemma 2.5. The proof of the Theorem 2.14 can be seen as a more careful examnaton of the lnear unpotence establshed n Theorem Hence, before comng nto the detals of the proof, we wll frst reformulate lemma 2.5 n a more precse way. In the followng, we suppose that the (part of the) generators {y 1,, y t } s properly chosen, namely, these generators {y 1,, y t } satsfy the propertes of the Theorem In partcular, the noton of reduced wrtten form (n R[y 1,, y t ]) can be appled. Moreover, because of Proposton 2.13, the degree functon wth respect to the varables u ( I) n the rng R[G] R K = K[G K ] s the same as the degree functon wth respect to the varables x as s consdered n 2.1. Proposton Keepng the notatons as before. Let m Z t 0 a multndex. Let (7) η(y m ) = α,β c α,β y α y β be the reduced wrtten form of η(y m ). 1. Suppose m = m(, r) wth [1, t] and r Z 1. If + 1 Ī [1, t], suppose further more that r p r(+1). In partcular, for any j [1, r 1], the monomal y j yr j s reduced. (a) If I. Then for any j [1, r 1], we have ( ) r c m(,j),m(,r j) =. j (b) If Ī. Then for any j [1, r 1], we have ( ) r c m(,j),m(,r j) mod π. j (c) If Ī and r = pa+1 for some nteger a 0. Then c m(,p a ),m(,(p 1)p a ) p mod πp.

23 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) Suppose m = m + m(, r) wth r > 0 and m < m(, 1) a mult-ndex dfferent from zero. Moreover, suppose that y m s reduced. Then c m,m(,r) 1 mod π. Proof. (1) Suppose frst of all I. Wth the notatons of Proposton 2.13, we have y t = a x ω() + f t (x 1,, x ω() 1 ) wth a K, and f K[x 1,, x ω() 1 ]. As a result, η(y ) = a η(x ω() ) + η(p (x 1,, x ω() 1 )) K[x 1,, x ω() 1 ]. by the defnton of the prmtve generators x 1, x n of K[G K ]. Hence As a result, we fnd η(y ) 0 mod M m(ω(),1) = M deg(y ). η(y r ) (y y ) r y r 1 1 y r mod M deg(y r ) r 1 ( ) r y j j yr j mod M deg(y r ). j=1 By our assumpton on m = m(, r), each monomal y j yr j n the prevous sum s reduced. By comparng the coeffcent wth (7), we fnd ( ) r c m(,j),m(,r j) =. Ths fnshes the proof of (1.a). Next, suppose Ī. Let η(y ) = α,β a α,β y α y β mod M deg(y ) be the reduced wrtten form for η(y ), such that for those a α,β 0, we have deg(y α ) < deg(y ), deg(y β ) < deg(y ) and deg(y α y β ) = deg(y ). hence we must have α, β < m(, 1) snce all the monomals y α, y β, y are reduced (Lemma 2.16 ()). In partcular, the monomal y α (resp. y β ) contans a factor y l for some l < (resp. a factor y l for some l < ). On the other hand, η(y r ) 1 y + y 1 + α,β a α,β y α y β r y r 1 1 y r mod M deg(y r ) r 1 ( ) r y j j yr j + b γ,δ y γ y δ mod M deg(y r ) j=1 γ,δ

24 24 JILONG TONG where for b γ,δ 0, we have ether y γ or y δ contan a factor y l for some l <. Hence, after we replace y γ y δ by ts reduced wrtten form, once a term of the form y j yr j appears, the coeffcent of the term y j yr j must be a multple of π. As a result, n the reduced wrtten form of η(y r ), the coeffcent of y j yr j s equal to ( r j) modulo π. Ths gves (1.b). To get (1.c), we need a more careful examnaton of the coeffcents of (7). Frst of all, snce for any nteger j [1, p a+1 ], the bnomal coeffcent ( ) p a+1 j s dvsble by p, the arguments before shows that the form y γ y δ whose reduced form gves the term y pa y r pa wth coeffcent not contaned n πp must be contaned n the followng sum (8) y pa y r pa + α,β a r α,β yrα y rβ Now, suppose y rα y rβ after reducton gves y pa y r pa, we clam that we have α = m( 1, p r() 1 ). Indeed, by Proposton 2.13, and snce the famly {y 1,, y t } s properly chosen, for each y j {y 1,, y t }, ts degree s of the form m(ω(j), p ). It follows that deg(y rα ) = deg(y pa ) = m(ω(), p b ) for some nteger b > 0. Hence deg(y α ) = m(ω(), p b a 1 ). Hence, accordng to the degree formula (Lemma 2.16 ()), there exsts some j <, such that y α = y pc j for some nteger c. Snce pdeg(y α ) = deg(y pα ) = m(ω(), p b a ) = deg(y ), and snce y α s reduced, we fnd that y α = y pr() 1 1. As a result, (9) y α y β = y pr() 1 1 For smplcty, we put y (p 1)pr() 1 1 λ := a m( 1,p r() 1 ),m( 1,(p 1)p r() 1 ) We wll prove (1.c) by nducton on the nteger ω() the followng two statements: (I) λ π d pr (II) The concluson (1.c) holds for the ndex. We clam frst that for an ndex, the statement (I) mples (II). Indeed, f we compute the coeffcent of the term y pa y r pa of the reduced wrtten of the term λ r y rpr() 1 1 y r(p 1)pr() 1 1

25 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 25 n (8), by (I), ths coeffcent s dvsble by Snce r = p a+1 p 2, ths gves (π dt p) r π dt pa π dt (pa (p 1)) = p r c m(,p a ),m(,r p a ) 1 mod πp. and hence the concluson of (1.c) holds for the ndex. That s (II) s verfed once (I) s proven. Suppose frst that ω = 1, hence 1 = ω() I. Moreover, the famly {y 1,, y t } s properly chosen, ths mples that π d y = y pr() 1 a,α y α Hence α<m( 1,p r() ) π d η(y ) η(y p() t 1 ) mod M deg(y ) Hence, accordng to (1.a), the coeffcent λ of the term y pr() 1 1 π d y pr() p r() 1 1 n the reduced form of η(y ) s equal to ( ) p r() p r() 1 In partcular, we get λ π d pr. As a result, the statement (I) s proven for, and so (II) for the same ndex. Suppose now ω() 2, and suppose that (I) and hence (II) have been shown for the ndex 1. We must show that the statement (I) holds for. Indeed, snce {y 1,, y t } s properly chosen, we have π d y = y pr() 1 + a,α y α α<m( 1,p r() ) we fnd π d η(y ) η(y r() 1 ) mod M deg(y p r() ). 1 Now applyng the statement (II) for the ndex 1, we fnd that the coeffcent of y pr() πp. Hence we have y (p 1)pr() 1 of the reduced form of η(y pr() 1 ) s congruent to p modulo p λ π d mod πp π d. Hence we get fnally λ π d pr. Ths proves (I) for, and hence fnshes the proof of (1.c)

26 26 JILONG TONG The proof of (2) s smlar to that of (1.b) Snce y m = y m y r, we have µ(y m ) = µ(y m ) µ(y ) r ( ) = y m y m + η(y m ) (y r y r + η(y r )) Hence Let = y m y m + y r y m + y m y r + (y m y m )η(y r ) +(y r y r )η(y m ) + η(y m )η(y r ). η(y m ) = y r y m + y m y r + (y m y m )η(y r ) +(y r y r )η(y m ) + η(y m )η(y r ). η(y r ) = α,β a αβ y α y β mod M deg(y r ), and η(y m ) = α,β b α β yα y β mod M deg(y m ) be ther reduced wrtten forms, such that deg(y α y β ) = deg(y r) for a αβ 0, and that deg(y α y β ) = deg(y m ) for b α β 0. We fnd η(y m ) y m y r + y r y m + ( a αβ y α+m y β + y α y β+m ) α,β + ) b α β (y α y r y β + y α y β y r α,β + a αβ b α β yα+α y β+β mod M deg(y m ). α,β,α,β Now let y γ y δ be any monomal n the prevous sum, whch s nether y r ym nor y m y r, then we have ether γ < m(, r), or δ < m. Hence, once a monomal of the form y r ym appears n ts reduced wrtten form, ts coeffcent must be a multple of π. Ths gves (2) Proof of the man result. Let us now begn the proof of the theorem. Let {y : 1 N} be an arbtrary famly of good generators, we wll prove by nducton on the nteger t 0 that up to modfy these generators, we can assume that the generators {y : 1 t} verfy the propertes requred by the theorem (.e., the famly {y 1,, y t } s properly chosen). Frst, snce 1 / Ī, hence we can keep y 1. Suppose now that the famly {y 1,, y t } has been properly chosen wth t < N. To fnsh the nducton, we only need to modfy y t+1 by y t+1 + f wth f R[y 1,, y t ] such that wth ths new y t+1, the famly {y 1,, y t, y t+1 } satsfes agan the property of the theorem. If t + 1 I, then there s nothng to do. Hence, n the followng, we

27 UNIPOTENT GROUPS OVER A DISCRETE VALUATION RING (AFTER DOLGACHEV-WEISFEILER) 27 may assume that t + 1 Ī. In partcular, accordng to 2.13, we have π d t+1 y t+1 = a y R[y 1,, y t ] Moreover, we may assume that the sum a y s reduced (Proposton 2.17). Lemma Keepng the notatons as above. Then a m R. Proof. Assume that a m = π b wth b A, and let z = π d t+1 1 y t+1 by m R[G] Then deg(z) < deg(y t+1 ), whch mples z R[y 1,, y t ] by the defnton of good generators. Hence π d t+1 y t+1 = πby m + πz, and we get π d t+1 1 y t+1 = by m + z R[y 1,, y t ] n vew of the flatness of A[G]. But ths equalty contradcts the defnton of the nteger d t+1. Therefore a m R. Accordng to ths lemma, up to replace y t+1 by a 1 m y t+1, we may assume that a m = 1, and that a / π d t+1 A for any < m. Lemma Wth the notatons as before. We have Proof. By Lemma 2.16, we have Hence π d t+1 η(y m ) M + M K,deg(y m ). π d t+1 η(y t+1 ) = η(π d t+1 y t+1 ) = η(y m ) + α<m a α η(y α ) η(y m ) mod M deg(y m ). π d t+1 η(y m ) η(y t+1 ) MK,deg(y m ). Hence the lemma follows once we remark that η(y t+1 ) M = R[G] R[G] snce y t+1 R[G]. Let m = (m 1,, m t, 0, 0), and η(y m ) = α,β a αβ y α y β

28 28 JILONG TONG be ts reduced wrtten form. Accordng to Lemma 2.21, for those (α, β) such that deg(y α y β ) = deg(y m ) = deg(y t+1 ), we must have (10) π d t+1 a αβ. In the followng, we wll show that m = m(t, p α ), and π d t+1 p. Frst of all, we clam that m t 0. Indeed, otherwse m < m(t, 1), and we fnd the followng contradcton wth the constructon of good generators (Corollary 2.10) deg(y t+1 ) = deg(y m ) < deg(y m(t,1) ) = deg(y t ) where the nequalty follows from the fact that the two monomals y m and y m(t,1) are reduced, and that m < m(t, 1) (Lemma 2.19). Hence, we fnd m t 0. Next, we clam that m = 0 for any < t. Otherwse, let m = (m 1,, m t 1, 0,, 0), then we have m 0, and y m = y m y m(t,mt) = y m y mt t Accordng to Proposton 2.19 (2), we have a m(t,mt),m 1 mod π. In partcular, π d t+1 a m(t,mt),m. Ths gves us a contradcton wth the dvsblty condton (10). Hence we must have m = 0. Now, we wll show that m t must be a power of p. Otherwse, let m t = p a b wth b > 1 prme to p. Let δ = m(t, m t p a ), and γ = m(t, p a ). Accordng to Proposton 2.19 (1.b), we must have ( p a ) b a γ,δ p a mod π. Snce, the bnomal coeffcent above s prme to p, we fnd π d t+1 a γ,δ. Thus, ths gves us a contradcton wth (10). Hence m t must be a power of p. Let m t = p a+1, wth a 0. To fnsh the proof, we only need to show π d t+1 p. We consder agan γ = m(t, m t p a ), and δ = m(t, p a ). Accordng to Proposton 2.19 (1.c), we have a γ,δ p Hence the dvsblty condton (10) mples π d t+1 p. mod pπ. Ths shows the exstence of y t+1 such that the famly {y 1,, y t, y t+1 } satsfes the relatons as ndcated n the theorem for all Ī [1, t + 1]. Now to complete the proof of Theorem 2.14, t remans to show that these relatons are the only ones verfed by y 1,, y t+1. Let B be the quotent of the polynomal rng R[Y 1,, Y t+1 ] by the deal generated by the relatons ndcated n the theorem for Ī [1, t + 1]. Then by Proposton 2.17 (1),