JOURNAL OF ALGEBRA 186, 182᎐206Ž. 1996 ARTICLE NO. 0369

The Forms of the Witt Group Schemes

Changchun Li

Department of , Uni¨ersity of Minnesota, Minneapolis, Minnesota 55455

Communicated by Susan Montgomery

CORE Received February 12, 1996 Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector

0. INTRODUCTION

Throughout this paper, let p be a fixed prime number and let Fp be the prime of p. We assume that all Witt group schemes are with respect to this p ŽwxwDem , DemGab x. and all rings are algebras over Fp. The purpose of this paper is to study the forms of Witt group schemes. The motivation for our work comes from the following aspects:

1. Witt group schemes can be defined over any algebra over Fp and play a crucial role in the theory of commutative unipotent group schemes and finite group schemes over a of char s p Žseewx Dem , wxwxDemGab , and Serre. ; but 2. over a non-perfect field, the Witt group schemes cannot describe the category of the commutative unipotent group schemeswx SchŽ Takeuchi made a correction and a generalization of Schoeller’s work. . 3. among the commutative group schemes whose underlying schemes are affine spaces, Witt group schemes are typical in the following sense: their Verschiebung morphisms are clear and any two of them have good relations; and 4. the structures of the forms of vector group schemes over a field or a discrete ring are well knownŽwxwxw Rus , KMT , WatWei xwx , Li , and wxLib. .

To begin with our work, we first fix some notations and definitions. We denote the set of all integersŽ.Ž. resp. positive integers by ޚ resp. ގ . For any ring R, R* means the set of all invertible elements in R; for any

182

0021-8693r96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. FORMS OF THE WITT GROUP SCHEMES 183

Ž. Ž. Ž ngގ,MRnnwGL R xindicates all n = n matrices resp. all n = n invertible matrices. with entries in R. The ¨ indicates the inclusion or equivalently injective map.

For any scheme S, we denote by WWn, S Ž.ϱ, S the Witt group scheme of a finite length n Ž.resp. of the length of infinity ; for each n g ގ, we indicate n the n-dimensional vector group schemeŽ. resp. affine space over S by G␣, S Ž n . resp. ށ S .IfSsSpec B for some B, then we write nn nn Wn,B Žresp. Wϱ,B, G␣,BBn, and ށ .Žfor W ,Sresp. Wϱ,S, G␣,SS, and ށ ..Aswe know, the three terms, the one-dimensional vector group scheme, the additive group scheme, and W1, indicate the same thing. For any affine scheme X, we denote by oXŽ.its coordinate ring, i.e., Ž. XsSpec oX. Let X12and X be two affine schemes and g be a morphism from X12to X . We denote by ogŽ.the corresponding ring morphism from oXŽ.21to oX Ž..IfGis a group scheme, G indicates its underlying scheme.

Let A be a commutative ring containing Fp. Without confusion, we always use f n to represent the morphism from A to A defined by n pn n faŽ. afor n ގ. For any scheme X over A, we denote X A Žf , A. sn g m by X Ž p .. Suppose that G is an affine flat commutative group scheme over A. Then there exists a group morphism from GŽ p. to G that is called Verschiebung morphismwx DemGab . We always indicate this morphism by p ␯G. We define a ring AFwxby F␣ s ␣ F for ␣ g A. It is obvious that both the terms left AFwx-module and commutative p-Lie algebra over A have the same meaningwx Jac . For a left AF wx-module M and n g ގ, we define Ž p n. Ž n. Ž p n. M sA,F mA M. Then M is still a left AFwx-module in an obvious way.

For any commutative ring B over Fp, the ‘‘BFwx-module’’ in Section 1 always means the ‘‘left BFwx-module’’ and in Sections 2, 3, and 4 always means the ‘‘right BFwx-module.’’ Let A be a commutative ring containing Fp.If His a group scheme Ž. Ž . over A, we define ᒍ H s Hom A H, G␣ , A , which is a commutative p- Lie algebra over A. Conversely, for any commutative p-Lie algebra M, we always denote by UMŽ p.Ž.its universal enveloping algebra over A ŽwxwxwxDemGab , Li , and Lib. ; it is in fact a Hopf algebra over A. We denote the corresponding group scheme by uMŽ.. Let A be an integral domain containing Fp with fraction field K. We indicate an algebraic closure of K by K and the integral closure of A in K by A. For n g ގ, define pppynnynpyn KsÄ4xgKNxgK,AK lA. Now let S be a scheme. Suppose that G and GЈ are two group schemes over S. We say that GЈ is an S-form of G provided that there exists a X faithfully flat and quasi-compact morphism SЈ ª S s.t. GSЈ , GSЈ as SЈ-group schemes. Assume that H and HЈ are two schemesŽ resp. sheaves of modules or group schemes. . If there exists an open covering of S, say 184 CHANGCHUN LI

Ä4S , s.t. H H Xfor i I, we say that H is locally isomorphic to HЈ, iigISSii, g or equivalently H is locally a trivial form of H. Let A be an integral domain with fraction field K and G be an affine flat group scheme over

A. We say that G is a model of GK .

DEFINITION 0.1. Let A be a commutative algebra over Fp. It is called an F-ring, provided that A is integrally closed domain and the injection py1 map A ¨ A is flat. If furthermore, the PicŽ.A s 0, we call A an FP-ring.

As for the basic examples of F-rings, we have the following lemmas.

LEMMA 0.2. If A is a regular ring o¨er Fp with the Krull F 2, then A is an F-ring.

Proof. First we can assume that A is localwx Mat, Chap. 2 . If the dimension is 0, it is a field; if the dimension is 1, it is a discrete valuation ring. In these cases, A obviously is an F-ring. Now let the dimension of A be 2. Choose a g A s.t. ²:a is a of A and A s Ar²:a is a py1py1 discrete valuation ringwx Mat, Chap. 7 . We claim that A raA ,asan y1 A-module, is torsion-free and hence flat. Then A p is flat over A ŽwHar, Chap. III, Lemma 10.3.Axw , Bour, Chap. III, Sect. 5 x. . Indeed, let x, y g A py1 py1 pp s.t. xy g aA . Then xysazfor some z g A.If xf²:a , since A ppy1 ppy1 y1 is a UFDwx Mat , then y g ²:a . Hence y s 0in A raA . This proves our claim.

LEMMA 0.3. Let A be an F-ring. Then the polynomial algebra of any number of indeterminates o¨erAisanF-ring.

Proof. Let E be a polynomial algebra over A with a set of indetermi- Ä4 nates xiigI.If Ais an integrally closed domain, then so is E wMat, Chap. py1 7x . Note that the injection E ¨ E is the composition of the following Ä py1 4 Äpy14 py1 two flat morphisms: E ¨ Axw iiigIixwand Ax gIAx¨A m Ä4py1 AxwiigIx. Hence E is an F-ring.

LEMMA 0.4. If X s Spec B is a smooth integral scheme o¨er a field k of char s p, then B is an F-ring.

Proof Ž.By W. Messing . It suffices to prove the following statement: Ž.py1 for any P g Spec B, there exists an element b g B _ P s.t. Bb is flat over Bb. FORMS OF THE WITT GROUP SCHEMES 185

Let P g Spec B. Then there exist an affine open neighborhood of P, say Spec Bb, and a commutative diagram

Spec B t 6 Spec k

b 6 6 , g h n ށ k where t is the structure morphism, g is etale, and h is the canonical projectionwx GroDie, Chap. IV . n n py1py1 py1 y1 Ž. Write ށ k s Spec kxwx1,..., xn and ށŽkp. s Spec kxw1 ,..., xn x. y1 p n 1 Let gЈ be the morphism from SpecŽ.Bb to ށŽ k py. induced from g, i.e., Ž.Žpy1.ŽŽ.Ž..py1 ogЈ ␣ s og ␣ for ␣ g kxwx1,..., xn . Then gЈ is an etale morphism. By Lemma 0.3, the natural imbedding kxwx1,..., xn ¨ py1 py1 py1 Žkx.w 1 ,..., xn x is flat. We indicate by q1 the corresponding mor- n n py1 y1 Ž. phism ށŽk p . ª ށ k . Similarly, let q2 indicate the morphism Spec Bb Ž.py1 ªSpec Bbbinduced by the natural imbedding B ¨ Bb. Then we have the commutative diagram

1 py gЈ 6 n 1 Spec Ž.Bbkށ6py

p1 r6

q2Xq1,

6

p2 6 6 g6 n Spec Bb ށk n where X SpecŽ.B = npށ y1, p is the canonical projection, and r is s b ށ k Žk . i given by Ž.q21, gЈ . Then p is etale because g is etale. Note that gЈ is etale; hence r is etale. But p21is flatŽ. since q is flat ; therefore q 2is flat. For FP-rings, the basic examples are any field of char s p, any polyno- mial algebra of a finite number of indeterminates over a field of char s p, and any local F-ring. This paper consists of four sections. In Section 1, we study the forms of vector group schemes, especially the forms of the additive group scheme. This section generalizes some results inwx Li and w Lib x . Our main result in this section is THEOREM 1.9. Let A be an FP-ring and let G be a form of the additi¨e group scheme o¨er A. Then either G s G␣, A or G can be expressed as a 2 closed scheme of G␣, A s Spec Axwx,y with ideal generated by an equation p n p pr y s x q ax1 qиии qaxr , p where a1,...,arigA with some a f A . 186 CHANGCHUN LI

In Section 2, we study the forms of the Witt group schemes with dimension G 2. In this section, we first describe the forms of Witt group schemes from different points of view and then study the forms of Witt group schemes over a field. We prove

THEOREM 2.4. Let A be an integral domain containing Fp and let G o¨er A be a form of Wn, A. Then the following two conditions are equi¨alent:

1. G s Wn, A; 2. there exists a composition series of G,

0 - G1 - иии - Gn s G,

whose quotients are isomorphic to G␣, A. If we further assume that A is an FP-ring, then the abo¨e conditions are equi¨alent to: n 3. G s ށ A. THEOREM 2.5. Let A be an F-ring and let G be a group scheme o¨er A. m m Then G is a form of Wn, AAn iff locally G py , W ,Apy for some m g ގ. If we further assume that A is an FP-ring, then G is a form of Wn, A iff m m GAnpy , W ,Apy for some m g ގ. THEOREM 2.8. Let A be an F-ring and let G be a group scheme o¨er A. Then G is a form of Wn, A iff there exists a unique composition series

0 s G01- G - иии - Gns G such that: Ž. 1. G1rGiy1 is a form of G␣ , A i s 1,...,n and 2. the Verschiebung morphism of G G , ␯ : Ž.G G Žp.G G , r iGrGiir ª r i has the factorization

Ž.pp␶ 6 Ž. Ž.GrGGii ŽrGq1 .

␯

GrGi X 6

⑀6

GrGi6 Gny1rGi for i s 0,...,ny2, where ␶ is the natural projection and ⑀ is the natural injection. COROLLARY 2.9. Let k be a field of char s p and let G be a form of Wn, k . Then under one of the following conditions, G , Wn, k:

1. Gln, W,l for some separable field extension l of k. 2. k is a perfect field.

In Section 3, we consider the models of the forms of Wn, where n ) 1 over a discrete valuation ring A. We assume that Fr A s K and the residue field of A is k. Then we obtain FORMS OF THE WITT GROUP SCHEMES 187

COROLLARY 3.2. Let G be a smooth group scheme o¨er A. If GKk and G are forms of finite dimensional Witt group schemes, then so is G o¨er A. In Section 4, we consider the existence and uniqueness of some kinds of forms of Witt group schemes. First, over an FP-ring A, we study the forms of W2, A and construct all forms of W2, A of height 1. Then we prove

THEOREM 4.4. Let A be an FP-ring and let H be a form of G␣, A of height ގ 1. Then for each n g , there exists at most one form of Wnq1, A of type H. For the definitions of ‘‘height’’ and ‘‘type,’’ see the following sections.

1. THE FORMS OF THE ADDITIVE GROUP

In this section, we study a class of forms of vector group schemes over an F-ring.

DEFINITION 1.1. Let A be a commutative ring containing Fp and let M be an AFwx-module.

1. If there exists a faithfully flat algebra extension B of A s.t. MB is a free BFwx-module of finite rank, we call M a form of some free AFwx-module of finite rank. 2. If inŽ. 1 , we further assume that B is the direct sum of a finite number of A , where a A, we call M a locally free AF-module of aii g wx finite rank.

PROPOSITION 1.2. Let A be a commutati¨e ring containing Fp. Then the G ¬ ᒍŽ.G and M ¬ u Ž M . are anti-equi¨alences between the cate- gory of forms of ¨ector group schemes o¨er A and the category of forms of free AFwx-modules of finite ranks. Proof. Let A [ the category of the forms of vector group schemes over A and B [ the category of the forms of AFwx-modules of finite ranks. For G g A, let A ª B be a faithfully flat algebra extension of A n Ž. Ž . s.t. GBs G␣,BBB. Then ᒍ G s ᒍ G wxwxLib, Lemma 1.5 is a free BF- module of rank n, i.e., ᒍŽ.G g B. Conversely, for any M g B, let A ª C be a faithfully flat algebra extension of A s.t. MC is a free CFwx-module of Ž. Ž . m Ž. rank m. Then uMCCsuM sG␣,CwxLib, Lemma 1.3 , i.e., uM is a m form of G␣, A. For H A, N B, we claim uŽŽᒍ H .. H and ; g g , NªᒍŽŽuN ... Let H s Spec D. We have ᒍŽ.H ¨ D. Hence there exists Ž p. a morphism U ŽŽᒍ H ..ª D, i.e., H ª uŽŽᒍ H ... As we know, this map is compatible with base extensionswx Lib . We let A ª AЈ be a faithfully flat algebra extension s.t. HAЈ is a vector group scheme over AЈ. Then ; ŽŽ .. ŽŽ .. ŽŽ .. HAЈ ªuᒍHAЈ . But u ᒍ HAЈ , u ᒍ H AЈ wxLib , and hence H , 188 CHANGCHUN LI uŽŽᒍH ... For the second , first there exists a natural mor- phism N ᒍŽŽuN ... After some faithfully flat extension of A, it becomes ª ; an isomorphism, so N ª ᒍŽŽuN ... This proves our claim. The rest of the argument for this proposition is trivial.

PROPOSITION 1.3. Let A be an F-ring. Suppose that M is an Awx F -module pyn pyn s.t.locally A mA M is a free Awx F -module of finite rank for some n g ގ. Then: 1. M, as an A-module, is projecti¨e and Ž n.Ž.n 2. tnA: A, f m M ª M, ta nmmsaF m is an injecti¨eAFwx- linear map, where f: A ª A is the Frobenius map of A. Proof. Ž.1 Obvious. Ž. 2 We only need check that tnnis injective. Let K s Fr A and T s ker t . Then

n K mAKnAAAT s kerŽ. id m t : K m Ž.A, f m M ª K m M i.e.,

X n K mAnKKAAT s kerŽ.t : Ž.K , f m Ž.K m M ª K m M

XŽ.nŽ. nn where tknKm1mmskF 1 m m s k m Fmand f : K ª K is in- n duced from f .So KmA Ts0wx KMT . Note that T is a submodule of n Ž A, f . m M that is a projective A-module. Then T is a submodule of some free A-module. Hence K mA T s 0 implies that T s 0. pyn PROPOSITION 1.4. In Proposition 1.3, if A mA M is locally a free pyn n n AFwx-module of rank m, then MrMwx is a projecti¨eAFwxr²F :-module w nx Ž. of rank m, where M s im tn . Proof. For any t G 1, we have an exact sequence

t t 0 ª M w x ª M ª MrM w x ª 0.

Then for any s G 1, we get the following commutative diagram with an exact row: 6 6 6 6 s wtx s s wtx 0 Ž.A, f mAAAMA Ž.,fmMA Ž.,fmŽ.MrM 0. 6

X 6

s wxt Ž.Ž.A,f mAM FORMS OF THE WITT GROUP SCHEMES 189

Taking s s n, t s 1, we know that

n w1x Ž.A, f mAŽ.MrM

1 is a locally free A Žs AFwxr²:.F -module of rank m. Hence mrM wxis a n projective A-module of rank m. We claim MrM w x is a finitely generated n AFwxr²F :-module. Indeed, since

n ; n n py w nx ppy y wnx AmAAŽ.MrMªŽ.Ž.AmMrAmAM

ppyn yn wxn sŽ.Ž.AmAAMrAmM

n n n py Žpy.Ž py .wnx as A -modules and A mAAM r A m M is a locally free pyn n A-module of rank mn, then MrM w x is a projective A-module of rank n mn.So MrMw x is a finitely generated A-module and hence a finitely n generated AFwxr²F :-module. Without loss of generality, we may assume 1 n that MrM w x is a free A-module of rank m. Since ²:²F r F :is a n n ideal of AFwxr²F :, by Nakayama’s lemma, MrM wxis gener- n ated by m elements as an AFwxr²F :-module. Therefore, we have an n exact sequence of AFwxr²F :-modules

n 0 ª T ª X ª MrM w x ª 0,

n where X is a free AFwxr²F :-module of rank m. From the condition that A is a domain and

w nx rank AAX s mn s rank Ž.MrM ,

n n we know that T s 0. So MrM wxis a free AFwxr²F :-module of rank m. By the same argument as that ofwx Lib, Propositions 3.18 and 3.20 , we obtain the following result:

THEOREM 1.5. LetAbeanF-ring, MbeanAFwx-module, and n g ގ. pyn pyn Then locally A mA M is a free Awx F -module of finite rank iff for some m g ގ locally there exists

r AsA01qAFqиии qArmF g MAFŽ.wx, where

A0g GL mimŽ.A , A g MAi Ž.Žs1,...,rs ..t. mm M AF x AF y ²:FYnX , sž/ž/[[wxii[ wx r yA is1 is1 190 CHANGCHUN LI where

tt XsŽ.x1,..., xm and Y s Ž.y1,..., ym

n are column ¨ectors and² F Y y AX: indicates the submodule generated by n the entries in the column ¨ector F Y y AX. COROLLARY 1.6. Let A be an F-ring, G be a group scheme o¨er A, and n n g ގ. Then locally GA py is isomorphic to the m-dimensional ¨ector group 2 m scheme iff locally, G is a subgroup scheme of G␣, A with ideal generated by u1,...,um, where

uyx111 ...n ...sFyA 0uyxmmm 0 0 for some pairŽ. n, A described in Theorem 1.5. DEFINITION 1.7wx KMT . Let K be a field of char s p and G be a form of the additive group scheme over K. We define the height of G, written

Ž. yn yn as ht G , to be the smallest integer n s.t. GK pp, G␣ , K . THEOREM 1.8. Let A be an F-ring with K s Fr A and G be a form of the additi¨e group scheme o¨er A. Then:

1. GK s G␣ , K iff there exists a projecti¨eA-module I of rank 1 s.t. Ž. Ž. oG sSymm A I , the symmetric algebra of I o¨er A with comultiplicity m ¬ m m 1 q 1 m m for m g I; and

2. the height of GK is a positi¨e number n iff G can be expressed locally 2 as a closed subgroup scheme of G␣, A s Spec Axwx,y with ideal generated by an equation

p n p pr y s x q ax1 qиии qaxr ,

p where a1,...,arigA with some a f A .

Ž. Ž. Ž. yn Proof. 1 has been proved inwx WatWei . We prove 2 . By 1 , GK p , n n n G␣, KAppy iff locally G y , G␣,Apy . By Corollary 1.6 andwx KMT , we win.

THEOREM 1.9. Let A be an FP-ring and G be a form of the additi¨e group scheme o¨er A. Then either G s G␣, A or G can be expressed as a closed 2 subgroup scheme of G␣, A s Spec Axwx,y with ideal generated by an equation

p n p pr y s x q ax1 qиии qaxr ,

p where a1,...,arigA with some a f A . FORMS OF THE WITT GROUP SCHEMES 191

Ž. Ž. Proof. Let ht GK s n, where K s Fr A. By Theorem 1.8, oG s pyn ynŽ. Symm A p I for some projective A -module I of rank 1 with I : pyn Ž yn . yn yn ᒍGA p . Since Pic A s 0, then I is free, and so GA pps G␣ , A . pyn Ž. pyn Ž. Hence, A mA ᒍ G is a free AFwx-module of rank 1. Set ᒍ G s M. By the same argument as in the proof of Proposition 1.4, first we know 1 that MrM w x is a projective A-module of rank 1, hence a free module of 1 rank 1. This means that MrM w x can be generated by one element. Then n n we know that MrM w x is a free AFwxr²F :-module of rank 1. Applying the proof ofwx Lib, Proposition 3.18 to our situation and by w Lib, Proposi- tion 3.20x , we know that

; ²:nr MªAFwx xmAF wx yrFyyŽ.1qaF1qиии qaFr x

p as an AFwx-module, where a1,...,arigA with some a f A . This implies our statement.

2. THE FORMS OF Wn WHERE n ) 1 AND THE FORMS OF THE Wϱ

DEFINITION 2.1. Suppose that A is a commutative ring containing Fp. Ž. 1. Let Wn, A s Spec Axwx1,..., xn . We call x1,..., xn a system of standard coordinates of Wn, A provided

oŽ.␯:Ax,..., x Ax,..., x Ž.f, A Wn,A wxwx1 nª 1 nAm is given by

x121¬ 0, x ¬ x m 1,..., xnn¬x y1m1.

2. Let Ž.x1,..., xmmbe a system of standard coordinates of W ,Afor ms1, n, and n q 1. We denote by enq1 the following exact sequence of group schemes over A:

tr 0ªG␣, AnªWq1, AnªW,Aª0, where

otŽ.Ž xnq11 .sx,ot Ž.Ž xi .s0 for i - n q 1 and

orŽ.Ž. xjjsx for j s 1,...,n. 192 CHANGCHUN LI

LEMMA 2.2. Let A be an integral domain containing Fp. Then

Ext A-commŽ.WnA,,G␣,A

Ž.where comm indicates commutati¨e is a free Awx F -module with the basis Ž. enq1. Proof. The statement is correct for the case that n s 1Žwxw Laz , CemGab, Chap. II, Sect. 3x. . From

tr enq1:0ªG␣ , AnªWq1, AnªW,Aª0, we have the long exact sequence

␦ иии ª Hom AŽ.G␣,A, G␣,AAª Ext -comm Ž.Wn, A , G␣ , A fg ªExt A-commŽ.Ž.Wnq1, A, G␣, AAª Ext -comm G␣ , A , G␣ , A . By induction on n, ␦ is an isomorphism, and then g is an injection. But from

G␣ , A

t 6 6 6 6 6

enq2:0 GW␣,Anq2, An Wq1, A 0 we obtain 6 6 6 6 e2: 0 GWG␣,A 2, A ␣, A 0

t

6 6 6 6 6 6

enq2 : 0 G␣, A WWnq2, Anq1, A 0, Ž. i.e., genq22se; hence g is an isomorphism. This proves our state- ment.

PROPOSITION 2.3. Let a be a commutati¨e ring containing Fp and let G be a group scheme o¨er A. If G is a form of Wn, A, then there exists a unique composition series

0 s G01- G - иии - Gns G such that: Ž. 1. GiirGy1 is a form of G␣,A i s 1,...,n ; FORMS OF THE WITT GROUP SCHEMES 193

2. the Verschiebung morphism of G G , ␯ : Ž.G G Žp.G G , has r iGrGiir ª r i the factorization

Ž.pp␶ 6 Ž. Ž.GrGGii ŽrGq1 .

␯

GrGi X 6

6 ⑀

GrGi6 Gny1rGi for i s 0,...,ny2, where ␶ is the natural projection and ⑀ is the natural injection; and Ž.Žp. 3. GiirG y1, G␣,An, where i s 1,...,ny1, pro¨ided that GrG y1 ,G␣,A. Ž. Ž. Proof. We first consider items 1 and 2 . For the case that G s Wn, A, the result is well knownŽwx Dem and w DemGab x. . For the general case, we let B be a faithfully flat algebra extension of A s.t. GBns W ,B. Write Ž. Wn, B sSpec Bxwx1,..., xn , where x1,..., xn is a system of standard coor- Ž. ŽŽ .. dinates of Wn, Bn; hence ᒍ W ,Bs BFxwx1. Let h: G ª u ᒍ G by the Ž Ž Ž ... natural morphism and Gny1 s ker G ª u ᒍ G . Then we have an exact sequence of group schemes over A:

h 0 ª Gny1ª G ª uŽ.ᒍŽ.G . Note that

GB sSpec Bxwx1,..., xn andŽ bywx Li .

uŽ.ᒍŽ.G BsuŽ.ᒍ Ž.GBsuŽ.ᒍ ŽGB .sSpec Bxwx1,G␣,B.

Then hB is faithfully flat; so is h. This means that

h 0 ª Gny1ª G ª uŽ.ᒍŽ.G ª 0 Ž. is an exact sequence of group schemes over A and Gny1 Bn, W y1, B.By the induction on n, we have provedŽ. 1 . For Ž. 2 , we only need to consider the case that i s 0. First we have the following commutative diagram with exact rows: 6 6 6 Žp.Žp. Ž.p 6 0GG1 Ž.GrG1 0

␯0 ␯ ␯ s 6

6 6 6 6 6 6 0 G11GGrG0. 194 CHANGCHUN LI

Since the Verschiebung morphism of G1 is zero, then we have a unique factorization in the above diagram:

Ž.p h:Ž.GrG1ªG.

Let A ª B be a faithfully flat extension of A s.t. GBns W ,B. Then by the base change of A to B, the above diagram becomes

6 Ž p. 6 Ž p. 6 Ž p. 6 0 GW␣,Bn,Bn Wy1, B 0

␯s0 ␯

6 6 6 6 6 6 6

0 G␣, BnWW,Bny1, B 0. Ž p. Ž.Žp. Under hBn, W y1, Bnis a subgroup of W ,B. Hence under h, GrG1is a subgroup of G. Note that

Ž.p 0ªŽ.GrG1BªGBBªSpec B ᒍŽ.G ª 0 is exact, where BwᒍŽ.GBBxindicates the subalgebra of oGŽ.generated by ᒍŽ.GB . But in our situation, we havewx Lib

B ᒍŽGBA .s B ᒍ Ž.G B s A ᒍ Ž.G m B.

Therefore, we obtain the exact sequence

Ž.p 0ªŽ.GrG1ªGªSpec A ᒍŽ.G ª 0.

Ž.Žp. This means GrG1 , Gny1. To proveŽ. 3 , by induction on n, it suffices to consider the case n s 2. Ž. But taking n s 2 in 2 , we have G111, GrG , i.e., G , G␣,A.

THEOREM 2.4. Let A be an integral domain containing Fp and let G o¨er A be a form of Wn, A. Then the following two conditions are equi¨alent:

1. G s Wn, A. 2. There exists a composition series of G,

0 - G1 - иии - Gn s G, whose quotients are isomorphic to G␣, A. If we further assume that A is an FP-ring, then the abo¨e conditions are equi¨alent to: n 3. G s ށ A. FORMS OF THE WITT GROUP SCHEMES 195

Proof. We only need to prove thatŽ. 2 « Ž.1 and under the assumption that A is an FP-ring,Ž. 3 « Ž.2. Ž.2« Ž.1 : Let

0 - G1 - иии - Gn s G

be a composition series whose quotients are isomorphic to G␣,A.By

induction on n, we can assume that GrG1 is isomorphic to Wny1, A. Write G1 s Spec Aywx, where y is a primitive element of G and GrG1 s Ž. Spec Axwx1,..., xny11, where x ,..., xny1is a system of standard coordi- nates of GrG1. Then we have the exact sequence of group schemes over A

st ⑀:0ªG11ªGªGrG ª0, where

GsSpec Axwx1,..., xny1, y ,

osŽ.Ž. y sy, os Ž.Ž. xi s0, i s 1,...,ny1,

and

otŽ.Ž xjj .sx, js1,...,ny1.

By Lemma 2.2, we may construct the commutative diagram with exact rows 6 6 6 6 en:0 G␣,AnsSpec Axwx W,An Wy1, A 0

␥ ␶ 6 6

⑀ 6 6 6 6 :0 G␣, AnsSpec Aywx G Wy1, A 0,

where oŽ.Ž.␥ y s gx Ž.is some p-polynomial in Axwxand

oŽ.Ž.␶ xiis x , i s 1,...,ny1, and oŽ.Ž.␶ y s xn.

From the Verschiebung morphism of Wn, A we know that the Ver- schiebung morphism of G, ␯G,is

␯ oŽ.G:Axwxwx1,..., xny11, y ªAx,..., xny1, y mAŽ.f, A , ␯␯ oŽ.Ž.G x1s0, o Ž.Ž.Gix s x iy1m 1 ␯ for i s 1,...,ny1 and oŽ.Ž.Gny s gx Žy1 .m1. 196 CHANGCHUN LI

By Proposition 2.3,

Axwx1,..., xny1mAŽ.f, A sAx1,..., xny2, gxŽny1 .Ž.mAf,A. Then,

Axwx1,..., xny11sAx,..., xny2, gxŽ.ny1. Ž. Since A is an integral domain, this implies that gxny1 saxny1 for some a g A*. Hence ␥ is an isomorphism. Therefore G , Wn, A. Ž. Ž. The proof of 3 « 2 : Since G is a form of Wn, A, by Proposition 2.3, there exists a composition series

0 - G1 - иии - Gn s G ށn Ž. s.t. G1rGiy1 is a form of G␣ , AAn.IfGs ,GrGy1K,G␣,K, where Ž. KsFr A wxDemGab . By our assumption that A is an FP-ring, GrGny1 A Ž .Ž. ,G␣, A Theorem 1.9;wx WatWei . By Proposition 2.3 3 , we obtain condi- tionŽ. 2 . Remark 1. It seems to us, in the above result, the assumption that ‘‘ A is an FP-ring’’ is unnecessary.

Remark 2. Let G12and G be group schemes over a field k and n G12,G,ށk. Suppose that G12and G are forms of each other. We conjecture that G12, G . In another paper we will study this problem in the following cases:

1. Gi is an extension of two vector group schemes. 2. char k / 0 and Ž.␯ ny1/ 0. G i THEOREM 2.5. Let A be an F-ring and G be a group scheme o¨er A. Then m m G is a form of Wn, AAn iff locally G py , W ,Apy for some m g ގ. If we m further assume that A is an FP-ring, then G is a form of Wn, AA iff G py , m Wn, A pyfor some m g ގ. Proof. We only need to prove the necessity for the first part.

Let G be a form of Wn, A and 0 - G1 - иии - Gn s G be a composi- Ž.Ž tion series of G s.t. GiirG y1is a form of Gi␣,As1,...,n Proposition 2.3. . By Theorem 1.8, there exists a positive number m s.t. locally

Ž.ppym ym pymŽ. ppym ym GrGny1 A sG␣ , AAn, that is, G r G y1As G␣,A. By Theo- Ž.pppym Ž.ym ym pym rem 2.4, locally all GiAr G iy1A s G␣,AA. Then locally G , m Wn,Apy. DEFINITION 2.6. Let A be an F-ring with K s Fr A. Suppose that Ž. GsSpec B is a form of Wn, AK. Then the subalgebra of B , Kwᒍ B x, generated by ᒍŽ.B is a Hopf algebra over K and Spec KwᒍŽ.B xis a form of G␣, K . We define the height of Spec KwᒍŽ.B xas the height of G and write it as htŽ.G . FORMS OF THE WITT GROUP SCHEMES 197

From the results above, we obtain:

COROLLARY 2.7. Let A be an FP-ring and G be a form of Wn, A with Ž. ym ym ht G s m ) 0. Then m is the smallest integer s.t. GAnpp, W ,A. Now we consider the converse of Proposition 2.3 and characterize the

forms of Wn, A for some kinds of A. THEOREM 2.8. Let A be an F-ring and let G be a group scheme o¨er A. Then G is a form of Wn, A iff there exists a unique composition series

0 s G01- G - иии - Gns G such that: Ž. 1. GiirGy1 is a form of G␣,A i s 1,...,n and 2. the Verschiebung morphism of G G , ␯ : Ž.G G Žp.G G , has r iGrGiir ª r i the factorization

Ž.pp␶ 6 Ž. Ž.GrGGii ŽrGq1 .

␯

GrGi X 6

6 ⑀

GrGi6 Gny1rGi

for i s 0,...,ny2, where ␶ is the natural projection and ⑀ is the natural injection. Proof. We prove the sufficiency by induction on n, and so we may

assume that GrG1 is a form of Wny1,A . Then there exists an integer m s.t. Ž.ppym ym pymŽ. GrG1 Anis locally W y1, A and G1 is locally G␣, A Theorem 2.5 . Let

u m m Spec A ppy Spec A y , where a A s D aii g is1

and for each i,

ym m p Ž.py pym Ž.G1Aa,G␣,AasSpecž/Ay, iaii wx

ym m p Ž.py pym Ž.GrG1Aa,Wn1, AasSpecž/Ax1,..., xn1, iy aii wxy

where y, x1,..., xny1 are indeterminates. Then we might write

ym m p GŽApy .sSpecž/Axa 1,..., xn1, y aii wxy 198 CHANGCHUN LI and have the exact sequence

m m ⑀:0 Spec Aypy Spec Axpy ,..., x , y iaª ž/iiwxª ž/ awx1ny1

m Spec Axpy,..., x 0. ªž/ai wx1 ny1ª Ž. Ž. By Lemma 2.2, for each i, ⑀iins hFe for some hFisc0, i qcF1, i ym li иии cFli AFp . Write gyŽ. cy cyppиии cy. qq lii,iag wx is0, i q1, ilq q i,i Then we obtain the following commutative diagram for each i:

ym 6 ym m p m p G␣,Aanppy sSpecž/AyW,Aay sSpec ž/Ax1,..., xn1, y aiiiwx aiwy x

␶i ␦i

6 6

ym6 ym mppm G␣,AappysSpecž/AyGŽAy.sSpec ž/Axa1,..., xn1, y , aiiiwx a iwy x where

oŽ.Ž.␶iiy s gy Ž.,

oŽ.Ž.␦ijxsx j,js1,...,ny1,

oŽ.Ž.␦iiysgy Ž..

Note that the Verschiebung morphism of Wn is given by

x121¬ 0, x ¬ x m 1,...,

xny1 ¬xny2 m1, y ¬ xny1 m 1.

m Then the Verschiebung morphism of GŽ A py . is given by ai

x121¬ 0, x ¬ x m 1,...,

xny1¬xny2m1, y ¬ gxinŽ.y1m1. But by our hypothesis,

m m Axppy ,..., x , gxŽ. Axy ,..., x , x . ž/ai 1 ny2 iny1sž/ai wx1 ny2ny1

m Since ŽA py.Žis an integral domain, then gx .bx for some aii ny1siny1 ym p m biaŽA .*. This implies that ␶iis an automorphism of Ga,Apy and so g i ai

m m GAnpyW,Apy,i1,...,u. aaii, s

This means that G is a form of Wn, A. FORMS OF THE WITT GROUP SCHEMES 199

COROLLARY 2.9. Let k be a field of char s p and let G be a form of Wn, k . Then under one of the following conditions, G , Wn, k: 1. Gln, W,l for some separable field extension l of k. 2. k is a perfect field.

Proof. Let 0 - G1 - иии - Gn s G be a composition series s.t. Ž. GiirGy1is a form of Gi␣,ks1,...,n . Then under either condition, GiirGy1,G␣,kn. By Theorem 2.4, G , W ,k. COROLLARY 2.10. Let k be a field of char s p and let G be a form of Wn, k . Then there exists a purely inseparable extension l of k s.t. Gln, W ,l. COROLLARY 2.11. Let A be an FP-ring with K s Fr A and let G be a form of Wϱ, A. We denote by m the height of Spec KwᒍŽ.G x. Then m is the m m smallest number s.t. GA py , Wϱ, A py and for any l G m, we ha¨e the following exact sequence of group schemes o¨er A:

0 ª Wϱ, Alª G ª G ª 0, where Gll is a form of W,A of height m.

Proof. Let B be a faithfully flat algebra extension of A s.t. GB , Wϱ, B. n For n g ގ we denote by ␯G the nth Verschiebung morphism of G. Since coker ␯ n exists, then coker ␯ n exists. Write N coker ␯ n. Then we have GGnB s G the following commutative diagram with exact rows:

nq1

␯ 6 6 6 Ž p nq 1. G GGNnq1 0

Žpn. ␥ ␯G nq1, n

6 n 6

␯ 6 6 6 Ž pn. G GGNn 0.

Ž. Ž. Ž␥ . Then Nnq1 Bn, W q1, BnBn, N , W ,Bn, and q1, nB is the projection from Wnq1, Bnto W ,B. From

lim Ž.Nnq1BBs lim Ž.Nnq1s lim Wnq1, B s Wϱ, B , ␥Ž␥ . ¤nq1, n ¤¤nq1, nB we conclude that

G , lim Nn 1 . ␥ q ¤ nq1, n

Ž. Ž . Note that ᒍ G s ᒍ Nnnfor any n g ގ. So the height of N is indepen- Ž. Ž.ym ym dent of n. Set m s ht Nnn. By Corollary 2.7, N Apps Wn,Afor 200 CHANGCHUN LI n g ގ. Hence

py m pym py m GAns lim Ž.N q1As Wϱ,A. ¤␥nq1, n

m m Obviously, this m is the smallest one s.t. GA py s Wϱ, A py . Note that for any l ) m y 1,

␯ l Ž p l . G Ž.@:0ªG ªGªNlª0

Ž p l . is an exact sequence of group schemes over A, and by the above G , Wϱ, Aland N is a form of W l,A. So we obtain the proof of the second part. Because of Corollary 2.11, we could extend the concept ‘‘height’’ to the forms of Wϱ. That is DEFINITION 2.12. Let A be an F-ring with K s Fr A. Suppose that G s Spec B is a form of some Witt group scheme. Then the subalgebra of BK , KwᒍŽ.B x, generated by ᒍŽ.B is a Hopf algebra over K and Spec KwᒍŽ.B xwis a form of G␣, K . We define the height of Spec K ᒍŽ.B xas the height of G and write it as htŽ.G .

COROLLARY 2.13. Let A be a discrete ¨aluation ring of char s p with a perfect residue field k and let G be a group scheme o¨er A. If G is a form of Wn, A Ž. resp. Wϱ, An, then G is already W,AŽ. resp. Wϱ,A.

3. THE MODELS OF WITT GROUP SCHEMES

In this short section, we study the models of Witt group schemes. Let A be a discrete valuation ring of char s p. We denote its fraction field and residue field by K and k, respectively.

THEOREM 3.1. Let G be a smooth group scheme o¨er A. If GKn, W ,K and Gkn, W ,kn, then G , W ,A.

Proof. Let H1 be the unique subgroup scheme of GK that is isomor- phic to G␣, K and let G11be the schematic closure of H in G wxSim . By wxRay , the quotient GrG1 exists, and byw Simb x , it is affine. Denote it by G22. Then G is smooth with connected fibers. This implies that dimŽ.G2k Ž. Ž. Ž. sdim G2 K s n y 1. Hence G2 kn, W y1, k, G2 Kn, W y1, K , and G1 , G␣, A. By induction on n, G2 , Wny1, A. Therefore, we obtain an exact sequence of group schemes over A:

0 ª G␣, Anª G ª W y1, A ª 0. FORMS OF THE WITT GROUP SCHEMES 201

Ž. Let x1,..., xny1 be a system of standard coordinates of Wny1, A and G␣, A sSpec Aywx, where y is a primitive element of G␣ , A. Then G s Ž.Ž.␯ Ž . Spec Axwx1,..., xny1, y with o Gny s gxy1m1 for some p-poly- Ž. Ž. nomial gxny1 . Since GKn, W ,Kn, then gxy1saxny1for some a g A Ž. Theorem 2.8 . Note that Gkn, W ,k, by the same reason, the image of a in k is nonzero. Hence a is invertible in A and G is a form of Wn, A Ž. n Theorem 2.8 . But the underlying scheme of G is already ށ An,soG,W,A Ž.Theorem 2.4 .

COROLLARY 3.2. Let G be a smooth group scheme o¨er A. If GKk and G are forms of finite dimensional Witt group schemes, then so is G o¨er A.

n Proof. By Theorem 2.5, there exists a positive integer n s.t. both GK py yn yn n p p and Gk py are finite dimensional Witt group schemes over K and k , n respectively. By Theorem 3.1, GA py is a finite dimensional Witt group n scheme over A py .

Remark 1. In the above results the restrictions on Gk are necessary. We give two examples below.

EXAMPLE 1. Let ␲ be an uniformizer of A and let Ž.x1,..., xn be a system of standard coordinates of Wn, A, where n ) 1. Then G s y1 n Spec Awx␲ x12, x ,..., xnKnis a group scheme with G , W ,KAand G s ށ . But Gkn` W ,k, since there are two primitive elements in Gkthat are algebraically independent.

p EXAMPLE 2. Let ␲ be an uniformizer of A and a g A and f A .We define a group scheme G over A as

Awx¨,x,y GsSpecpp , ž/²:yyŽ.xqax where x and y are primitive elements and

1 py1 a py1 jj pyjjjpyj ⌬Ž.¨s¨m1q1m¨qÝÝCyppmy q Cxmx . pp js1 js1

We will see in Section 4 that H is a form of W2, A with height 1. Set p p B s Ž Awx¨, x, y .²r y y Žx q ax .:. Then H s SpecŽBr ²␲ , x, y :. is a subgroup scheme of Gk that is isomorphic to G␣ , k . The blow up of G along H, G H, is still a group scheme over A wxWatWei . It is given by

H 1 1 G s Spec A ¨ , ␲y x, ␲y y , 202 CHANGCHUN LI

y1 y1 where Aw¨, ␲ x, ␲ yx indicates the sub-A-algebra of B mA K generated y1 y1 H by ¨, ␲ x, and ␲ y. It is easy to show that GK is a form of W2, K of H 2 height 1 and Gk is a form of G␣ , k of height 1. Remark 2. Let G be a group scheme over A. We conjecture that:

1. G , Wϱ, AKif G , Wϱ,Kkand G , Wϱ,k; 2. G is a form of Wϱ, AKif G is a form of Wϱ,Kkand G is a form of Wϱ,k.

4. THE FORMS OF W2 AND THE FORMS OF WITT GROUP SCHEMES OF HEIGHT 1

The purpose of this section is to study some simple cases of forms of Witt group schemes over an FP-ring.

Let A be a commutative ring containing Fp and let G s Spec B over A be a form of some Witt group scheme. Then the subalgebra of B, AwᒍŽ.B x, generated by ᒍŽ.B is a Hopf algebra over A and H s Spec AwᒍŽ.B xis a m form of G␣, A. If we further assume that A is an FP-ring, then HA py , m G␣, A py , where m is the height of G Ž.see Section 1 . For our convenience, we make the following definition.

DEFINITION 4.1. Let A be a commutative ring containing Fp and let G s Spec B over A be a form of some Witt group scheme. We say that G is of type Spec AwᒍŽ.B x. Let H be a form of the additive group scheme over A. By the discussion in Section 2, we know that any form of W2, A of type H is an extension of H by H Ž p.. Now we assume that A is an FP-ring. Because of Theorem 1.9, we can use the same argument as that inwx KMT, 3.4.1 TheoremŽ also see wxLic, Theorem 2.1.4. to obtain the following exact sequence:

Ž p.Žp.Ž␴ p. 0ªExt␯s0commŽ.H, H ª Ext Ž.Ž.H, H ª End H ª 0, ŽŽp.. where Ext␯s0 H, H indicates the set of all extensions with zero Ver- schiebungs. Assume that

Žp. A:0ªH ªTªHª0 Žp. is an element in Ext commŽH, H .whose image under ␴ is idH Žp. . Then we have

THEOREM 4.2. Let A be an FP-ring and let H be a form of the additi¨e group scheme o¨er A. Then the set of forms of W2, A of type H is just

Ž p. Ž p. Ä4G N 0 ª H ª G ª H ª 0 g A q Ext␯s0Ž.H, H . FORMS OF THE WITT GROUP SCHEMES 203

Ž p. Žp. Proof. First we note that Ext commŽ H, H .Žis a right End H .- Ž Ž p.. module in an obvious way and Ext␯s0 H, H is its submodule. For any exact sequence of group schemes over A,

Žp. B:0ªH ªGªHª0, Žp. Gis a form of W2 iff ␴ Ž.B is an automorphism of H Ž.Theorem 2.8 . Set ␴ Ž.B s ␣. Then ␴ Ž.A␣ s ␣, i.e., B s A␣ q C for some C g Ž Ž p..Ž␣ Žp..Ž␣␣y1. Ext␯s0H, H .If is invertible in End H , then B s A q C . 1 Hence the middle terms of both B and A q C␣y are isomorphic. This finishes our proof.

THEOREM 4.3. Let A be an FP-ring and let H be a form of the additi¨e group scheme o¨er A with height 1. Then there exists a unique form of W2, A of 2 type H. We suppose that as a subgroup of G␣,As Spec Axwx,y, His expressed by the ideal generated by an equation

p p pr y s x q ax1 qиии qaxr , p Ž. where a1,...,arigA with some a f A Theorem 1.8 . Then this unique group scheme G is gi¨en as

Awx¨,x,y GSpecr , s ²:ppp ž/yyŽ.xqax1qиии qaxr where x and y are primiti¨e elements and

1 py1 jj pyj ⌬Ž.¨s¨m1q1m¨y Ýcyp my p js1

r p 1 a y jpj ijpiy1 piy1y qÝÝCxpŽ.Ž.mx , p is1 js1

m where Cn indicates the number of combinations of m elements from a set of n elements. Proof. Ž. Claim 1. Ext␯s0 G␣ , A, G␣ , A s 0. Let

Ž.␤ :0ªG␣,AªGªG␣,Aª0 Ž. Ž.Ž.Ž.␤ be an element of Ext␯s0 G␣ , A, G␣ , A . By Lemma 2.2, s egF2 for Ž. Ž. Ž . some gFgAFwx. But ␯G s 0, then gFs0, hence ␤ s 0. This proves our Claim 1. 204 CHANGCHUN LI

Ž. Claim 2. Ext␯s0 H, G␣ , A s 0. To prove our Claim 2, we let

h ␲ :0ªG␣,A ªGªHª0 Ž. py1 Ž be an element in Ext␯s0 H, G␣ , AB. Set B s A . Then H , G␣,BTheo- .2 rem 1.9 . By Claim 1, GB , G␣ , B. This implies that

ᒍŽ.hB 0ªᒍŽ.HBBªᒍ Ž.G ªᒍ ŽG␣,B .ª0 is an exact sequence of BFwx-modules. That is to say that, after a faithfully flat algebra extension of A, the sequence of AFwx-modules

ᒍŽ.h 0ªᒍŽ.H ªᒍ Ž.G ªᒍ ŽG␣,A .ª0 becomes exact. Then it is already exact. Therefore, there exists an AFwx- morphism gЈ: ᒍŽ.Ž.G ᒍ G s.t. gЈиᒍ Ž.h id . Since G is a form ␣, A ª s ᒍŽG␣, A. 2 Ž. of G␣, A, there exists a unique morphism g: G ª G␣ , A s.t. ᒍ g s gЈ and then h и g idŽ. Proposition 1.2 . This means ␲ 0 and so s G␣ A s Ž., Ext␯s0 H, G␣ , A s 0. By Claim 2 and Theorem 4.2, we get the proof of the uniqueness. Ž. Now let W2, A s Spec Awx¨, z , where z, ¨ is a system of standard Ž. Žp. coordinates Definition 2.1 . Then for the G in the theorem, G , W2, A, Ž ppry1. where ¨ ¬ ¨ and z ¬ y y a12q ax qиии qaxr . But obviously, G is p an extension of H by H .SoGis a form of W2, A of type H.

THEOREM 4.4. Let A be an FP-ring and let H be a form of G␣, A of height ގ 1. Then for each n g , there exists at most one form of Wnq1, A of type H. X Proof. Let Wnq1 be a form of Wnq1, A of type H. Then we have an exact sequence of group schemes over A:

XXtr enq1:0ªG␣ , AnªWq1ªTnª0. We will prove the following statement that implies our original one: X Ž X . Wnq1 is the unique form of Wnq1, Anof type H and Ext W , G␣,Ais a free ŽX . right AFwx-module with the basis enq1 . We prove this statement by induction on n. For the case that n s 1, it has been provedŽ. see Theorems 4.2 and 4.3 . Suppose that the statement is correct for n - k and consider the case that n s k. By our hypothesis, Tiiis the unique form of W of type H for X i-kq1. We write Wiifor T . From

XXtrX ek :0ªG␣ , AkkªW ªWy1ª0, FORMS OF THE WITT GROUP SCHEMES 205 we obtain the following long exact sequence of abstract groups:

␦ иии X ª Hom AŽ.G␣,A, G␣,AAª Ext -comm Ž.Wky1 , G␣ , A g h X ª Ext A-commŽ.Wk , G␣ , AAª Ext -comm ŽG␣ , A , G␣ , A ..

By induction, ␦ is an isomorphism; then g is an injection. It is easy to see that g is a right AFwx-map. Ž X . ␶ Now let gekq1 s . This means that we obtain the following commu- tative diagram of group schemes over A with exact row: 6 6 6 6 ␶ :0GXG␣,A ␣,A0

t

6 6

XX6 6 6 X6 ekq1:0 GW␣ , Akq1 Wk0.

X X Ž.ppy1 y1 Ž. pppy1 y1 y1 Note that Wkq1 Ak, W q1, AkAkand W , W ,AA. Then X , 2 y1 Ž.Ž. W2, AAp. But X s ށ , then X , W2, A Theorem 2.4 . So ␶ is a basis of X X Ext A-commŽ.G␣, A, G␣, Akand hence Že 1.Žis a basis of Ext A-comm Wk , G␣, A.. Y q If Wkq1 is another form of Wkq1, A of type H, then by repeating the above Ž Y . argument, we have another basis ekq1 , where

YYtrX ekq1:0ªG␣ , AkªWq1ªWkª0.

Y X This means ek 1 s ek 1 a for some invertible element a g A. Hence YXq q Wkq1,Wkq1.

COROLLARY 4.5. Let A be an FP-ring and let H be a form of G␣, A of height 1. Then the following are equi¨alent: 1. For each n g ގ, there exists a form of Wn, A of type H; 2. there exists a form of Wϱ, A of type H; and 3. there exists a unique form of Wϱ, A of type H.

ACKNOWLEDGMENT

This paper was carried out under the supervision of Professor William Messing. I express my deepest gratitude to him. 206 CHANGCHUN LI

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