ANNALES SCIENTIFIQUES DE L É.N.S. JOHN TATE FRANS OORT Group schemes of prime order Annales scientifiques de l É.N.S. 4 e série, tome 3, n o 1 (1970), p

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ANNALES SCIENTIFIQUES DE L É.N.S. JOHN TATE FRANS OORT Group schemes of prime order Annales scientifiques de l É.N.S. 4 e série, tome 3, n o 1 (1970), p Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1970, tous droits réservés. L accès aux archives de la revue «Annales scientifiques de l É.N.S.» (http://www., implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Ann. sclent. EC. Norm. Sup., 4 s6rie, t. 3, 1970, p. i a 21. GROUP SCHEMES OF PRIME ORDER BY JOHN TATE AND FRANS OORT (*) INTRODUCTION. Our aim in this paper is to study group schemes G of prime order p over a rather general base scheme S. Suppose G===Spec(A), S=:Spec(R), and suppose the augmentation ideal I==Ker(A ^R) is free of rank one over R (so G is of order p = 2), say I = Rrc; then there exist elements a and c in R such that x 2 = ax and such that the group structure on G is defined by sx=x(^)i-{-i(^)x cx(^)x. One easily checks that ac=i\ conversely any factorization ac = 2 R defines a group scheme of order 2 over R. In this way all R-group schemes whose augmentation ideal is free of rank one are classified, and an easy sheaf-theoretic globalization yields a classification of group schemes of order 2 over any base S. case p 2 the difficulty is to find a good generator for the ideal I. To this end we prove first (theorem 1) that any G of order p is commutative and killed by p, i. e. is a ( module scheme 59 over F^==Z/pZ. In order to exploit the action of F^ on G, we assume in section 2 that the base S lies over Spec (Ap), where ^^'i^-d^' ^ being a primitive (p i)-th root of unity in the ring of p-adic integers Zp. For S over Spec (Ap) we prove (theorem 2) that the S-groups of order p are classified by triples (L, a, &) consisting of an invertible 0s-module L, together with sections a and b of L 0^ ^ and L 0^ such that a(^) b = Wpy where Wp is the product of p and of an invertible element of Ap. Since the p-adic completion of A.p is Zpy this structure theorem In (*) Work on this paper was partially supported by the National Science Foundation. 2 J. TATE AND F. OORT. applies in particular to a base of the form S==Spec(R), where R is a complete noetherian local ring of residue characteristic p : for such an R, the isomorphism classes of R-groups of order p correspond to equivalence classes of factorizations p = ac of p R, two such factorizations p = ai Ci and p = a^c^ being considered equivalent if there is an invertible element u in R such that a^=u p ~^a^ and c^=u i ~ P Cl (c/*. remark 5 at the end of section 2). In section 3 we apply this theory to obtain a classification (theorem 3) of group schemes of order p defined over the ring of integers in a number field, in terms of idele class characters. As a special case we recover an unpublished result of M. Artin and B. Mazur, to the effect that the only group schemes of order p over Z are the constant group (Z/pZ)^ and its Cartier dual pi^' Our original proof of theorem 1, insofar as the killed by p 9? part is concerned, was intertwined with the proof of theorem 2. We can now avoid this procedure thanks to P. Deligne, who communicated to us a direct proof of the fact that, for any integer m^i, a commutative group scheme of order m is killed by m. We give Deligne's proof in section PRELIMINARIES AND TWO GENERAL THEOREMS. Let S be a prescheme and T a prescheme over S. We say that T is of finite order over S if T is of the form T==Spec(A), where A is a sheaf of g-algebras which is locally free of constant rank r; and then we say that T is of order r over S. If S is locally noetherian and connected, then T is of finite order over S if and only if it is finite and flat over S. Suppose G-='Spec{A.) is a group scheme of finite order over S. We denote by ( J ) s^=zs: A- A(g)^A respectively t^== t : A (g)^ A - A the homomorphisms of s-algebras which correspond to the law of composition, respectively the diagonal map GxsG-.G, Ao : G- GxsG. Let A' denote the (^-linear dual of A : A'=S om^(a, 0s); this is a locally free sheaf of the same rank as A. of finite rank, the natural map A'(g)^A^ (A(g)^Ay As A' is locally free is an isomorphism, and we obtain (2) t^=(s^': A'^A^A 7 and ^= W ; A'-^A^^A', GROUP SCHEMES OF PRIME ORDER, 3 which makes A' into an associative, coassociative, and cocommutative Hopf-algebra over Og, with unit and counit the analog of the group algebra of G. The map (3) G (S) == Hom^,y ^-algebras (A, ^) ^~ T (S, A') is an isomorphism of G(S) onto the multiplicative group of invertible elements ger(s, A') such that s^,{g} === g0g. The group scheme G is commutative if and only if the ring A' is commutative. Suppose this is the case. Then the S-prescheme G'= Sp^c(A'), with the law of composition induced by s^y is a commutative S-group of the same finite order as G, the Cartier dual of G. As there is a canonical isomorphism A^AT, we have G===(G')', and (3) can be interpreted as an isomorphism (4) G(S) ^ Homs-g^G^ Gr/^s), where G,^g is the multiplicative group over S. Viewed symmetrically, (4) gives a bimultiplicative morphism of schemes over S, (5) GxsG^G^s, which we call the Cartier pairing. Let G ^-S be an S-group scheme. For each integer m Z we denote by me: G-^G the morphism obtained by raising to the m-th power all elements of the group functor G, i. e. for all T - S, and any ^eg(t), m^^)=^1. Suppose G=Spec(A), then we use [m]: A - A for the corresponding 0s-algebra homomorphism. The ( laws of exponents 9) (^n) m =^nm and (^m) (^) == ^m+ /^ amount to the identities [m ].[/?] == [mn\ and t^o {\m\ (g) [ n]) o s^== \m 4- n\ Of course [i]====ida, and [o] = 1*0^ where : A (9g corresponds to the neutral element of G(S), and where i: (9s-^-A corresponds to the structure morphism G - S. The ideal I = Ker(s) = Ker[o] is called the augmentation ideal of A (or of G). If m^2, clearly [m]== (A-^A^-^A), 4 J. TATE AND F. OORT. the first arrow being defined by iteration of s^, and the second being the multiplication. We thank P. Deligne for letting us present here his proof of : THEOREM (Deligne). A commutative S- group of order m is killed by m (i.e. 171^=0^). The proof of the theorem is inspired by the following : let F be a finite commutative (abstract) group of order m, and let x^t. Then n^n^^fn^)^ yer T r \er / and hence x m = e. In order to be able to apply this idea to group schemes Deligne defines the following trace map : let G be a commutative group scheme of finite order over S, and suppose 1 T= t pec(b) is of order m over S, with structure morphism f:t-^s. Then Tr/ is the unique map such that the diagram G(T)C ^(T^^A^^B ^) I Try 4- ' 4- G(S) c ^r(s, A') is commutative, where the (injective) horizontal arrows are as in (3), and where N denotes the norm map for the A'-algebra B(g)^A', which is locally free of rank m over A' (here we use the commutativity of A'). From this definition we easily deduce that Try is a homomorphism, and that (6) Tiy(/^)==:^ forall^eg(s), where f =G{f): G(S) - 'G(T). Suppose t: T - T is an S-automorphism; then (7) Tiy(p)=Tiy(T^T_iG) for all (3eG(T); this follows immediately from the properties of a norm map. Proof of the theorem. In order to prove that a group scheme H - U of order m is killed by m, it is sufficient to show that for any S - U, each element of H(S) has an order dividing m; as H(S)= Horns (S, HxuS), it suffices to prove that for any f: G-^S, a commutative group scheme of order m, and for any section u G(S) we have 1^=1. We denote N GROUP SCHEMES OF PRIME ORDER. 5 by ta: G - G the translation on G by u, i. e. ^= (G ^ G XsS^ G XsG- G). We consider (the analog of JJn? ^d using (7) we note that TCP As Tiy(i,0 ==Tiy(G-^G-^G). IGO^=IGX (fu) : G G (o means composition, and X means multiplication), using (6) we obtain Tiy(iG) =Tiy(iGX (/^)) = Try^) X Try (/^) == Try (i(,) X ^, and the theorem is proved. Remark. A group scheme of order m over a reduced base is killed by its order [cf. [I], 11^.8.5), however we do not know whether this is true for (non-commutative) group schemes over an arbitrary base. Example and notation. Let F be a finite group, R any commutative ring with identity element, R[F] the group ring ofr, and R 1 = Map(F, R) the ring of functions from F into R. The constant group scheme defined by r over R is 1^= Spec(R 1 ). Elements of R[F] are R-linear functions on R 1, and we see that R 1 and R[F] are in duality. In particular the dual of the constant cyclic group scheme (Z/nZ)^ is ^,R, the group scheme representing the n-th roots of unity for any R-algebra B. ^,R(B)==:J^[^ B, ^==1 $ Let p be a prime number. For the rest of this section we will be concerned exclusively with groups of order p. THEOREM 1. An S-group of order p is commutative and killed by p. By Deligne's theorem we need only prove commutativity. It is clear that it suffices to treat the case S = Spec(R), where R is a local ring with algebraically closed residue class field. LEMMA 1. Let k be an algebraically closed field, and suppose G = Spec (A) is a k-group of order p. Then either G is the constant group scheme, or the characteristic of k equals p and G == ^p,k or G = a?^. In particular, G is commutative and the k-algebra A is generated by a single element. Postponing the proof of the lemma for a moment we first show how theorem 1 follows from it. Let tilda (i. e. ) denote reduction modulo Ann. J?c. Norm., (4), III. FASC. 1. 2 6 J. TATE AND F. OORT. the maximal ideal of R. Then G = GXsSpec(/c) is commutative by lemma 1, and we can apply that lemma to its Cartier dual G'= Spec (A'). Let rcea' be such that its residue class e(a') == (A)' generates the /c-algebra A'. Then {R[x]f= k\x~\ = (A'f, and by Nakayama's lemma (which is applicable because A' is a free R-module of finite rank p) we conclude that A'==R[^]. Hence A' is commutative, and this means G is commutative. For the convenience of the reader we include a proof of the well-known lemma 1. Recall first that the connected component H of a finite /c-group H is a (normal) subgroup scheme, and if Hi is a subgroup scheme of H, then the order of H equals the product of the orders of Hi and of H/Hi (c/ 1. [I], 1^.3.2 (iv)). Since G is of prime order, its connected component is either Spec(/c) or all of G, and, accordingly, G is either etale or connected. If G is etale, then it is constant because k is algebraically closed, hence it is isomorphic to (Z/pZ)/,, and A, the /c-algebra consisting of all /c-valued functions on Z/pZ, is generated by any function which takes distinct values at the points of Z/pZ. Suppose G=Spec(A) is connected, i.e. the /c-algebra A is a local artin ring. Its augmentation ideal IcA is nilpotent. By Nakayama's lemma 17^ I 2, hence there exists a non-zero /c-derivation d : A - A*. This means that the element de. I'C A' has the property s^ {d) = d(^) i + i(g) d (as =iea'). Thus /c[c?]ca' is a /c-sub-bialgebra of A', and as k[d] is a commutative ring, we obtain a surjective /c-bialgebra homomorphism A ^ A» {k[d]y ; as the order p of G is prime, this implies that the order of k[d] equals p, and hence k[d] = A'. As before we conclude that G'=Spec(A') is either etale or connected. If G' is etale this means G'^ (Z/pZ)/,, and thus G^^,/,; as G was supposed to be connected this implies char(/c)=p. If G' is connected, d is nilpotent, and, as k[d] is of rank p, we must have d^1^ o and d 1 ' = o; as s^, is a ring homomorphism this implies p = o in /c, hence char(/c) = p ; moreover we already know that s^, {d) = d (g) i + i (g) d, hence G' ^ a^/,, and thus G ^ a^/, which proves the lemma. Note that the last part of the proof could have been given using p-lie algebras (c/*. [I], VIL. 7). Remark. In contrast with group theory there exists a group scheme of rank p which acts non-trivially on another group scheme of rank p, namely ^ resp. Up. Hence there exist group schemes of rank p 2 which are not commutative. For example, let R be any F^-algebra, and define A=R[T,a], with T^==I, 0-^=0, ST== T^T, and 50- = T(g)G-4-cr(g) i. The R-group G==Spec(A) is isomorphic to the semi-direct product GROUP SCHEMES OF PRIME ORDER. 7 of the normal subgroup scheme defined by T==I, which is isomorphic to a^r, and the subgroup scheme defined by o- === o, which is isomorphic to ^R. 2. A CLASSIFICATION THEOREM. We denote by Z^, the ring of p-adic integers, and by x : y/.- zp the unique multiplicative section of the residue class map Z^ ^-F^== Z^/pZ^. For any a Z^,, we have ^W= lim [ap-\, V- - os where 2 is the residue of a modp. Thus, y(o)==o, and for m F^, y^(m) is the unique (p-i)-th root of unity in Zp whose residue (mod?) is m. The restriction of ^ to F^ is a generator for the group Horn (Fj^, Z^) of ( multiplicative characters of F^??. Let ^-^^'^-Ty]^-' the intersection being taken inside the fraction field Qp of Zp. Thus Ap is the ring of elements in the field of (p i)-th roots of unity Q^^))? which are integral at all places not dividing p(p i) and also at one place above p, namely that given by the inclusion Q(y,(F/ ))CQ^. The prime ideal in A.p corresponding to this last place is A/.npZ/,==pA^ and Zp is the p-adic completion of Ap. Examples : p==2, A^=Z; p=3, A^=Z ^j; p=5, A^zp,^^], where ^=^(2) is the unique element of Z,^ such that i 2 :^ -!, and 1^2 (mod 5). In this section we fix a prime number p, we write A == A^,, and we assume our ground scheme S is over Spec (A). We shall often view ^ as taking values in the A-algebra r(s, 0s)? writing simply %.(m) instead of %.(^).i(9 $ tor example if p0s==o? then ^(m) = m. Let G== Spec(A) be an S-group of order p. By theorem 1, the group F* operates on G, and we can therefore regard A, and the augmentation 8 J. TATE AND F. OORT. ideal I of G, as sheaves of modules over the group algebra (9s [F^]. For each integer i, let I,==^I, where ei is the (?s-linear operator (8) ^^-S^^W^F;]. ^ F/S Clearly e^ hence also I,, depends only on i (mod(p--i)). { \ LEMMA 2. We have I =^, I^, direct sum. For each i, I, is an!==! iwertible Q^module^ consisting of the local sections of A. such that [m]f=^{m) f for all m^tp. We have IJyC!,+/ for all i andj, and I; == I, for i^i^p i. Proof. For i^^'^p i, the elements e, are orthogonal idempotents in the group algebra A[F^] whose sum is i and which satisfy [m}ei=^(m)ei for m F^. Hence I is the direct sum of the! for i^i^p i, and \i consists of the local sections f of I such that [m]f= ^{m)f for all m F^, or, what is the same since y/(o) ==o, of the local sections f of A such that [m]f==^{m)f for all m F^. From this and the rule [ m }{fg)={[ m \f} ( \[ 1n \g) we see that IJyCl^. Since the 9s-module Its locally free of rank p i, its direct summands I, are locally free over 0s of ranks r, such that r^ ^_i = p i. To prove that n=i for each i, and that 1[ = I, for i^^p i, it suffices to examine the situation in case S = Spec(/c), where the A-algebra k is an algebraically closed field, and to exhibit in that case a section fi of Ii such that f[^o for i^i^p i; then kf[ C I, shows r^i for all i, hence ^=i, and kf[ = I;. By lemma 1 there are only three cases to consider, namely G ^ (Z/pZ)^ x^/,, or ^,/,, and the last two only for char(/c) = p, in which case %(m) = m. If G ^ (Z/pZ)/,, then A is the algebra of /c-valued functions on Z/pZ = Tp, and ([m]/*) {n) = f{mn) for jfea and m, n F^; hence we may take fi=^ If G^a^/, (resp. ^/,), then A = k[t] with tp= o, ^ = ((g)i + i (g) ^ so [m]( = m ^ [resp. 5(14- () = (i+ ) (i+ (), so [m]t = (i+ ^ i]; hence in both cases [m]t= mt = ^(m)t (modt 2 ), we have eit=t^o (modt 2 ), and can take f^=eit. This completes the proof of lemma 2. Example and Notation. The group p^,a. We have p^,a= Spec(B), where B = A[js], with ^==1. The comultiplication in B is given by sz == z(^)z, and [m]z = z 1 for all m F^. The augmentation ideal L = J of B is J = B{z i), and has a A-base consisting of the elements 7^ i for m F^ : B (z i) = J === A (z i) A (zp- 1 i). GROUP SCHEMES OF PRIME ORDER. 9 For each integer i we put (9) yi==(p-i)ei(i-z) ^) ::=: == ^ r^h r 1^) (i-^ 1 -^) m F? [ 7^ ^1 ^//t ^/n if^==o if^=o mod (/? i), (p i), } m Fp w Fp ^ ^ ^'(/n)^ -'(/n)^ if?-^o if^o mod(/? i) mod(p-i). Note that yi depends only on ^mod(p i). Then p-i (10) 1 ^^==p i-^^:^-^.^^(^)^ for m F;, i=l and ^ J^i i j.= ^ /^'(w) {(i ^).(g)(i ^) ^ F^ /? l p \ ^ (^ z'tyi^z ^^)^ 2% 7 ( /n ) z^^)^- ^ /^ /=1 k=l = '^ 2j ^^^' /+^E^ mod(p l) hence p-\ ( 11 ) ^ =^ I + I 0^+Y- L ^J/0J^-7 Formula (10) shows that /=1 Jr=Aji+...4-A^_i. Hence J,=^J==A?/, for each i, because ^,...,^_i are orthogonal idempotents. Putting y = y^ we can therefore define a sequence of elements i = w^ ^2,... in A by (12) y^^-jz. PROPOSITION. The elements wi are invertible in A for i^i^p i, and Wp= pwp^. We have B == A[y], w^ ^ = w^2/, anrf /? -v^- ^' (13) ^J=J0i+i0j y + ^-^.^^(g)^-!; , J - - : /^-l ^-P ^ Wi (t^) I.^Jj^ZC^)^ ^or m F/,; (i^) Wi===i\ (mod?) for i^^^ i; (. .).=,+ fj+^ ^). I /?V tva W,,_J Z=1 Wp-i' 10 J. TATE AND F. OORT. Proof. By lemma 2, AY=(\yY=^^=^=\y^ thus the Wi are invertible for i^^^p i. thus 2/^=0 (modp); moreover 5=i+J ^ ^2 ^ {) } Clearly [z iy==o (modp), (mod/,); comparing the coefficients of y'^y, i^=^ ^p i? m both sides of / y/^1 \ - / y^- 1 \ i + y (g) i 4- y \ w p-^) \ ^-l/ ^i4-5j+_(.^)2_^ ^. (^)/.-i ^2 W^-l (mod/^), we obtain w,+i=(i+i)-^ (modp) for i^^ p i, which proves (i5). The other formulas have been proved already, except for the identity Wp= pwp_y. To this end, choose an embedding A=Ap c -^K, where K is some field containing a primitive p-th root of unity ^ =K (e.g. ACZ^, and choose for K an algebraic closure of Qp, or choose an embedding of A into the field C of complex numbers). Extend the embedding A C ^K to a homomorphism A[jz] - K by z(- c; let yi ^- r\i and r\=r\^ under this homomorphism. Then by (9) we find rjp_i = p; as p ^ o and w?^^ o in A, we see YJ ^ o, and using (12) we have Y^ pwp^ == ^^-i W/,-1 == Y^- 1 == = W,,,. which concludes the proof of the proposition. Remark. The wi^a can be computed inductively from Wi == i and the relations p if (E== o or / = o, ( 1?) S^-^ (- 1 )' if^o, j^o but,+./=o, ( i^^-'j^ ^./), if^o, y^o and?+y^o, where the congruences are mod(p i), and where ^ denotes the Jacobi sums g(i,./)= V ^J m 4- TI = 1 /«,/i F? ^(^)y/(^). Choose an embedding A = A^ A^C K as in the proof of the proposition; then ^z+/ _ f}jrij WiWf ~~ Yj^/ GROUP SCHEMES OF PRIME ORDER. 11 The first case of (17) is clear, because ^_i=p. Suppose i^o and j^omod(p i); then p ^ 2, and ^( i)= i; letting Z, m, and n run through Fp we have ^/ = j 2 ^(m)^ \ \ 2 ^/(n) s// } ( ^7^0 ) ( fl-^0 ) = ^J X -/ ( m ) % --/ (^)S ///+// //(7l-^:( =2^ 2 y. - '( /w )x -/ ( / ') l /n-{- /f,=l mn^:0 = S^' (- ) 7-7 (/() ^2 2j ^ ^t ( - /w) ^-/ (- //^ ^7^0 /^:0 fti-+-n=: 1 7/l/i T^: 0 = =(- I )^2^(^+7) ( /2 ) + (- I )^+/ (2 c/ '~ (^+7)(/) i //-^o ( /^o ) X ( ^ /.- / (^)x -/ ( /^) /// + n-=-- [ inn-^-o i (-i)^+' Y^. ;} (-,, -y), if / +y ^ o, = (-i)^-i)- ^ ^Y^V in-+./=o.,/n^ // + // = i //mt'^o This proves the third case of (17), and the second results on replacing n by mn in the last formula 2 ''-'W^ 2 t'( )=-7.'(-')=-(-i)'- //^+/?.== 1 //f(l4-/z)== 1 inn-^.0 n-^q, l Many of the facts established in the last few paragraphs are essentially equivalent to well-known properties of the Jacobi sums ^(^j), and the Gauss sums g[i)= r\_i attached to the multiplicative character y/ of the prime field (e. g. see [5]). As examples we mention p=2, A2==Z, Wi==I, W2==2; p=3, A3=:Z - 1, wj=i, ^== 1, w;;=: 3, 7?=5, Ag==Z^\ ^/^-., where?=:%(2) and ^^r i, Wi=I, W2= ^(2+0, W3==(24-0 2^ W4== (2 4-Q 2, w,= 5(24-?) 2. 12 J. TATE AND F. OORT. We now continue our d
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