On the surjectivity of localisation maps for Galois cohomology of unipotent algebraic groups over fields

Transcription

1 On the surjectivity of localisation maps for Galois cohomology of unipotent algebraic groups over fields Nguyêñ Quôć Thǎńg and Nguyêñ DuyTân Institute of Mathematics, 18 - Hoang Quoc Viet Hanoi, Vietnam Abstract We show that the localisation map for Galois cohomology of smooth unipotent algebraic groups over fields is always surjective and we give some applications. AMS Mathematics Subject Classification (2000): Primary 11E72, Secondary 18G50, 20G10 1 Introduction Let k be a field, S a finite set of (non-equivalent) valuations of k and for v S, let k v be the completion of k at v. We consider only usual linear algebraic k-groups G, which thus are smooth (i.e., absolutely reduced) algebraic k- groups G as in [Bo], [Oe], and by convention, a k-group is understood as a linear algebraic k-group. For a field extension K/k we denote by K s the separable closure of K in an algebraic closure K of K, andbyh 1 (K, G) := H 1 (Gal(K s /K),G(K s )) the usual first Galois cohomology set. We have a Regular Associate of the Abdus Salam I. C. T. P. Support in part by Fund. Res. Prog. Vietnam and Abdus Salam I. C. T. P. nqthang@math.ac.vn (corresponding author). 1

2 localisation map ϕ v :H 1 (k, G) H 1 (k v,g) and we denote by ϕ S := v S ϕ v the product of localisation maps ϕ S :H 1 (k, G) H 1 (k v,g). v S If k is of characteristic 0 (resp. perfect), then a well-known result of Rosenlicht says that any unipotent (resp. connected unipotent) (smooth) k-group has a composition series, each factor of which is k-isomorphic to the additive group G a (see [Ro1], [Ro2]). In particular, it follows that (see, e.g., [Se1], Ch. III) the first Galois cohomology for such unipotent groups is trivial. However, in the case k is non-perfect, this is no longer the case, and the cohomology set may be non-trivial (see [Se1], Ch. III, [SGA3], Exp. XVII, Sec. 5.9), or even be infinite (see [Oe], Ch. IV, Sec. 2.2) (see next section). In general, one may expect non-trivial behaviour regarding the first Galois cohomology of unipotent groups over non-perfect fields and it is worth of studying the Galois cohomology of unipotent groups in this case. In this note we are interested in the surjectivity of the above localisation map for smooth unipotent algebraic groups which are not necessarily connected. Namely we have the following Theorem. With above notation, if G is a (smooth) unipotent group over k, then the localisation map ϕ S is always surjective. An application of the surjectivity of the localisation map is devoted to a local - global principle over global fields of characteristic p>0in[th]. 2 Preliminaries We recall some facts regarding unipotent groups over a field k of characteristic p (see [Oe], Chap. V, [Ro1, Ro2], [SGA3], Exp. XVII). Let α p be the affine group scheme represented by the affine k-algebra k[t ]/(T p )andletf p := Z/pZ be the constant group scheme. Then, by definition, an affine algebraic group scheme G defined over k is unipotent if it has a composition series, each factor of which is isomorphic over k to a closed subgroup scheme of G a. (If G is smooth, then G is unipotent if it is isomorphic to a closed subgroup of the group T n consisting of upper triangular matrices with 1 s on the main diagonali for some n.) 2

3 In general, G is unipotent if and only if it is so over an algebraic closure k of k. Over k, G is unipotent if and only if it has a composition series, each factor of which is isomorphic G a (resp. either to G a,α p, or F p )ifchar.k = 0 (resp. char.k = p > 0). According to M. Raynaud ([SGA3], Exp. XVII, Theorem 3.5, iii)), an algebraic k-group scheme is unipotent if and only if it has a central composition series, each factor of which is isomorphic to G a (resp. to a (twisted) k-form of either αp r, Gr a or Fr p ), if char.k = 0 (resp. char.k = p>0), for some r 0. So from this (resp. Proposition 4.1.1) of (loc. cit.), we see that an algebraic k-group is unipotent and smooth (resp. unipotent, smooth and connected) if and only if it has a central composition series, each factor of which is a (twisted) k-form of either G r a or F r p (resp. of G r a ), for some r 0. Moreover, if k has characteristic 0 (resp. is perfect), then any unipotent (resp. unipotent, smooth and connected) k-group has a central composition series, each factor of which is isomorphic over k to G a (see [Ro1], Prop. 5, Corol. 2, [Ro2], p. 98, [SGA3], Exp. XVII, Corol ). Next we consider some examples of unipotent groups in characteristic p>0. Examples. 1. ([Ro1], Sec. 3.) Let k 0 be any field of characteristic p>2, k = k 0 (t) the rational function field in variable t, P = P (X, Y ):=X p X ty p, G := {(x, y) G 2 a : P (x, y) =0}. Then it is easy to see that G is unipotent, smooth, connected and is isomorphic over k(t 1/p ) (but not over k) tog a. Moreover, one can check that the group of k-rational points G(k) is finite and has exactly p elements. 2. ([Se], Chap. III, [SGA3], Exp. XVII, Sec. 5.9.) We use again the example given by Rosenlicht above to compute the first Galois cohomology of G. Here the field k = k 0 ((t)) the field of Laurent power series in t. One checks that H 1 (k, G) =k/p(k 2 ), and if p>2, then t 2 P (k 2 ), so H 1 (k, G) ([Oe], Chap. IV, Sec. 2.2.) Let k = F q ((t)) be the field of Laurent power series of characteristic p>2. Let G = {(x, y) G 2 a : P (x, y) := y p x tx p =0}. Oesterlé observed that the elements {t 2,t p+2,..., t np+2,...} (mod. P (k 2 )) are linearly independent over F p. Hence H 1 (k, G) is infinite. Recall that (see [KMT], Sec. 6, [Oe], Ch. V) a unipotent k-group G is called k-wound if any k-morphism of varieties A 1 G is constant. 3

4 We proceed by induction on the dimension of G. Weneedthefollowing lemmas. Let Γ be the Galois group Gal(k s /k), where k s denotes the separable closure of k. Lemma 1. Let G be a (smooth) linear algebraic group defined over a field k. Assume that 1 K G G/K 1 is an exact sequence of k-groups, where K is a smooth commutative normal k-subgroup of G such that for any 1-cocycle b = (b s ) s Γ with values in G(k s ) with its image c = (c s ) s Γ in (G/K)(k s ),forthetwistedform b K of K (obtained by twisting by b) and for any extension L/k, we have H 2 (L, bk) =0. If the assertion of the theorem (i.e., the surjectivity of the localisation map) holds for all such b K and c(g/k), then the same holds for G. Proof. We consider the exact sequence of k-groups 1 K G G/K 1 and commutative diagram of Galois cohomology with exact lines H 1 (k, K) f H 1 (k, G) g H 1 (k, G/K) 0 α β γ v S H 1 (k v,k) f S v S H 1 (k v,g) g S v S H 1 (k v,g/k) 0 Take any element z S v S H 1 (k v,g). We need to show that z S Im(β). By assumption γ is surjective, so there is y H 1 (k, G/K) such that g S (z S )= γ(y). Since g is surjective, there is x H 1 (k, G) such that g(x) =y, hence g S (β(x)) = g S (z S ). Let b =(b s ) s Γ denote the 1-cocycle representing x H 1 (k, G), and let c =(c s ) s Γ be the image of b in G/K. We consider the twisting b G of the group G by the cocycle b. Since K is commutative and normal in G, this twisting does not affect the commutativity and the normality of the subgroup b K in b G. Hence we get the following exact sequence 1 b K b G c (G/K) 1 by twisting the initial sequence. Now we consider the similar commutative diagram with exact lines 4

5 H 1 (k, b K) f H 1 (k, b G) g H 1 (k, c (G/K)) 0 α β γ v S H 1 (k v, b K) f S v S H 1 (k v, b G) g S v S H 1 (k v, c (G/K)) 0 Recall that there is a functorial bijection between H 1 (L, H) andh 1 (L, b H) for any k-group H, 1-cocycleb : Gal(k s /k) H(k s ), and field extension L/k (see [Se1]). Thus in the following comutative diagram, the maps p, q, p,q are all bijective. p H 1 g (k, G) q H 1 (k, G/K) H 1 (k, b G) g H 1 (k, c (G/K)) β γ β γ v S H 1 (k v,g) g S v S H 1 (k v,g/k) p q v S H 1 (k v, b G) g S v S H 1 (k v, c (G/K)) Since we twist by the cocycle representing x, wehavex = p(1),β(x) =p (1 S ), where by abuse of notation, 1, 1 S denote the trivial cohomology class correspondingly. Since p is a bijection, there is z S v S H 1 (k v, b G) such that z S = p (z S ). Since g S(z) =g S (β(x)), it follows that g S (z S )=g S (p (z S )= g S (p (1 S )) = q (g S (1 S)) = q (g S (z S )). Since q is bijective, it follows that 1 S = g S(1 S )=g S(z S), hence z S Im(f S), z S = f S(k S), k S v S H 1 (k v, bk). Since α is surjective by assumption, there is k H 1 (k, bk) such that k S = α (k ), thus z S = f S (α (k)) = β (f (k)). Therefore z S = p (z S ) = 5

6 p (β (f (k))) = β(p(f (k))), i.e., z S Im(β) as required. Lemma 2. The assertion of the theorem holds for smooth commutative unipotent k-wound groups of exponent p. Proof. By Tits theory on unipotent k-wound groups (see e.g. [KMT], Sec. 6, [Oe], Chap. 5), any commutative unipotent k-wound group G of exponent p is k-isomorphic to the k-group given as the kernel of a separable p-polynomial F in d +1 variables, where d = dim(g) andf is considered as a k-homomorphism : F : G d+1 a G a. One thus finds that 1-Galois cohomology H 1 (K, G) K/F(K d+1 ) for any field extension K/k. ForK = k v, one checks that the subgroup F (kv d+1 ) k v is open, since F is a separable morphism and we may use the implicit function theorem (see [Se2]) in this case. The assertion now amounts to saying that the canonical homomorphism ϕ S : k/f(k d+1 ) v S k v /F (k d+1 v ) is surjective. Considering k as embedded into v S k v,thisisequivalentto saying that k + v S F (kv d+1 )= v S k v, which is true since k is dense in the product v S k v and v S F (kv d+1 )isopenthere. Lemma 3. The assertion of the theorem holds for any smooth commutative unipotent k-group. Proof. Let G be a smooth unipotent group defined over k. Denote by G s the maximal connected normal k-split (in the sense of [Bo]) subgroup of G. Then, due to the triviality of the H 1 and H 2 for commutative k-split unipotent groups ([Se1]), for any field extension L/k we have a canonical isomorphsim H 1 (L, G) H 1 (L, G/G s ). Since G/G s is unipotent k-wound group ([Oe], Chap. V), we may assume from the beginning that G is k-wound. If G is of exponent p, then we may refer to Lemma 2. If G is not of exponent p, then it is well-known (see [Oe], Chap. V, Sec. 3.3), that there is a series of smooth k-subgroups of G : G = G 0 >G 1 > >G n 1 >G n =1,where G i /G i+1 is of exponent p for 0 i n 1, and G G 1. Now we may use induction on n (which can be taken as the smallest number such that there 6

7 exists the above series). Then we consider the exact sequence 1 G 1 G G/G 1 1 and the related commutative diagram of abelian groups H 1 (k, G 1 ) f H 1 (k, G) g H 1 (k, G/G 1 ) 0 α β γ v S H 1 (k v,g 1 ) f S v S H 1 (k v,g) g S v S H 1 (k v,g/g 1 ) 0 Now we may finish the proof by using Lemma 1. Namely by induction hypothesis, α and γ are surjective. By chasing on the diagram one checks that β is surjective, too. 3 Theorem and its consequences Theorem. With above notation, if G is a (smooth) unipotent group over k, then the localisation map ϕ S is always surjective. Proof. We use induction on the length s of the central composition series of G (see Sec.1). If the length is 1, i.e., G is commutative, then Lemma 3 gives us the base of induction. Assusme now G is not commutative. Let G = G 0 >G 1 > >G s 1 >G s = 1 be the central composition series for G, where all these groups are smooth. Here K = G s 1 1 is a central, hence commutative, k-subgroup in G, K G and the length of the derived series of G/K is less than that of G. Also, all these facts remain true if we twist G, K, G/K with cocycles with values in G(k s ). Thus by using the induction hypothesis and Lemma 1 in combining with Lemma 3, the theorem follows. As a direct consequence of the theorem, it follows that for unipotent groups G, in order to check the triviality of H 1 (k, G) one should first check the triviality H 1 (k v,g) for all valuations v of k. Corollary 4. Let k = F q (t) be a global function field of char. p and let 7

8 G be the smooth unipotent group over k defined by the equation y p = x + tx p. Then H 1 (k, G) is infinite. Proof. Indeed, by Theorem, there is a surjective homomorphism of localisation H 1 (k, G) H 1 (k v,g), where v is the valuation with uniformizing element t, and the latter cohomology group is infinite by [Oe], Chap. IV, 2.2. Next we consider some applications of the theorem just proved. The following statement and the idea of its proof is close to the one given by Kneser in [Kn], Sec. 3. Proposition 5. Let k be a field, S be a finite set of (non-equivalent) valuations of k, G a smooth k-subgroup of a smooth k-group H. If the localisation map γ : H 1 (k, G) v S H 1 (k v,g) is surjective (resp. η : H 1 (k, H) v S H 1 (k v,h) is injective), and H has weak approximation with respect to S, i.e., via the diagonal embedding, H(k) is dense in v S H(k v ),thenweak approximation with respect to S also holds for H/G. Proof. We consider the exact sequence 1 G H H/G 1 and the related commutative diagram with exact lines H(k) π (H/G)(k) δ H 1 (k, G) f H 1 (k, H) α β γ η v S H(k v ) π S v S(H/G)(k v ) δ S v S H 1 (k v,g) f S v S H 1 (k v,h) With the usual v-adic topology on H(k v )and(h/g)(k v ) we may endow the sets H 1 (k,.), H 1 (k v,.) appearing in this diagram with the weakest topology such that all maps appearing there are continuous. Let x v S(H/G)(k v ), 8

9 x =(x v ) v S. Then δ S (x) =γ(y), y H 1 (k, G), since γ is surjective. Then 0=f S (γ(y)) = η(f(y)). Since η is injective, it follows that y Im (δ), y = δ(z), z (H/G)(k). Therefore δ S (x) =δ S (β(z)). Let h H(k s ) (resp. h v H(k v,s ) such that π(h) =z (resp. π(h v )=x v ). Denote by [.] the cohomology class of (.). Then we know that for Γ := Gal(k s /k), Γ v := Gal(k v,s /k v )then δ(z) =[(h 1 s h) s Γ ],δ v (x v )=[(h 1 v t h v ) t Γv ]. Since δ S (x) =δ S (β(z)) we see that for all v S, [(h 1 v t h v ) t Γv ]=[(h 1 t h) t Γv )] (the equality holds in H 1 (k v,g)). Therefore there is a v G(k v,s ), v S, such that for all t Γ v we have h 1 t h = a 1 v h 1 v t h t v a v, i.e., h(h v a v ) 1 = t (h(h v a v ) 1 ), t Γ v. Therefore h(h v a v ) 1 =: h v H(k v ), for all v S. Let h := (h v ) v S H(k v ), then h β(z) =x. (H/G may not be a group, but we still have an action of H(k) on(h/g)(k) via multiplication on the left.) By assumption, H has weak approximation in S, so h = lim n h n,h n H(k). Therefore x = lim n h n β(z) =lim n β(π(h n ))β(z) Cl S (β(h/g)(k)) as required. Corollary 6. For any field k and smooth unipotent k-group G, the classifying variety of G has weak approximation over k (i.e., it has weak approximation with respect to any finite set S of non-equivalent valuations). Proof. Recall that the classifying variety B G of G with respect to a closed embedding G H, whereh is a smooth k-rational k-group satisfying H 1 (L, H) = 0 for all field extension L/k, is defined as the quotient space H/G. By [Me], Sec. 2.1, all classifying variety of G defined over k are birationally stably equivalent to each other over k. Therefore if one of such varieties satisfies weak approximation with respect to S then the other do, too (see [Kn], 2.1). Hence we need only prove the assertion for a specified classifying variety of G. We may assume that G GL n is a closed k-embedding. Since the group GL n has weak approximation over k (because it is rational over k) and it has trivial Galois cohomology (Hilbert 90), the assertion follows from our Theorem, Proposition 5 and the result of Merkurjev just mentioned. In fact, Corollary 6 can be deduced from the following stronger assertion, which was noticed after the first version of the paper had been written, in combining with the referee s remarks. 9

10 Proposition 7. If G is a smooth affine algebraic k-group which is isomorphic to a k-subgroup of a connected smooth solvable k-split group H, then any classifying variety B G of G is stably rational over k. Moreover, if G is a smooth connected unipotent group, which is split over k (e. g. if k is perfect), then its classifying varieties are rational over k. Proof. We may assume that G is a k-subgroup of H. By [Ro1], Prop. 8, (or [Ro2], Theorem 5) since H is a k-split connected solvable k-group, any homogeneous (with respect to an action of H) k-variety V of H, isrational over k. In particular, the quotient space H/G is rational over k. Since any two classifying varieties are stably birationally equivalent over k (see above), the first part of the proposition follows. For the second, just observe that G has a central composition series, each factor of which is isomorphic over k to G a. Now by induction on the length of the composition series, proving the rationality of classifying varieties of G is reduced to proving the rationality of the quotient of an affine space by G a, which is known to be rational. Remarks. 1. As the referee pointed out, from Proposition 7 one deduces also another (shorter) proof of the theorem. 2. It is interesting to know whether the classifying varieties of a smooth unipotent (or solvable) k-group are always rational over (non-perfect field) k, or at least to give examples of rationality (or non-rationality) in the case of k-wound unipotent groups. Acknowledgements. Thanks are due to the referee for his/her careful reading and very helpful suggestions regarding the paper which help to remove the inacuracies of the paper and improve its readability. References [Bo] A. Borel, Linear algebraic groups (second and enlarged version), GTM No. 126, Springer - Verlag, [KMT] T. Kambayashi, M. Miyanishi and M. Takeuchi, Unipotent algebraic groups, Lecture Notes in Math. v. 414, Springer - Verlag,

11 [Kn] [Me] [Oe] [Ro1] [Ro2] M. Kneser, Schwache Approximation in algebraischen Gruppen, in Colloque sur la théorie des groupes algébriques, CBRM, Bruxelles, 1962, A. Merkurjev, Unramified cohomology of classifying varieties for classical simply connected groups, Ann. Scient. Éc. Norm. Sup. t. 35 (2002), J. Oesterlé, Nombre de Tamagawa et groupes unipotents en carateristique p, Invent. Math. v. 78 (1984), M. Rosenlicht, Some rationality questions on algebraic groups, Annali di Mat. Pura ed Appl. 43 (1957), M. Rosenlicht, Questions of rationality for solvable algebraic groups over non-perfect fields, Annali di Mat. Pura ed Appl. v. 61 (1963), [Se1] J. -P. Serre, Cohomologie Galoisienne, Lecture Notes in Math. v. 5, Springer - Verlag, [Se2] J. -P. Serre, Lie algebras and Lie groups, Harvard Univ. Lecture Notes, Benjamin, [SGA3] M. Raynaud, Groupes algébriques unipotents. Extensions entre groupes unipotents et groupes de type multiplicatif, Exp. XVII, in: Schémas en groupes, SGA 3 (2) = Lectune Notes in Math., v. 152, Springer - Verlag, 1970, p [Th] N. Q. Thang, A local - global principle for Galois cohomology of algebraic groups over function fields. Preprint,