DSpace at VNU: On the topology on group cohomology of algebraic groups over complete valued fields

DSpace at VNU: On the topology on group cohomology of algebraic groups over complete valued fields tài liệu, giáo án, bà...

Journal of Algebra 399 (2014) 561–580 Contents lists available at ScienceDirect Journal of Algebra www.elsevier.com/locate/jalgebra On the topology on group cohomology of algebraic groups over complete valued fields ´ b,∗,2 Dào Phuong Bˇac´ a,1,2 , Nguđ Qc´ Thˇang a b Department of Mathematics, VNU, Univ of Science, 334 Nguyen Trai, Hanoi, Viet Nam Institute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 12 September 2012 Available online November 2013 Communicated by Gernot Stroth a b s t r a c t We introduce some topologies on the group cohomology of algebraic groups over complete valued fields and consider some applications © 2013 Elsevier Inc All rights reserved MSC: primary 14L24 secondary 14L30, 20G15 Keywords: Topology Affine group schemes Galois cohomology and flat cohomology Introduction Let G be a affine algebraic group scheme over a field k One may define the flat cohomology sets (or groups) Hif lat (k, G ), i = 0, 1, and if G is commutative, also the groups Hif lat (k, G ) for i If, moreover, G is a smooth (i.e., absolutely reduced) k-group scheme, these cohomologies are isomorphic to Galois cohomology sets (or groups) Hi (k, G ) If k is a field endowed with a topology, say a v-adic topology, where v is a non-trivial valuation, then H0f lat (k, G ) = G (k) has induced v-adic topology Due to the need of duality theory over local fields, a natural topology on the groups of cohomology has been introduced for commutative group schemes (only) and it has shown to have * Corresponding author ´ nqthang@math.ac.vn (N.Q Thˇang) ´ E-mail addresses: bacdp@math.harvard.edu, daophuongbac@yahoo.com (D.P Bˇac), Current address: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA This research is funded by VIASM and Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2011.40 0021-8693/$ – see front matter © 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.jalgebra.2013.08.041 562 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 many applications (e.g Tate–Nakayama theorems) in duality theory for commutative group schemes over local and global fields (see [19,20,13]) There the following natural (and basic) question was discussed: are the connecting maps continuous? (Here the connecting maps are the ones induced from a short exact sequence of algebraic groups.) Also, it is natural to rise the following question: what can one say in the non-commutative case? In [22] a natural topology, called special topology, on the sets of cohomology has been introduced, which has applications in various arithmetic problems In this paper, we continue this study more systematically by establishing a relation between the special topology and the natural (which is called here canonical) topology introduced before by Shatz [19,20] and find further applications One of main applications of our approach is to prove some theorems on the topology of orbits of algebraic groups (see [2–4]) and also some applications to weak approximations on higher cohomology groups (see [21]) This is done via an introduction of some specific topologies on the (Galois or flat) group cohomology and their relation with the problem of detecting the closedness of a given relative orbit The main ingredient is the following theorem, where we refer to Section for the notion of (adèlic) special and canonical topology on the cohomology set H1f lat (k, G ) Theorem Let G be an affine group scheme of finite type defined over a ring k, which is either a field, complete with respect to a non-trivial valuation of real rank 1, or the adèle ring of a global field Then 1) The (adèlic) special and canonical topologies on H1f lat (k, G ) coincide 2) Any connecting map appearing in the exact sequence of cohomology in degree induced from a short exact sequence of affine group schemes of finite type involving G is continuous with respect to (adèlic) canonical (or special) topologies Recall that the assertion 2) in Theorem is known in the case G is commutative, k is a local field, but the proof given in [13] is too short, so it is appropriate to give a full account here Some preliminary results on this topic are presented in Section In Section we discuss the notion of special and canonical topologies on group cohomology In Section we consider a relation between the special and canonical topologies on group cohomology and prove the main theorem In Section we consider the twisting effect on the topology and finally in Section we prove the continuity of the connecting maps in the topologies considered Some of earlier results have been published in [1–3] and the results of the present paper improve some of main results obtained there Notations and conventions In this paper we consider strictly only affine group schemes of finite type (i.e., algebraic) defined over a field k By a smooth k-group G we always mean, by conventions, a smooth affine k-group scheme (i.e., linear algebraic k-group, as defined in [5]) For an affine k-group scheme G of finite type, Hif lat (k, G ) denotes the flat cohomology of G of degree i, whenever it makes sense We always denote by {1} the set consisting of the trivial cohomology class in Hif lat (k, G ) When G is smooth, one may consider Galois cohomology of G of degree i, denoted by Hi (k, G ) We refer to [5] for other terminologies and basic facts of algebraic groups used here, and to [17] for basic facts concerning Galois cohomology of linear algebraic groups over fields, and [12,13,19,20,23] for étale and flat cohomology of group schemes Finally, if ϕ : X → Y is a map of sets, where X is given a topology, then we denote by Tϕ the quotient topology on Y with respect to ϕ : X → Y Preliminaries 1.1 Galois and flat cohomology The general case of rings We need in the sequel several facts concerning Galois and flat cohomology of affine algebraic groups (Cf [17] for Galois cohomology and [12,13,19,20] for étale and flat cohomology of group schemes.) Let R be a commutative ring with unity, G an affine R-group scheme of finite type For any covering S / R, we set S ⊗n := S ⊗ R · · · ⊗ R S (n-times) Let D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 563 e i : S ⊗n → S ⊗(n+1) be the map s1 ⊗ · · · ⊗ si −1 ⊗ si · · · ⊗ sn → s1 ⊗ · · · ⊗ si −1 ⊗ ⊗ si ⊗ · · · ⊗ sn For any group (covariant) functor G from the category Com Alg R of commutative R-algebras to the category G r of groups, we denote the corresponding morphism by G (e i ) : G S ⊗n → G S ⊗(n+1) ˘ If G is commutative we consider the following Cech–Amitsur complex related with faithfully flat extension S / R (see, e.g., [10], Exp 190, [12], Chap III, Sec 2, or [16], Exp V) d G ,0 d G ,1 d G ,2 d G ,3 → G ( R ) −→ G ( S ) −→ G S ⊗2 −→ G S ⊗3 −→ G S ⊗4 → · · · , (1) where G is considered as a covariant functor from the category Com Alg R to the category G r of groups and the differential di := d G ,i are given by the formula (written additively in the commutative case, for simplicity) d G ,i = −G (e ) + G (e ) − · · · + (−1)i +1 G (e i +1 ) In particular, we have d0 ( f ) = f (due to the embedding R ⊂ S), d1 ( f ) = − f + f , for all f ∈ G ( S ), and for f ∈ G ( S ), f ∈ Im(G ( R ) → G ( S )) if and only if f ∈ Ker(d1 ) By convention, for x ∈ G ( S ⊗n ), we denote xi it := G (e it ) ◦ G (e it −1 ) ◦ · · · ◦ G (e i )(x) whenever it makes sense ˇ cohomology The cohomology group Hr ( S / R , G ) := Ker(dr +1 )/ Im(dr ) of this complex is called Cech ˘ cohomology of G with respect to the covering (or layer) S / R Then we define the Cech–Amitsur p p H f lat ( R , G ) := lim H f lat ( S / R , G ), →S/R p 0, where the limit is taken over all faithfully flat extensions S / R ˘ complexe for a If G is non-commutative, then we may consider the non-abelian Cech–Amitsur faithfully flat extension S / R d G ,0 d G ,1 d G ,2 → G ( R ) −→ G ( S ) −→ G S ⊗2 −→ G S ⊗3 , (2) where the differentials d G ,i are given by the formulas (written multiplicatively) d G ,0 = id, d G ,1 = G (e )−1 G (e ), d G ,2 = G (e )−1 G (e )G (e )−1 One defines Z ( S / R , G ) := g ∈ G S ⊗2 g 1−1 g g 3−1 = ⊂ G S ⊗2 , and for a, b ∈ Z ( S / R , G ), a ∼ b in Z ( S / R , G ) if a = c 1−1 bc for some c ∈ G ( S ), and define (3) 564 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 H0f lat ( S / R , G ) = G ( R ), H0f lat ( R , G ) := G ( R ), H1f lat ( S / R , G ) = Z ( S / R , G )/ ∼, (3.1) H1f lat ( R , G ) := lim H1f lat ( S / R , G ), (3.2) →S/R where the limit is taken over all faithfully flat extensions S / R ¯ 1.2 Galois and flat cohomology The case of fields Let L /k be a normal field extension (resp L = k) ˘ The Cech–Amitsur cohomology is defined via the complex (following (1)) → G (k) → G ( L ) → G ( L ⊗k L ) → · · · → G ⊗kr L → · · · , where the complex may go on to infinity One defines for commutative group schemes G the groups of cocycles and the groups of cochains Z r L /k, G ( L ) := Ker(d G ,r +1 ), B r L /k, G ( L ) := Im(d G ,r ) ˘ Then we define the Cech–Amitsur cohomology Hrf lat ( L /k, G ) = Z r L /k, G ( L ) / B r L /k, G ( L ) ˘ One may use this Cech cohomology to obtain two types of cohomology for G: the Galois cohomol¯ and ogy Hr (Gal(k s /k), G (k s )), by taking L = k s the separable closure of k in a fixed algebraic closure k, ¯ If G is a smooth k-group the flat cohomology Hr (k¯ /k, G ) (denoted also by Hrf lat (k, G )) by taking L = k scheme, then it is known [20, Theorem 43] that Hr Gal(k s /k), G (k s ) Hr (k¯ /k, G ) If G is not commutative, one uses again (2), (3), (3.1), (3.2) above to define the H0f lat (k, G ), H1f lat (k, G ) for G Topology on Galois or flat cohomology sets and groups In many problems related with cohomology, one needs to consider various topologies on the group cohomology, such that all the connecting maps are continuous Of course, the weakest (coarsest) topology is not interesting since it does not give anything, thus it is excluded from consideration 2.1 Special topology Assume that G is an arbitrary affine group scheme of finite type defined over a field k, complete with respect to a non-trivial valuation v of real rank It seems that not very much is known about how to endow canonically a topology on the set H1f lat (k, G ) such that all connecting maps are continuous First we recall a definition of a topology on H1f lat (k, G ) via embedding of G into special k-groups given in [22] Recall that a smooth (i.e linear) algebraic k-group H is called special (over k) (after Grothendieck and Serre [15]), if the flat (or the same, Galois) cohomology H1f lat ( L , H ) is trivial for all extensions L /k Given a k-embedding G → H of G into a special group H , we have the following exact sequence of cohomology δ → G (k) → H (k) → ( H /G )(k) −→ H1f lat (k, G ) → Here H /G is a quasi-projective scheme of finite type defined over k (cf [8, Proof of Theorem 5.4, p 341]) Since δ is surjective, by using the natural (Hausdorff) topology on ( H /G )(k), induced from that of k, we may endow H1f lat (k, G ) with the strongest topology such that δ is continuous D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 565 Definition For the moment, we call it the topology just defined the special topology on H1f lat (k, G ) with respect to the embedding into the special group H , or just the H -special topology on H1f lat (k, G ) for short 2.1.1 Theorem Let k be a field which is complete with respect to a non-trivial valuation of rank and G an affine k-group scheme of finite type Then the special topology on H1f lat (k, G ) does not depend on the choice of the embedding into special groups and it depends only on the k-isomorphism class of G Proof It was mentioned in [22], that when G is smooth, the H -special topology does not depend on H In fact, the proof given there (p 4293, lines 6–7) did not use the fact that G is smooth and with a small modification, it also holds for any affine group k-scheme as follows The following well-known argument (à la Speiser) (see [7, Prop 4.9], [11, Sec 1]) is quite short, so we give here for the convenience of the readers With notation as in [22], we take two embeddings f : G → H , f : G → H , and form another one ϕ = ( f , f ) : G → H ×k H , g → ( f ( g ), f ( g )) We set L = H / f (G ), L = ( H ×k H )/ϕ (G ) and consider the corresponding projections π : G → L = H / f (G ), π : H ×k H → L := H ×k H /ϕ (G ) Denote the special topology on H1f lat (k, G ) for the embedding defined by G → H (resp G → H × H ) by τ (resp τ ) We show that τ = τ The projection p : H ×k H → H clearly induces a surjective k-morphism of k-varieties q : L → L, which makes the following diagram commutative → ϕ → H×H G ↓= → G π → p↓ f → H L → ↓q π → L → From this we derive the following commutative diagram G (k) δk → H (k) × H (k) → L (k) → H1f lat (k, G ) ↓= G (k) ↓ pk → H (k) ↓ qk → L (k) ↓= δk → H1f lat (k, G ) Then the natural surjective projection q : L → L makes L → L a H -torsor In particular, we may identify L = L / H Since H has trivial degree Galois cohomology over any field extension k ⊂ K , we have H1f lat ( K , H ) = 0, hence the map L ( K ) → L ( K ) is surjective for any such K Applying to the case K equal to rational function field k( L ) of L, we see that there exists a k-rational section ψ to q defined over an open set U ⊂ L hence it defines an open embedding H × U → L Thus π defines a birational equivalences of varieties H × U ∼ H × L ∼ L Since H is special, it is known also that it is rational over k as k-variety [11, Sec 1] Hence the function field k( L ) is a purely transcendental extension of k( L ) and it follows that q is a separable morphism Since q is a separable k-morphism, qk : L (k) → L (k) is an open mapping by Implicit Function Theorem If U ∈ τ , then V := δk−1 (U ) = qk−1 (δk−1 (U )) is open in L (k), since qk , δk are continuous Thus U ∈τ Conversely, if U ∈ τ , then W := δk −1 (U ) = qk−1 (δk−1 (U )) is open in L (k) Since qk is an open map, qk ( W ) is open in L (k) But qk ( W ) = qk (qk−1 (δk−1 (U ))) = δk−1 (U ) since qk is surjective, thus U ∈ τ Therefore τ = τ , and since this argument also holds for H instead of H , it implies that the special topology defined by using H and the one defined by using H are the same as desired 566 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 Next we show that if ϕ : G G is a k-isomorphism, then ϕ induces a homeomorphism H1f lat (k, G ) with respect to special topology Since the special topology does not depend on the choice of H , we may take an arbitrary embedding : G → H Set L := H / (G ), L := H /( ◦ ϕ )(G ) Then we have also an embedding ◦ ϕ : G → H which induces a k-morphism ψ : L → L Then we have the following commutative diagram ϕ : H1f lat (k, G ) → G1 → ϕ↓ → H π1 → ↓= → G H L1 → ↓ψ π → L → and we derive from this the following commutative diagram G (k) ϕ↓ G (k) π1,k → H (k) → ↓= L (k) δ1,k → H1f lat (k, G ) ↓ ψk πk k → H (k) → L (k) ↓ϕ δk → H1f lat (k, G ) Here we obtain a bijection ϕ by functoriality (Thus, if z1 = [( g i )i ] is the class of a 1-cocycle ( g i ) of G , then z = ϕ ( z1 ) = [( f ( g i )i )].) Since ψk is clearly a homeomorphism, the above diagram shows that so is ϕ with respect to H -special topology ✷ 2.1.2 Definition The topology just defined is called the special topology on H1f lat (k, G ) and denoted by Ts Notice that if k ⊂ L (⊂ k¯ ) is a normal extension, then we have canonical embedding f L : H1f lat ( L /k, G ( L )) → H1f lat (k, G ) and next we identify H1f lat ( L /k, G ( L )) with a subset (denoted by R L for short) of H1f lat (k, G ) Then we may regard H1f lat (k, G ) = f L H1f lat L /k, G ( L ) L /k = RL L /k Note that each element of H1f lat (k, G ) is coming from H1f lat ( L /k, G ( L )) for some finite normal extension L /k Fix a special embedding G → H Consider the following commutative diagram with exact rows and exact last column D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 567 ↓ ϕ → H1f lat ( L /k, G ( L )) D L /k ↓ fL ∩↓ν → → πk G (k) → H (k) → ( H /G )(k) ↓α ↓β ↓γ G (L) → H (L) → ( H /G )( L ) ↑ξ ↑μ πL D L /k Here δk → H1f lat (k, G ) → ↓ψ δL → H1f lat ( L , G ) → q → π L ( D L /k ) = D L /k α , β, γ , ξ, μ, ν are just embeddings, and we let D L /k := δk−1 f L /k H1f lat L /k, G ( L ) = Ker(ψδk ) = Ker(δ L γ ) = π L H ( L ) ∩ ( H /G )(k), and D L /k := h ∈ H ( L ) d H ,1 (h) ∈ Z L /k, G ( L ) We set q = π L | D L /k = π L−1 ( D L /k ) Let ϕ := d H ,1 | D L/k : D L /k → ϕ D L /k ⊂ H L ⊗2 , h → h− h2 (In terms of Galois cohomology, if L /k is a finite Galois extension, D L /k := {h ∈ H ( L ) | h−1s h ∈ G ( L ), for all s ∈ Gal( L /k)}.) Then we have q ( D L /k ) = D L /k and δk ( D L /k ) = f L (H1f lat ( L /k, G ( L ))) and δL (γ ( D L /k )) = ψ(δk ( D L /k )) = {1}, and γ ( D L /k ) ⊆ πL ( H ( L )) = H ( L )/G ( L ) The latter set has the quo- tient topology induced from that of H ( L ) and since D L /k = π L−1 ( D L /k ), it follows that the topology on D L /k (⊂ π L ( H ( L ))) is the quotient topology of the topology on D L /k with respect to the map q With notation as in the diagram above, we have the following commutative diagram G ( L ⊗2 ) ∪↑r D L /k ϕ → q ↓ D L /k Z ( L /k, G ( L )) θ L /k ↓ ϕ → H1f lat ( L /k, G ( L )) → H1f lat (k, G ) where θ L /k denotes the natural projection Z ( L /k, G ( L )) → H1f lat ( L /k, G ( L )) 568 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 2.1.3 Definition On H1f lat ( L /k, G ( L )) we may consider two topologies: the first is the topology induced from the special topology on H1f lat (k, G ), and the second is the quotient topology with respect to the map ϕ The first topology is called induced special topology (denoted by Ts, L /k ) and the second topology is called H − L /k-special topology (denoted by Tϕ ) on H1f lat ( L /k, G ( L )) ¯ the H − L /k-special topology is just the special topology Ts (related with the emThus, if L = k, bedding G → H ) A relation between these two topologies is given by the following 2.1.4 Lemma With notation as above, the topology Ts, L /k is weaker than the H − L /k-special topology Tϕ and they coincide if G is smooth Proof Let U = U ∩ R L , where U is open in special topology in H1 (k, G ) Then V := δk−1 (U ) is open in ( H /G )(k) by definition, thus ϕ −1 (U ) = V ∩ D L /k is open in D L /k , i.e., U is open in the quotient topology with respect to the map ϕ = δk | D L /k Further, we assume that G is smooth Then D L /k is open in ( H /G )(k) In fact, γ ( D L /k ) = γ (( H /G )(k)) ∩ πL ( H ( L )), thus D L /k = γ −1 γ ( H /G )(k) ∩ πL H ( L ) = ( H /G )(k) ∩ γ −1 πL H ( L ) is open in ( H /G )(k) Let U ∈ Tϕ , i.e V := ϕ −1 (U ) is open in D L /k Since D L /k is open in ( H /G )(k), it follows that so is V = δk−1 (U ) Hence U itself is also open in H1 (k, G ), which is what we need ✷ 2.1.5 Lemma The action of H (k) on ( H /G )(k) gives rise to the action of H (k) on D L /k ⊗2 ) We Proof In fact, if z ∈ H (k), x = hG ∈ D L /k , where we may assume that h ∈ H ( L ), h− h2 ∈ G ( L define for z ∈ H (k) z.x := zh.G, and then ( zh)1 −1 ( zh)2 = ( z1 h1 )−1 ( z2 h2 ) −1 = h− z1 z2 h 1 ⊗2 = h− h2 ∈ G L Since z ∈ H (k), so z.x ∈ D L /k Therefore we obtain the quotient space ( D L /k / ∼) of H (k)-orbits, which gives clearly a bijection ϕ0 : ( D L /k / ∼) ( Z ( L /k, G ( L ))/ ∼) ✷ 2.1.6 Lemma We have π L ( D L /k ) = γ ( D L /k ), thus we have a surjective map γ −1 ◦ q : D L /k → D L /k Proof Indeed, it is clear that D L /k ⊆ π L ( D L /k ) On the other hand, if h ∈ D L /k , then x = hG ∈ ( H /G )(k), and it is clear that δk (x) = 1, thus x ∈ D L /k , i.e., πL ( D L /k ) ⊆ D L /k ✷ 2.1.7 Lemma The map d H ,1 : H ( L ) → H ( L ⊗2 ), h → h− h defines a continuous and surjective map ϕ : D L /k → Z L /k, G ( L ) where we consider the induced topology on Z ( L /k, G ( L )) (as a subset of G ( L ⊗2 )) (In term of Galois cohomology, if [ L : k] = n, ϕ : h ∈ D L /k → (h−1 s h)s∈Gal( L /k) ∈ G ( L )n gives rise to a surjective and continuous map ϕ : D L /k → Z ( L /k, G ( L )), with induced topology on Z ( L /k, G ( L )) as a subset of G ( L )n ) D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 569 Proof It is clear that ϕ is continuous with respect to topologies on H ( L ⊗2 ) and G ( L ⊗2 ) To check that it is surjective, let g ∈ Z ( L /k, G ( L )) be any element Its cohomology class in H1f lat ( L /k, G ( L )) is the image of an element x = hG from D L /k , where as above, we may assume that h ∈ H ( L ), h1−1 h2 ∈ G ( L ⊗2 ) Since h1−1 h2 and g have the same image in H1f lat ( L /k, G ( L )), there exists t ∈ G ( L ) such that we have g = t 1−1 h1−1 h2 t But the latter can be written as (ht )1−1 (ht )2 ∈ Im(ϕ ), so ϕ is surjective ✷ 2.2 Canonical topology Commutative case 2.2.1 Let G be an affine commutative k-group scheme of finite type As in [13], Chap III, Section 6, or [19], Section one may define a natural topology on the flat cohomology groups of commutative group schemes of finite type G, which is in a sense induced from the topology on k as follows With r ¯ k by taking notation as in 1.1, if k is equipped with a v-adic topology, we can define a topology on the topology induced from the complete tensor product ( ˆ k k¯ ) (see [6, Chap III, Sec 2, Excer 28]) ¯ then the corresponding ¯ v denotes the maximal ideal of the ring O¯ v of integers of k, In particular, if m ¯ v -adic topologies are m ¯ There is a natural topology on r L, induced from Let L be an algebraic extension of k (inside k) k r ¯ r r k, thus also on G ( k L ) The set Z ( L /k, G ( L )) (resp B r ( L /k, G ( L ))), being considered as a that of r subset of G ( k L ), has induced topology 2.2.2 Definition The induced topology is called the L /k-canonical topology on Z r ( L /k, G ( L )) (resp ¯ it is called the canonical topology on Z r (k¯ /k, G (k¯ )) (resp B r (k¯ /k, G (k¯ ))) B r ( L /k, G ( L ))) If L = k, r For L as above, it is clear that Z r ( L /k, G ( L )) is a closed subgroup of G ( k L ) Then we equip the quotient group Hrf lat ( L /k, G ( L )) = Z r ( L /k, G ( L ))/ B r ( L /k, G ( L )) with the quotient topology, which may not be Hausdorff 2.2.3 Definition This quotient topology is called the L /k-canonical topology on Hrf lat ( L /k, G ( L )) (de- ¯ it is called the canonical topology on Hr (k, G ) (denoted by Tc ) noted by T L /k,c ) If L = k, f lat Next, due to Hrf lat (k, G ) := each Hrf lat ( L /k, G ( L )) Hrf lat ( L /k, G ( L )) we may consider the topology (denoted by Tc , L /k ) on which is induced from Tc 2.2.4 When we are in the category of commutative group schemes of finite type, with canonical topology on their flat cohomology groups, all the connecting homomorphisms appearing in any long exact sequence of flat cohomology involving commutative groups are continuous, see [13], Chap III, Sec 6, [19] In fact, regarding the connecting maps Hrf lat (k, A ) → Hrf lat (k, B ), on the level of cocycles, these maps are given by polynomials, induced from the morphism A → B Thus the induced maps are continuous Below we will give a conceptual proof for these facts 2.3 Canonical topology Non-commutative case 2.3.1 Let G be a non-commutative affine k-group scheme of finite type We define canonical topology on the set H1f lat (k, G ) by using the method given for commutative case (2.2.1) of complete valued field k Let k ⊂ L ⊂ k¯ be a normal extension, θ L /k : Z ( L /k, G ( L )) → ( Z ( L /k, G ( L ))/ ∼) the quotient map, where ∼ is as in 1.1 2.3.2 Definition The topology on Z ( L /k, G ( L )) induced from that of G ( L ⊗k L ) is called the the L /k-canonical topology on Z ( L /k, G ( L )) The corresponding quotient topology on H1f lat ( L /k, G ( L )) with respect to the projection θ L /k : Z ( L /k, G ( L )) → Z ( L /k, G ( L ))/ ∼ is called the L /k-canonical topol- ¯ it is called the canonical topology on ogy on H1f lat ( L /k, G ( L )) (and is denoted by T L /k,c ) If L = k, 570 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 Z (k¯ /k, G ( L )) (resp on H1f lat (k, G ) and is denoted by Tc ) Especially, we may also consider the topology on H1f lat ( L /k, G ( L )) induced from canonical topology Tc on H1f lat (k, G ) and it will be denoted by Tc, L /k 2.3.3 If K /k is another extension, K ⊂ L, then by functoriality there is a natural map g L / K : Z K /k, G ( K ) → Z L /k, G ( L ) which induces the following commutative diagram where f L / K is just an embedding and the inflation map for the extension L / K Z ( K /k, G ( K )) g L/ K → ↓ θ K /k H1f lat ( K /k, G ( K )) Z ( L /k, G ( L )) ↓ θL /k f L/ K → H1f lat ( L /k, G ( L )) 2.4 Adèlic topology Now let k be a global field and let A be the ring of adèles of k Let k¯ be an alge¯ Then A¯ has a natural topology and (see [6, Chap III, Sec 2, Excer 28]) we ¯ = A ⊗k k braic closure of k, A ¯⊗ ¯ ˆ k A¯ and the induced topology on A := A¯ ⊗k A¯ ⊂ A¯ ⊗ ˆ k A may consider the complete tensor product A 2.4.1 Notice that if H is a special k-group, then we also have H1f lat (A, H ) = Therefore we may introduce the special topology on H1f lat (A, G ) as it is done in the case of fields (see 2.1.2) The same proof as in the case of fields (see 2.1.1) shows that the definition of special topology does not depend on the choice of embedding G → H into a special k-group H 2.4.1.1 Definition This topology is called adèlic special topology on H1f lat (A, G ) 2.4.2 If G is a commutative affine k-group scheme, we can endow G ( A ) with the topology induced ¯ ⊗r ) and consider the adèlic from A , thus we may consider the complex {C r , d G ,r } where C r := G (A r r topology on C and thus also on H f lat (A, G ) like in 2.3.2 2.4.2.1 Definition This topology is called adèlic canonical topology on Hrf lat (A, G ) If G is not commutative, then we may restrict only to the case of H0 and H1 and the definition is as in the previous case We have the following analog of Theorem 2.1.1 2.4.3 Theorem The adèlic special topology on H1f lat (A, G ) does not depend on the choice of embedding of G into a special k-group H The proof is almost verbatim, except that we need the following version of Implicit Function Theorem in the adèlic setting: 2.4.4 Proposition (See [14, Chap I, Sec 3.2, p 20].) Let f : V → W be a smooth morphism of k-varieties with non-empty absolutely integral fibers, all are of the same dimension d Then the induced mapping f A : V (A) → W (A) is a continuous open mapping Note We call such a topology “canonical” since it is defined intrinsically only in term of G It is clear that when G is commutative, this is just the definition we gave above D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 571 2.5 Embedding of commutative and nilpotent groups into special groups It was also mentioned in [22], that in the commutative case, the special topology and the canonical (as introduced in [13], Chap III, Section 6) coincide In fact, this remark (regarding the statement in [13], Remark 6.6) should be understood in the sense, that we assume that there exist special commutative k-groups H containing G as a closed k-subgroup Thus we are led to the problem of finding a special commutative k-group H containing a given group G Since we cannot find a proof for this fact in the literature, we propose here a proof We have the following simple assertion, which perhaps is known to experts 2.5.1 Proposition Let G be a commutative (resp nilpotent) affine group scheme of finite type defined over a field k Then there exists a connected smooth affine special k-group H containing G as closed k-subgroup, which is also commutative (resp nilpotent) with unipotent part H u defined and split over k ¯ then it also holds Proof Notice that if the statement holds for some finite extension k ⊂ K ⊂ k, for k Indeed, let G K = G ×k Spec( K ), and G K → H , where H is a smooth affine commutative (resp nilpotent) special K -group Then R K /k ( H ) is also a smooth affine commutative (resp nilpotent) special k-group and we have a sequence of embeddings over k G → R K /k (G K ) ⊂ R K /k ( H ), which gives us the required embedding By the structure theorem of commutative (resp nilpotent) affine k-group (see [8], Chap IV, Section 3, no 1, Theorem 1.1), there exists a maximal k-subgroup D of multiplicative type; in fact, D = G s is the set of all semisimple elements of G It is well-known that we may embed D into an induced k-torus T (i.e., a torus which is k-isomorphic to a direct product of k-groups of the form R K i /k (Gm ), where K i /k is a separable finite extension of k) With D as above, it is well-known that the unipotent part U := G u (the set of all unipotent elements) of G is k-closed and defined over a finite purely inseparable extension k ⊂ L ⊂ k¯ of k As we have noticed, we may assume (by passing to L) that G is already the direct product D × U over k Since G u is commutative (resp nilpotent) unipotent k-group, there exists a (smooth) k-split commutative unipotent group (resp k-split unipotent) Q containing U (see [14], Chap V, Sec 4, Prop 4.1) By taking an embedding D → P to an induced k-torus P , we have G → P × Q =: H , where the latter is commutative (resp nilpotent), smooth group and special k-group The proposition follows ✷ Relation between canonical and special topologies The following theorem shows that the two topologies introduced above are in fact the same 3.1 Theorem Let k be a field, which is complete with respect to a non-trivial valuation v of real rank Then for any affine k-group scheme of finite type G and any special k-embedding G → H , the H-special topology on H1f lat (k, G ) coincides with the canonical topology there In particular, the H-special topology does not depend on the choice of the embedding G → H Proof We will show that For any normal extension k ⊂ L (⊂ k¯ ), the L /k-canonical topology on H1f lat ( L /k, G ( L )) coincides with the H − L /k-special topology T L /k,s and is stronger than the topology Ts, L /k , induced from the special topology on H1f lat (k, G ) They coincide if either L = k¯ or G is smooth We consider again the following commutative diagram 572 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 G ( L ⊗2 ) ∪↑r Dk ϕ → q ↓ Dk Z ( L /k, G ( L )) (4) θL ↓ ϕ → H1f lat ( L /k, G ( L )) → H1f lat (k, G ) Recall that on Z ( L /k, G ( L )) we have two topologies, namely the canonical topology (obtained as induced topology via embedding Z ( L /k, G ( L )) ⊂ G ( L ⊗2 )) and also the quotient topology Tϕ of the v-adic topology on D L /k with respect to the map ϕ By definition, the quotient of the first topology, with respect to the projection θ L , gives us the canonical topology T L /k,c on H1f lat ( L /k, G ( L )); the quotient topology Tϕ (of v-adic topology on D L /k ) with respect to ϕ gives the H − L /k-special topology on H1f lat ( L /k, G ( L )) Thus to show that the topology Tϕ and T L /k,c coincide, it is sufficient to show that the quotient topology on Z ( L /k, G ( L )) with respect to ϕ and the canonical topology on Z ( L /k, G ( L )) coincide This is equivalent to showing that ϕ : D L /k → Z ( L /k, G ( L )) is an open mapping, i.e., for any open subset U ⊂ D L /k , ϕ (U ) is an open set in Z ( L /k, G ( L )) Since the map 1 ⊗2 ), so we have a d H ,1 : H ( L ) → H ( L ⊗2 ) is given by h → h− h and Z ( L /k, G ( L )) = d H ,1 ( H ( L )) ∩ G ( L ⊗ map ϕ : D L /k → d H ,1 ( H ( L )) ∩ G ( L ) Each open subset U ⊂ D L /k has the form U = U ∩ D L /k , where U is an open subset of H ( L ) First we need the following 3.1.1 Lemma We have ϕ (U ∩ D L /k ) = d H ,1 (U ) ∩ Z ( L /k, G ( L )) ϕ (U ∩ D L /k ) ⊂ d H ,1 (U −1 ( Z ( L /k, G ( L ))), so if x = d Proof of 3.1.1 On the one hand we have obviously the other hand, by definition we have D k = ϕ u ∈ U , then u ∈ D L /k thus u ∈ D L /k ∩ U Hence ϕ U ∩ D L /k = d H ,1 U ∩ Z L /k, G ( L ) as desired ) ∩ Z ( L /k, G ( L )) On H ,1 ( u ) ∈ Z ( L /k, G ( L )), (5) ✷ Therefore we need only show that the map d H ,1 : H ( L ) → d H ,1 H ( L ) is an open map with respect to the v-adic topology on H ( L ) and the induced (from H ( L ⊗2 )) topology on d H ,1 ( H ( L )) In fact we have the following more precise statement 3.1.2 Lemma Let k be a non-archimedean completely valued field, H a special k-group, L /k a normal exten1 sion Let d H ,1 be the map d H ,1 : H ( L ) → H ( L ⊗2 ), h → h− h2 a) The image of d H ,1 is a closed subset in H ( L ⊗2 ) b) The surjective map d H ,1 : H ( L ) → d H ,1 ( H ( L )) is an open mapping, where the topology on H ( L ) (resp d H ,1 ( H ( L ))) is the usual v-adic topology induced from that on k (or L) (resp H ( L ⊗2 )) D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 573 Proof of 3.1.2 a) First we show that Im(d H ,1 ) is closed in H ( L ⊗2 ) Let yn := d H ,1 (hn ) → x ∈ H ( L ⊗2 ) Then we have ( yn )1 ( yn )2−1 ( yn )3 = Since the maps d H ,i , i = 0, are continuous in v-adic topol- ogy, it follows that x1 x2−1 x3 = in H ( L ⊗2 ), i.e., x ∈ Z ( L /k, H ( L )) Since H is special, we have H1f lat ( L /k, G ( L )) = 0, thus x is a trivial 1-cocycle, i.e., there is h ∈ H ( L ) such that x = h1−1 h2 , or x ∈ Im(d H ,1 ) (In the case L /k is a finite Galois extension of degree n with Galois group Γ L , we may identify H ( L ⊗2 ) and H ( L )n If limn d H ,1 (hn ) = x = (xs )s ∈ H ( L )n , then since f s (hn )s f t (hn ) = hn−1 s hn s hn−1 t hn = hn−1 st hn = f st (hn ), it follows that for all s, t ∈ Γ L , we have obviously xst = xs s xt , thus (xs )s is a 1-cocycle from Z (Γ L , H ( L )) Since H is special, it follows that H1f lat ( L /k, H ( L )) = 1, so (xs )s is a 1-cochain Therefore there exists h ∈ H ( L ) such that xs = h−1 s h for all s ∈ Γ L , i.e., x ∈ Im(d H ,1 ).) Thus Im(d H ,1 ) is a closed subset in H ( L ⊗2 ) b) Notice that two elements u , v ∈ H ( L ) are in the preimage of an element x ∈ d H ,1 ( H ( L )) if and only if u 1−1 u = v 1−1 v , i.e., ( vu −1 )1 = ( vu −1 )2 , thus v = zu, z ∈ H (k) Thus elements of d H ,1 ( H ( L )) correspond bijectively with the set of right cosets H (k) \ H ( L ) of H (k)-orbits of elements from H ( L ) The topology on the quotient space is the quotient topology, so the projection p is open Thus we have the following commutative diagram H (L) d H ,1 → ↓= H (L) d H ,1 ( H ( L )) ↓r p → H (k) \ H ( L ) where r is the map (h1−1 h2 ) → H (k)h, which is a bijection (r −1 is the map : H (k)h → ( z1−1 z2 ) for any z ∈ H (k)h), and p is just the projection We check that r is a homeomorphism To this we show that r is an open map Pick any open subset U˜ ⊂ H ( L ⊗2 ) We may assume that U˜ ∩ d H ,1 ( H ( L )) = ∅ Then ˜ d− H ,1 U ∩ d H ,1 H ( L ) is open in H ( L ) since d H ,1 is continuous and U˜ is open in H ( L ⊗2 ) Since the projection p is always open, it follows that r (U˜ ∩ d H ,1 ( H ( L ))) is open in H (k) \ H ( L ) In a similar way r −1 is also open In fact, let U be any open subset of H (k) \ H ( L ), then U = p ( V ), where V is an open subset of H ( L ), namely V = p −1 (U ) We set V˜ = d H ,1 ( V ) Then r −1 ( U ) = r −1 p ( V ) = r −1 r d H ,1 ( V ) = V˜ Thus d H ,1 induces an analytic isomorphism d H ,1 ( H ( L )) H (k) \ H ( L ) H ( L )/ H (k) Therefore we may regard d H ,1 ( H ( L )) as the image of H ( L ) via the projection r : H ( L ) → H ( L )/ H (k) Since H (k) is a closed analytic subgroup of H ( L ), this gives rise to a surjective differential dd H ,1 : Lie ( H ( L )) → Lie (d H ,1 ( H ( L ))) [18] Thus by Implicit Function Theorem, d H ,1 is an open map Lemma 3.1.2 is proven ✷ 574 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 3.1.3 Lemma a) Let notation be as above Then the canonical topology on Z ( L /k, G ( L )) (denoted also by T L /k,c ) and the topology, which is the quotient topology with respect to the surjective map d H ,1 : D L /k → d H ,1 ( D L /k ) (denoted also by Tϕ ), coincide b) The quotient topology of T L /k,c and Tϕ on H1 ( L /k, G ( L )) with respect to the surjective map θ L coincide This topology is stronger than the topology, induced from the special topology on H1 (k, G ) and they coincide if ¯ G is smooth or L = k Proof of 3.1.3 a) Let U ∈ T L /k,c be an open subset of Z ( L /k, G ( L )) We need to show that d− H ,1 ( U ) is open in D k We have U = U ∩ Z ( L /k, G ( L )), where U is an open subset in v-adic topology of H ( L ⊗2 ) Since the map ϕ is continuous with respect to the v-adic topology on H ( L ), H ( L ⊗2 ), −1 −1 −1 V := d− H ,1 (U ) is open in H ( L ), and d H ,1 (U ) = d H ,1 (U ) ∩ d H ,1 ( Z ( L /k, G ( L ))) = V ∩ D k is open in D k Thus we see that T L /k,c ⊂ Tϕ Conversely, let U be an open set in Tϕ , i.e., V := ϕ −1 (U ) is open in D k , V = V ∩ D k , where V is an open subset in H ( L ) We show that U is open in Z ( L /k, G ( L )) in canonical topology, i.e., for some open subset U in H ( L ⊗2 ) we have U = U ∩ Z L /k, G ( L ) = U ∩ (d H ,1 H ( L ) ∩ G L ⊗2 = U ∩ d H ,1 H ( L ) ∩ G L ⊗2 On the one hand, by Lemma 3.1.2, b), we know that d H ,1 is a surjective open map H ( L ) → d H ,1 ( H ( L )), so U := d H ,1 ( V ) is an open subset of d H ,1 ( H ( L )), i.e., U = U ∩ d H ,1 ( H ( L )), U is open in H ( L ⊗2 ) On the other hand, since ϕ is surjective, by using (5) we have U = d H ,1 ( V ) = d H ,1 V ∩ D k = d H ,1 V ∩ d H ,1 D k = d H ,1 V ∩ Z L /k, G ( L ) = U ∩ d H ,1 H ( L ) ∩ Z L /k, G ( L ) as desired Thus the image of an arbitrary open subset of D k is open in canonical topology in Z ( L /k, G ( L )), so Tϕ ⊂ T L /k,c and these two topologies coincide b) The first assertion follows from a) Next we show that, the special L /k-topology on H1f lat ( L /k, G ( L )) (i.e obtained as the quotient topology with respect to the map ϕ ), in general, is stronger than the topoplogy T H , L /k induced from H1 (k, G ) (where H1 ( L /k, G ( L )) is considered as a subset of H1 (k, G )) and they coincide if G is smooth Let U ∈ T L /k,c Then U = U ∩ H1 ( L /k, G ( L )), where U is an open subset in special topology of H1 (k, G ) It means that δk−1 (U ) is open in ( H /G )(k) Therefore δk−1 (U ) = δk−1 U ∩ δk−1 H1 L /k, G ( L ) = δk−1 U ∩ D k is open in D k Hence U ∈ T L /k,c Now assume further that G is smooth Then the map π is separable, so π L is an open map Let U ∈ T L /k,c , ϕ −1 (U ) = U ∩ D k , where U is open in ( H /G )(k) We show that D k is open in ( H /G )(k) In fact, γ ( D k ) = γ (( H /G )(k)) ∩ π L ( H ( L )), thus D k = γ −1 γ ( H /G )(k) ∩ πL H ( L ) = ( H /G )(k) ∩ γ −1 πL H ( L ) D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 is open in ( H /G )(k) (noticing that the image of π L is open in ( H /G )( L ) Hence also open in ( H /G )(k), so U ∈ T Lemma 3.1.3 is proven ✷ 575 ϕ −1 (U ) = δk−1 (U ) is ¯ then from (5), it follows that the topology on D is just the quotient topology with respect If L = k, to the map q Hence, due to the commutativity of the above diagram, we have f := θk¯ ◦ ϕ = ϕ ◦ q , so the special topology on H1f lat (k, G ) is the quotient topology with respect to f Thus it follows that the H − L /k-canonical topology is just the canonical topology on Z (k¯ /k, G ), which coincides with the quotient topology of ϕ , which in turn is the special topology With this, the proof of 3.1.3 is finished and thus the proof of Theorem 3.1 is complete ✷ 3.2 Adèlic special and adèlic canonical topology The same method of proof also shows that Theorem 3.1 still holds if instead of considering complete fields k, we take the adèle ring of a global field k Thus we have 3.2.1 Theorem Let k be a global field, A the adèle ring of k Then for any affine k-group scheme of finite type G and any special k-embedding G → H , the adèlic special topology on H1f lat (A, G ) coincides with the adèlic canonical topology there In particular, the adèlic special topology does not depend on the choice of the embedding G → H ✷ Topology and the twisting 4.1 Let G be an affine group scheme defined over a field k complete with respect to a non-trivial real valuation v of rank It is natural to ask about the relation between the special (or canonical) topology on H1f lat (k, G ) and the special (or canonical) topology on H1f lat (k, a G ), where a G is the affine k-group scheme group obtained by twisting G with a cocycle a of G 4.2 We consider first the special case where G is smooth, since the notation is simple 1) Smooth case Let a = (as ) ∈ Z := Z (Γ, G (k s )) be a 1-cocycle of Γ := Gal(k s /k) with values in G (k s ), and let G := a G, the k-group twisted by a, Z := Z (Γ, G (k s )) A natural question arises as how the topologies (special or canonical) on H1 (k, G ) and H1 (k, G ) (or H1f lat (k, G ) and H1f lat (k, G )) are related to each other We consider only the canonical topology since the canonical and special topologies are the same by Theorem 3.1 Let G d (resp G v ) be the set G (k s ) equipped with the discrete (resp v-adic) topology Let M d := Cont Map (Γ, G d ) (resp M d := Cont Map (Γ, G d )) be the set of all maps Γ → G (k s ) (resp Γ → G (k s )) which are continuous with respect to the profinite topology on Γ and discrete topology on G (k s ) (resp discrete topology on G (k s )), and let M v (resp M v ) be similar sets, where the discrete topology is replaced by the usual v-adic topology on G (k s ) (resp G (k s )) Thus we have Z ⊂ Md ⊂ M v , Z ⊂ Md ⊂ M v Then it is clear that the quotient topology of Tc on H1 (k, G ) is the canonical one We denote an element from M v by ( f s )s∈Γ Then the translation map ta : M v → M v given by ta : ( f s )s → ( f s as )s = ( f s )(as ) are well-defined, and when restricted to Z it gives the usual translation map (denoted by the same symbol ta ) considered by Serre in [17, Chap I, Prop 35’] Also, it is clear that the bijective map ta is continuous and so is the inverse mapping Thus ta is a homeomorphism M v M v , so it Z also induces a homeomorphism Z 2) The general case follows the same pattern We denote Z := Z (k¯ /k, G ), a ∈ Z , G := a G, Z := Z (k¯ /k, G ) We give (using terminology from [9, Chap III, Sec 3.6]) only a sketch of the description of the bijection ta in the general case, where the (general) cohomology is defined via direct limit of ˘ ˘ cohomology groups) Cech cohomology (and we know that H1f lat (k, G ) is the direct limit of Cech Let E be a Grothendieck site where finite fiber products exist, ℵ = ( S i → S ) a covering of E, G a sheaf of groups on E, S i j = S i × S S j Let P be a G-torsor (where G acts on the right) trivialized 576 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 by ℵ, P ◦ the symmetric (left) G-torsor corresponding to P , and let F be another right G-torsor and G ◦ let F P := F P be the twisting of F by P By passing to a refinement of ℵ, we assume that P , F are trivialized by ℵ Denote by Z (ℵ, G ) the set of all 1-cocycles of G with values in ℵ If P is given by the cocycle a = (ai j ), j ∈ G ( S i j ) and F is given by b = (b i j ), then one checks that F P is given by ) Thus, as expected, there is a bijection the cocycle ba−1 = (b i j a− ij ta : Z (ℵ, a G ) → Z (ℵ, G ), u = (u i j ) → (u i j j ) Now, in the case G is an algebraic group scheme over a field k, equipped with a non-trivial valuation, we may consider ℵ = (Spec(k¯ ) → Spec(k)) and equip Z (ℵ, G ) = Z (k¯ /k, G ) → G (k¯ ⊗2 ) with the topology induced from G (k¯ ⊗2 ) Therefore the above formula for ta shows that ta : Z (ℵ, a G ) → Z (ℵ, G ) is indeed a homeomorphism Thus we have 4.2.1 Proposition Let k be a field complete with respect to a real valuation of rank The bijection ta : H1f lat (k, G ) H1f lat (k, G ) is a homeomorphism with respect to the special and canonical topologies on these sets ✷ 4.3 Adèlic case We have the following analog in the case of adèlic special topology 4.3.1 Proposition Let k be a global field, A the adèle ring of k, G an algebraic affine k-group scheme, G is as above Then the bijection ta : H1f lat (A, G ) H1f lat (A, G ) is a homeomorphism with respect to the adèlic special and canonical topologies on these sets ✷ Continuity of connecting maps Below, while we are discussing a property related with special topology without mentioning H , it means that there is no need to introduce a special group H , and the statement holds for any special groups H 5.1 Theorem Let k be either a field complete with respect to a valuation v of real rank 1, or an adèle ring of a global field Let be given an exact sequence of affine k-group schemes of finite type π → A → B −→ C → (6) 1) All connecting maps between cohomology sets in degree induced from (6) are continuous in their natural (i.e v-adic or adèlic) and the (adèlic) special (resp the canonical) topology on these sets 2) Let A be central in B If C is smooth then the coboundary map H1f lat (k, C ) → H2f lat (k, A ) is also continuous with respect to (adèlic) canonical topologies 3) If B is commutative, then all connecting maps in the exact sequence of flat cohomology, induced from above exact sequence, are continuous with respect to (adèlic) canonical topology Proof Since the (adèlic) special and canonical topology are the same, we will prove the theorem for either of these topologies Proof of 5.1, 1) For connecting maps between cohomology sets of degree If the connecting map is of type Hif lat (k, X ) → Hif lat (k, Y ), i 1, which is induced from a k-morphism X → Y , then the continuity follows from the nature of the morphism by using the canonical topology on both cohomology sets Below we will give another proof in this case by using special D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 577 topology Thus the corresponding assertion of Theorem 5.1 will follow from the following a bit more general 5.1.1 Lemma Let k be as in 5.1 1) If π : X → Y is a k-morphism of affine algebraic k-group schemes, then the induced map H1f lat (k, X ) → H1f lat (k, Y ) is continuous with respect to the special (or canonical) topology on H1f lat (k, X ) and H1f lat (k, Y ) 2) If X is a closed k-subgroup of an affine k-group scheme Y then the coboundary map δ : H0f lat (k, Y / X ) → H1f lat (k, X ) is continuous with respect to the special (or canonical) topology on H0f lat (k, Y / X ) and H1f lat (k, X ) Proof of Lemma 5.1.1 1) If π is the morphism in question, then we can regard π as the composition π = ϕ ◦ ψ, where ψ : X → X × Y , x → (x, π (x)), x ∈ X , is a closed embedding, and ϕ is the projection X × Y → Y It is clear that ϕ induces a continuous mapping H1f lat (k, X × Y ) = H1f lat (k, X )× H1f lat (k, Y ) → H1f lat (k, Y ) for any topology on the cohomology sets We show that with respect to the special topology on H1f lat (k, X ), H1f lat (k, Y ), the map ψ induces a continuous mapping ψ : H1f lat (k, X ) → H1f lat (k, X ) × H1f lat (k, Y ) We show a stronger statement, that if X is a k-subgroup of Y , then the embedding X → Y induces a continuous mapping (denoted by the same symbol) ψ : H1f lat (k, X ) → H1f lat (k, Y ) For, let Y → H be an embedding into a special k-group, considered as a closed subgroup, then we have a natural surjective morphism of quasi-projective k-schemes γ : H / X → H /Y , and also the following commutative diagram with exact rows → X (k) δX → H (k) → ( H / X )(k) → H1f lat (k, X ) → ↓ πk → Y (k) ↓ id ↓γ ↓ψ δY → H (k) → ( H /Y )(k) → H1f lat (k, Y ) → Since γ : H / X → H /Y is a morphism of k-schemes, it induces a continuous mapping (with the same notation) γ : ( H / X )(k) → ( H /Y )(k) We have ψ ◦ δ X = δY ◦ γ If U ⊂ H1f lat (k, Y ) is an open subset in the special topology, then, by definition, the set δY−1 (U ) is open in ( H /Y )(k) in the v-adic (resp −1 (U )) is adèlic) topology on ( H /Y )(k) Since γ is continuous in such topology, γ −1 (δY−1 (U )) = δ − X (ψ − also open in ( H / X )(k), which means that ψ (U ) is open in H f lat (k, X ) as desired 2) Assume that X → Y embedded as a closed k-subgroup, and Y → S is an embedding into a special k-group S Then we have the following commutative diagram with exact rows → X (k) δ1 → Y (k) → (Y / X )(k) → H1f lat (k, X ) → H1f lat (k, Y ) ↓ id → X (k) ↓ β1 → S (k) ↓ γ1 ↓ id δ2 → ( S / X )(k) → H1f lat (k, X ) → 578 D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 We have δ1 = δ2 ◦ γ1 Since γ1 is just the inclusion, thus continuous and δ2 is continuous by definition, so for an open subset U ∈ H1f lat (k, X ) δ2−1 (U ) is open for the S-special topology on H1f lat (k, X ) and δ1−1 (U ) = γ1−1 δ2−1 (U ) = δ2−1 (U ) ∩ (Y / X )(k) is open in (Y / X )(k) since (Y / X )(k) has the topology induced from that of ( S / X )(k) Thus δ1 is continuous for the S-special topology on H1f lat (k, X ) and the lemma is proved ✷ 5.1.2 Corollary With the above notation of 5.1, assume that for X → Y (as a closed subgroup scheme) Y / X is smooth, then the induced map H1f lat (k, X ) → H1f lat (k, Y ) is open in the special (thus also in canonical) (resp adèlic) topology on H1f lat (k, X ) and H1f lat (k, Y ) Proof We keep the notation as in the first diagram in the proof of 5.1.1 Let U ⊂ H1f lat (k, X ) be an open subset with respect to the special topology Let V = δ − X (U ) Then we have δ X ( V ) = U Notice that δY is an open mapping by definition and so is γ , since Y / X is smooth by assumption, thus the image γ ( V ) of V in ( H /Y )(k) is open, thus δY (γ ( V )) is open in H1f lat (k, Y ) On the other hand, δY (γ ( V )) is the image of U in H1f lat (k, Y ), since the diagram is commutative, and the assertion is proved ✷ From this we derive the following 5.1.3 Corollary With the above notation as in 5.1, for any smooth k-group G, the canonical (or special) (adèlic) topology on H1 (k, G ) is the discrete topology Proof 1) First we assume that k is a field complete with respect to a non-trivial valuation of real rank We note that the trivial cohomology class {1} in H1 (k, G ) is open in the special topology, by taking the embedding {1} → G in the above theorem (We can also argue directly as follows Take a k-embedding of G into a special k-group H Then H (k) → ( H /G )(k) is an open mapping (with respect to the v-adic topology) since G is smooth The image of H (k) there is just the preimage of {1} By the definition of the special topology, {1} is open in H1 (k, G ).) Next, let z ∈ H1 (k, G ) be any element and let a be a 1-cocycle representing z We know that z = ta (1), where ta is the bijection as in 4.2 By Proposition 4.2.1 (or 4.3.1), ta is a homeomorphism with respect to canonical topologies, thus also special topologies, by Theorem 3.1 (or 3.2.1) Hence { z} is also an open subset of H1 (k, G ), and thus so is any one-element subset of H1 (k, G ) Therefore, the topology on H1 (k, G ) is the discrete one 2) The method of the proof is similar to the proof of 1) Namely we apply the Implicit Function Theorem in the adèlic setting (see 2.4.4) From this we derive that {1} is an open subset of H1 (A, G ) and further we apply the same method of proof as above ✷ : H1f lat (k, C ) → H2f lat (k, A ) From Corollary 5.1.3, the continuis trivial since the topology on H1 (k, C ) is discrete ✷ Proof of 5.1, 2) Continuity of the map ity of Proof of 5.1, 3) Assume that B is commutative We show first that the conboundary map : H1f lat (k, C ) → H2f lat (k, A ) is continuous with respect to the canonical topology Since B is commutative, by 2.5.1, we see that one can embed B (as a closed k-subgroup) into a smooth k-split commutative k-group B Let C := B / A Then we have the following commutative diagram with exact rows D.P Bˇa´c, N.Q Thˇan ´ g / Journal of Algebra 399 (2014) 561–580 H1f lat (k, A ) → =↓ H1f lat (k, A ) → H1f lat (k, B ) ↓ H1f lat (k, B ) → H1f lat (k, C ) ↓f 579 → H2f lat (k, A ) ↓= → H1f lat (k, C ) → H2f lat (k, A ) Since C is smooth, f and are continuous by previous part, it follows that the same is true for = ◦ f Then by induction and by standard shifting dimension as above, given that all the coboundary maps between degrees and and the degrees and (i.e the maps δ : Hif lat (k, C ) → Hif+lat1 (k, A ) for various A , C , i = 0, 1) are continuous in canonical topologies, it follows that the same holds for other (“higher”) coboundary maps of the type δr : Hrf−lat1 (k, C ) → Hrf lat (k, A ) ✷ Proof of main theorem It follows from Theorems 3.1 and 5.1 ✷ 5.2 Remarks 1) In the case of commutative group schemes, the continuity of connecting maps, induced from an exact sequence of cohomology related with a short exact sequence of algebraic groups was mentioned as “obvious” in [13, Chap 3, Sec 6] However, this fact seems to be non-trivial and the proof we gave is through the study a relation between the special topology and the canonical topology 2) Corollaries 5.1.2 and 5.1.3 are not valid any more if we drop the smoothness assumption of L and G, respectively Indeed, let k be a local function field of char p > 0, G = α p the k-group scheme represented by k[ T ]/( T p ) Then it is known that H1f lat (k, G ) = k/k p , and that the subgroup k p is closed but not open in k Thus the trivial class of H1f lat (k, G ) (corresponding to the subgroup k p ) is merely closed and not open in the special topology Acknowledgments We thank P Deligne, J Milne and S Shatz for some email exchanges on the topic of the paper We thank 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Nguyen Q Thˇang, ´ On the topology on group cohomology of algebraic groups over local fields, in: Proc of the [1] Dao P Bˇac, International Conf on Research in Math and Education, 2009, pp 524–530... discuss the notion of special and canonical topologies on group cohomology In Section we consider a relation between the special and canonical topologies on group cohomology and prove the main theorem