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...

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Journal of Algebra www.elsevier.com/locate/jalgebra

On the topology on group cohomology of algebraic groups over complete valued fields

Dào Phuong Bˇa ´ca,1,2, Nguyêñ Quô ´c Thˇa ´ngb,∗,2

aDepartment of Mathematics, VNU, Univ of Science, 334 Nguyen Trai, Hanoi, Viet Nam

bInstitute of Mathematics, 18-Hoang Quoc Viet, Hanoi, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:

Received 12 September 2012

Available online 6 November 2013

Communicated by Gernot Stroth

MSC:

primary 14L24

secondary 14L30, 20G15

Keywords:

Topology

Affine group schemes

Galois cohomology and flat cohomology

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

Introduction

Let G be a affine algebraic group scheme over a field k One may define the flat cohomology

sets (or groups) Hi

f lat(k,G), i=0,1, and if G is commutative, also the groups H i

f lat(k,G)for i2

If, moreover, G is a smooth (i.e., absolutely reduced) k-group scheme, these cohomologies are

iso-morphic 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.

E-mail addresses:bacdp@math.harvard.edu , daophuongbac@yahoo.com (D.P Bˇa ´c), nqthang@math.ac.vn (N.Q Thˇa ´ng).

1 Current address: Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA.

2 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.

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 dis-cussed: 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 2for the notion of (adèlic) special and canonical topology on the cohomology set H1

f 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 degree2 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 prelimi-nary results on this topic are presented in Section1 In Section2we discuss the notion of special and canonical topologies on group cohomology In Section 3we consider a relation between the special and canonical topologies on group cohomology and prove the main theorem In Section4we consider the twisting effect on the topology and finally in Section5we 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, H i

f 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 Hi f lat(k,G) When

G is smooth, one may consider Galois cohomology of G of degree i, denoted by H i(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, ifϕ :X→Y is a map of sets, where X is given a topology,

then we denote byTϕ the quotient topology on Y with respect toϕ :X→Y

1 Preliminaries

1.1 Galois and flat cohomology The general case of rings We need in the sequel several facts

con-cerning 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 cover-ing S/R, we set S⊗n:=S⊗ · · · ⊗ S (n-times) Let

e i:S⊗n→S ⊗( n+1)

be the map s1⊗ · · · ⊗s i−1⊗s i· · · ⊗s n→s1⊗ · · · ⊗s i−1⊗1⊗s i⊗ · · · ⊗s n

For any group (covariant) functor G from the category Com.Alg R of commutative R-algebras to the

categoryGr of groups, we denote the corresponding morphism by

G(e i) :G

S⊗n

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)

1→G(R) −→d G ,0 G(S) −→d G ,1 G

S⊗2d G ,2

S⊗3d G ,3

S⊗4

where G is considered as a covariant functor from the category Com.Alg R to the category Gr of groups and the differential d i:=d G , are given by the formula (written additively in the commutative case, for simplicity)

d G , i= −G(e1) +G(e2) − · · · + (−1)i+1G(e i+1).

In particular, we have d0(f) =f (due to the embedding R⊂S), d1(f) = −f1+ f2 , 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

x i1 i t:=G(e i t) ◦G(e i t− 1) ◦ · · · ◦G(e i1)(x)

whenever it makes sense

The cohomology group Hr(S R,G) :=Ker(d r+1)/Im(d r)of this complex is called ˇCech cohomology

of G with respect to the covering (or layer) S/R Then we define the ˘Cech–Amitsur cohomology

Hp f lat(R,G) := lim

→S RHp f lat(S/ ,G), p0,

where the limit is taken over all faithfully flat extensions S/R.

If G is non-commutative, then we may consider the non-abelian ˘Cech–Amitsur complexe for a faithfully flat extension S/R

1→G(R) −→d G ,0 G(S) −→d G ,1 G

S⊗2d G ,2

S⊗3

where the differentials dG , are given by the formulas (written multiplicatively)

d G ,0=id, d G ,1=G(e1)−1G(e2), d G ,2=G(e1)−1G(e2)G(e3)−1.

One defines

Z1(S/ ,G) := g∈G

S⊗2 g−1

1 g2g−1

3 =1

⊂G

S⊗2

and for a,b∈Z1(S R,G), a∼b in Z1(S R,G)if a=c−1bc2 for some c∈G(S), and define

H0f lat(S/ ,G) =G(R), H1f lat(S/ ,G) =Z1(S/ ,G)/ ∼, (3.1)

H0f lat(R,G) :=G(R), H1f lat(R,G) := lim

→S RH1f lat(S/ ,G), (3.2)

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))

1→G(k) →G(L) →G(L⊗k L) → · · · →G

⊗r

k 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/ ,G(L) 

:=Ker(d G ,+1), B r

L/ ,G(L) 

:=Im(d G ,).

Then we define the ˘Cech–Amitsur cohomology

Hr f lat(L/ ,G) =Z r

L/ ,G(L) 

/ r

L/ ,G(L) 

.

One may use this ˘Cech cohomology to obtain two types of cohomology for G: the Galois

cohomol-ogy Hr(Gal(k s/ ),G(k s)), by taking L=k s the separable closure of k in a fixed algebraic closure k, and¯

the flat cohomology Hr(¯k k,G)(denoted also by Hr f lat(k,G)) by taking L= ¯k If G is a smooth k-group

scheme, then it is known[20, Theorem 43]that

Hr

Gal(k s/ ),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.

2 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 1 It seems that not very much

is known about how to endow canonically a topology on the set H1

f lat(k,G)such that all connecting maps are continuous First we recall a definition of a topology on H1

f 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

1→G(k) →H(k) → (H/G)(k) −→δ H1f lat(k,G) →0.

Here H/G is a quasi-projective scheme of finite type defined over k (cf.[8, Proof of Theorem 5.4,

p 341]) Sinceδis surjective, by using the natural (Hausdorff) topology on (H/G)(k), induced from

that of k, we may endow H1 (k,G)with the strongest topology such thatδis continuous

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 1 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

fol-lowing 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 oneϕ = (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 projectionsπ :G→L=H/f(G),

π:H×k H→L:=H×k H/ ϕ (G) Denote the special topology on H1

f 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

From this we derive the following commutative diagram

G(k) → H(k) ×H(k) → L(k) δ k

→ H1f lat(k,G)

G(k) → H(k) → L(k) →δ k 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 1 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 sectionψ to q defined over an open set U⊂L hence it defines an open embedding H×U→L Thusπ defines a

birational equivalences of varieties H×U∼H×L∼L Since His 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,

q k:L(k) →L(k)is an open mapping by Implicit Function Theorem

If U∈ τ, then V := δ−1

k (U) =q−1

k (δ−1

k (U)) is open in L(k), since q

k, δk are continuous Thus

U∈ τ

Conversely, if U∈ τ, then W := δ −1

k (U) =q−1

k (δ−1

k (U)) is open in L(k) Since q k is an open

map, q k(W) is open in L(k) But q k(W) =q k(q−1

k (δ−1

k (U))) = δ−1

k (U) since q k 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 de-sired

Next we show that if ϕ :G1 G is a k-isomorphism, then ϕ induces a homeomorphism

ϕ:H1

f lat(k,G1) H1

f 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),

L1:=H/(  ◦ ϕ )(G) Then we have also an embedding  ◦ ϕ :G1→H which induces a k-morphism

ψ :L1→L Then we have the following commutative diagram

and we derive from this the following commutative diagram

G1(k) → H(k) π→1, k L1(k) δ→1, k H1f lat(k,G1)

G(k) → k H(k) →π k L(k) →δ k 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 G1, then z= ϕ( 1) = [(f(g i)i) ].) Sinceψk is clearly a homeomorphism, the above diagram shows that so isϕ with respect to H -special topology. 2

2.1.2 Definition The topology just defined is called the special topology on H1f lat(k,G)and denoted

byTs.

Notice that if k ⊂L ( ⊂ ¯k) is a normal extension, then we have canonical embedding f L :

H1

f lat(L k,G(L))  →H1

f lat(k,G) and next we identify H1

f lat(L k,G(L)) with a subset (denoted by R L

for short) of H1

f lat(k,G) Then we may regard

H1f lat(k,G) = 

L k

f L



H1f lat

L/ ,G(L) 

L k

R L.

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 L k →ϕ H1f lat(L/ ,G(L))

1 → G(k) → H(k) →π k (H/G)(k) →δ k H1f lat(k,G) → 0

1 → G(L) → H(L) →π L (H/G)(L) →δ L H1

f lat(L,G) → 0

D

L k

q

→ πL(D

L k) =D L k

Hereα , β, γ , ξ, μ , ν are just embeddings, and we let

D L k:= δ−1

k



f L k

H1f lat

L/ ,G(L) 

=Ker(ψ δk) =Ker(δLγ ) = πL



H(L) 

∩ (H/G)(k),

and

D

L k:= h∈H(L) d H ,1(h) ∈Z1

L/ ,G(L) 

= πL−1(D L k).

We set q= πL|D

L/k Let

ϕ:=d H ,1|D

L :D

L k→ ϕ

D

L k



⊂H

L⊗2

, h→h−1

1 h2.

(In terms of Galois cohomology, if L/k is a finite Galois extension, D

L / := {h∈ H(L) |h−1s h∈

G(L), for all s∈Gal(L k) }.) Then we have q(D

L /) =D L / and δk(D L /) = f L(H1f lat(L k,G(L))) and

δL( γ (D L /)) = ψ(δk(D L /)) = {1}, and γ (D L /) ⊆ 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 / = πL−1(D L /), it follows that the topology on

D L /( ⊂ πL(H(L)))is the quotient topology of the topology on D

L / 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

→ Z1(L/ ,G(L))

q↓ θL k↓

D L k →ϕ H1f lat(L/ ,G(L))  → H1f lat(k,G)

whereθL / denotes the natural projection Z1(L k,G(L)) →H1 (L k,G(L))

2.1.3 Definition On H1f lat(L k,G(L)) we may consider two topologies: the first is the topology in-duced 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 byTs , L / ) and the second topology is

called H−L k-special topology (denoted by Tϕ ) on H1f lat(L k,G(L))

Thus, if L= ¯k, the H−L k-special topology is just the special topologyTs (related with the

em-bedding G →H ).

A relation between these two topologies is given by the following

2.1.4 Lemma With notation as above, the topologyTs , L / is weaker than the H−L k-special topologyTϕ 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:= δ−1

k (U)is open

in (H/G)(k) by definition, thus ϕ 1(U) =V ∩D L / is open in D L / , i.e., U is open in the quotient

topology with respect to the mapϕ = δk|D L/k

Further, we assume that G is smooth Then D L / is open in (H/G)(k) In fact, γ (D L /) =

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 / Since D L / is open in(H/G)(k), it

follows that so is V = δ−1

k (U) Hence U itself is also open in H1(k,G), which is what we need 2

2.1.5 Lemma The action of H(k)on(H/G)(k)gives rise to the action of H(k)on D L /

Proof In fact, if z∈H(k), x=hG∈D L / , where we may assume that h∈H(L),h−1

1 h2∈G(L⊗2) We

define for z∈H(k)z.x:=zh.G, and then



(zh)1

−1

(zh)2= (z1h1)−1(z2h2)

=h−1 1



z−1

1 z2



h2

=h−1

1 h2∈G

L⊗2

.

Since z∈H(k), so z.x∈D L / Therefore we obtain the quotient space(D L // ∼)of H(k)-orbits, which gives clearly a bijectionϕ0: (D L / / Z1(L k,G(L))/ ∼) 2

2.1.6 Lemma We haveπL(D

L /) = γ (D L /), thus we have a surjective mapγ−1◦q:D

L / →D L /

Proof Indeed, it is clear that D L / ⊆ πL(D

L /) On the other hand, if h∈D

L / , then x=hG∈ (H/G)(k), and it is clear that δk(x) =1, thus x∈D L /, i.e.,πL(D

L /) ⊆D L / 2

2.1.7 Lemma The map d H ,1:H(L) →H(L⊗2), h→h−1

1 h2 defines a continuous and surjective map

ϕ:D

L k→Z1

L/ ,G(L) 

where we consider the induced topology on Z1(L k,G(L))(as a subset of G(L⊗2)) (In term of Galois coho-mology, if[L:k] =n,ϕ:h∈D

L / → (h−1 s h)s∈Gal ( / )∈G(L)n gives rise to a surjective and continuous map

ϕ:D →Z1(L k,G(L)), with induced topology on Z1(L k,G(L))as a subset of G(L)n )

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∈Z1(L k,G(L))be any element Its cohomology class in H1

f lat(L k,G(L))is

the image of an element x=hG from D L / , where as above, we may assume that h∈H(L),h−1

1 h2∈

G(L⊗2) Since h−1

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 h−1

1 h2t2 But the latter can be written as(ht)−1

1 (ht)2∈Im( ϕ), soϕis surjective 2

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 4 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 notation as in 1.1, if k is equipped with a v-adic topology, we can define a topology on r k by taking¯

the topology induced from the complete tensor product( ˆk k¯ ) (see[6, Chap III, Sec 2, Excer 28])

In particular, ifm¯v denotes the maximal ideal of the ringO ¯v of integers ofk, then the corresponding¯

topologies arem¯v-adic

Let L be an algebraic extension of k (inside k) There is a natural topology on¯ r k L, induced from

that of r k, thus also on G¯ ( r

k L) The set Z r(L k,G(L))(resp B r(L k,G(L))), being considered as a

subset of G( r 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

B r(L k,G(L))) If L= ¯k, it is called the canonical topology on Z r(¯k k,G(¯k))(resp B r(¯k k,G(¯k)))

For L as above, it is clear that Z r(L k,G(L))is a closed subgroup of G( r k L) Then we equip the quotient group Hr f 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 H r f lat(L k,G(L)) (de-noted byTL / , c ) If L= ¯k, it is called the canonical topology on H r f lat(k,G)(denoted byTc)

Next, due to Hr f lat(k,G) := Hr f lat(L k,G(L))we may consider the topology (denoted byTc , L /) on each Hr f lat(L k,G(L))which is induced fromTc.

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 Hr f lat(k,A) →Hr f 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, θL / :Z1(L k,G(L)) → (Z1(L k,G(L))/ ∼) the quotient map, where∼is as in 1.1

2.3.2 Definition The topology on Z1(L k,G(L)) induced from that of G(L⊗k L) is called the the

L k-canonical topology on Z1(L k,G(L)) The corresponding quotient topology on H1f lat(L k,G(L))with respect to the projection θL / :Z1(L k,G(L)) →Z1(L k,G(L))/ ∼ is called the L/k-canonical topol-ogy on H1 (L k,G(L)) (and is denoted by TL / , c ) If L= ¯k, it is called the canonical topology on

Z1(¯k k,G(L))(resp on H1f lat(k,G)and is denoted byTc) Especially, we may also consider the

topol-ogy 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 /

2.3.3 If K/k is another extension, K⊂L, then by functoriality there is a natural map

g L K:Z1

K/ ,G(K) 

→Z1

L/ ,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

Z1(K/ ,G(K)) g→L K Z1(L/ ,G(L))

H1f lat(K/ ,G(K)) f→L K H1f lat(L/ ,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-¯

braic closure of k,A¯ =A⊗k k Then¯ A has a natural topology and (see¯ [6, Chap III, Sec 2, Excer 28]) we may consider the complete tensor productA¯ ˆ⊗k A and the induced topology on A¯ := ¯A⊗kA¯ ⊂ ¯Aˆ⊗kA.¯

2.4.1 Notice that if H is a special k-group, then we also have H1f lat(A,H) =1 Therefore we may introduce the special topology on H1

f 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

from A, thus we may consider the complex{C r,d G , r} where C r:=G( ¯A⊗r) and consider the adèlic

topology on C r and thus also on Hr f lat(A,G)like in 2.3.2

2.4.2.1 Definition This topology is called adèlic canonical topology on H r f lat(A,G)

If G is not commutative, then we may restrict only to the case of H0and H1and the definition is

as in the previous case We have the following analog ofTheorem 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 The-orem 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 fA:

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.