### WEAK CORESTRICTION PRINCIPLE FOR NON-ABELIAN GALOIS COHOMOLOGY

NGUYˆE ˜N QU ˆO ´C TH ˇA ´NG

(communicated by Hvedri Inassaridze)
*Abstract*

We introduce the notion of (Weak) Corestriction Principle
and prove some relations between the validity of this principle
for various connecting maps in non-abelian Galois cohomology
over fields of characteristic 0. We also prove the validity of
Weak Corestriction Principle for images of coboundary maps
H^{1}(k, G)*→*H^{2}(k, T), where*T* is a finite commutative*k-group*
of multiplicative type, *G* is adjoint, semisimple and contains
only almost simple factors of certain inner types.

### Introduction.

Let *G* be a commutative algebraic group defined over a field *k* of characteris-
tic 0. Let H* ^{i}*(k, G) denote the usual Galois cohomology H

*(Gal(¯*

^{i}*k/k), G(¯k)). It is*well-known that there exists corestriction homomorphism

*Cores*:=

*Cores*

*k*

^{0}*/k*: H

*(k*

^{i}

^{0}*, G)→*H

*(k, G) for any*

^{i}*i*>1 and any finite extension

*k*

*of*

^{0}*k, which gives rise*to a map of functors (G

*7→*H

*(k*

^{i}

^{0}*, G))→*(G

*7→*H

*(k, G)). In particular, if*

^{i}1*→A→*^{j}*B→*^{p}*C→*1

is an exact sequence of commutative algebraic *k-groups,* *{α*1*, α*2*, ...}*

(resp.*{α*^{0}_{1}*, α*^{0}_{2}*, ...}) denotes the sequence of homomorphisms appearing in the long*
exact sequence of cohomology deduced from (∗) as cohomology of*Gal(¯k/k)-modules*
(resp. as*Gal(¯k/k** ^{0}*)-modules), then we have

*Cores◦α*^{0}* _{m}*=

*α*

*m*

*◦Cores*

for all*m*>1. However, if in (∗) one of the groups is not commutative, then it turns
out that there is no corestriction map between these two long exact sequences in
general. (In [R1], C. Riehm has found some sufficient conditions for the existence
of corestriction map.) It leads us to the following definition. Let*A, B* be algebraic
groups defined over *k. Assume that we are given a map of functors* *f* : (L *7→*

H* ^{i}*(L, A))

*→*(L

*7→*H

*(L, B)), where*

^{j}*L*denotes a field extension of

*k, i.e., a collection*of maps of cohomology sets

*f*

*L*: H

*(L, A)*

^{i}*→*H

*(L, B), where*

^{j}*f*

*L*is functorial in

*L.*

Received March 21, 2003, revised June 30, 2003; published on July 18, 2003.

2000 Mathematics Subject Classification: Primary 11G72; Secondary 18G50, 20G10.

Key words and phrases: Corestriction maps, Norm maps, Non-abelian Galois cohomology.

c

*°*2003, Nguyˆe˜n Quˆo´c Thˇa´ng. Permission to copy for private use granted.

Assume further that among *A, B, only* *B* (resp. *A) is a commutative algebraic*
*k-group and 0*6*i*61 (resp. 06*j*61).

Definition. We say that the Corestriction Principle over *k* holds for the image
(resp. the kernel) of*f**k* if we have

*Cores*_{k}^{0}* _{/k}*(Im (f

*k*

*))*

^{0}*⊂*Im (f

*k*) (resp.

*Cores**k*^{0}*/k*(Ker (f*k** ^{0}*))

*⊂*Ker (f

*k*))

for any finite extension *k** ^{0}* of

*k. We call a map H*

*(k, A)*

^{i}*→*H

*(k, B)*

^{j}*connecting*if it is the usual connecting map appearing in the long exact sequence of Galois cohomology deduced from an exact sequence of

*k-groups involving*

*A*and

*B. It*is natural and important to investigate whether or not the Corestriction Principle always holds for connecting maps. In the case

*i*= 1, j = 2, Rosset and Tate [RT]

constructed an example showing that, in general, Corestriction Principle for the image (or kernel) does not hold. Namely, let

1*→µ**n**→*SL*n**→*PGL*n* *→*1

be the exact sequence of algebraic *k-groups, whereµ**n* denotes the center (=n-th
roots of 1) of the special linear group SL*n*. Then they showed the following. Assume
that *µ**n* *⊂k* and consider any finite extension*k*^{0}*/k. Any element of the image of*
the connecting (boundary) map ∆* ^{0}* : H

^{1}(k

^{0}*,*PGL

*n*)

*→*H

^{2}(k

^{0}*, µ*

*n*) will be called a (cohomological)

*symbol*over

*k*

*. Then via the corestriction map*

^{0}*Cores*

*k*

^{0}*/k*, the image of a symbol is a

*sum*(in the corresponding group) of elements from the image of ∆ and may

*not*be in the image. Thus the norm (corestriction) of a (cohomological) symbol may not be a symbol, though it is a

*sum*of symbols. In other words, a weaker statement holds true :

*the corestriction of a symbol overk*

^{0}*lies in the group generated*

*by the symbols over*

*k*

*.*(In view of Merkurjev and Suslin’s Theorem [MS], if

*k*contains a primitive

*n-th root of 1, then we have the following stronger statement :*the image of the connecting map ∆ : H

^{1}(k,PGL

*)*

_{n}*→*H

^{2}(k, µ

*) =*

_{n}

_{n}*Br(k)generates*the whole

*n-torsion subgroup*

*n*

*Br(k) of the Brauer groupBr(k) ofk.)*

So for a connecting map *α**k* : H* ^{p}*(k, G)

*→*H

*(k, T), where*

^{q}*G,*

*T*are connected reductive

*k-groups,*

*T*is a torus, 0 6

*p*6 1,0 6

*q*6 2 and for a collection of connecting maps

*α*

*k*

*: H*

^{0}*(k*

^{p}

^{0}*, G)→*H

*(k*

^{q}

^{0}*, T*) for finite extensions

*k*

*of a field*

^{0}*k, it*is natural to ask if the above generation phenomenon always holds.

Definition. We say that*Weak Corestriction Principle holds for the image of* *α**k**,*
if

*Cores**k*^{0}*/k*(Im (α*k** ^{0}*))

*⊂ hIm (α*

*k*)i,

where *hAi* denotes the *subgroup* generated by*A* in the corresponding group and
*Cores**k*^{0}*/k* denotes the corestriction map for the corresponding cohomology groups
for the finite extension*k*^{0}*/k.*

Similarly, one may consider (W eak) *Corestriction Principle for the kernel* of a
connecting map. Notice that some particular cases of the Corestriction Principle (in

other terminology, Norm Principle), was proved to hold in [Gi], [Me2], [T1], [T2]

under certain restrictions either on the group*G, or on the arithmetic nature of the*
field*k.*

In this paper we show that the above observation made by Rosset and Tate
holds in fact for a large class of groups. We would like to make a conjecture that
Weak Corestriction Principle always holds over any field. Our first main result
(see Theorems 2.10, 2.11) shows some interrelation between the validity of (Weak)
Corestriction Principle for different kinds of connecting maps. As applicationis, one
can get other counterexamples to the Corestriction Principle (e.g. for*i*=*j*= 1).

As a second main objective of this paper, we investigate the Weak Corestric-
tion Principle for images of connecting maps coming from Galois cohomology of
*reductive*groups. Our second main result (see Theorem 4.1) is the following Weak
Corestriction Principle.

Let *k* be a field of characteristic 0 and let ∆ : H^{1}(k, G) *→* H^{2}(k, F) be the
coboundary map between cohomology sets where *F* is the center of a semisimple
*k-group* *G*1,*G*=*G*1*/F* is the adjoint *k-group of* *G*1. Assume that*G*contains only
almost simple factors of classical, inner types ^{1}A, B,C, ^{1}D*n* (n even). Let *k** ^{0}* be
a finite extension of

*k*and assume that

*k*contains (m+ 1)th-roots of unity if

*G*contains a

^{1}A

*m*-factor. Then

*Cores*_{k}^{0}* _{/k}*(Im (∆

*⊗k*

*))*

^{0}*⊂ hIm (∆)i.*

After some preliminary results in Section 1, in Section 2 we prove some equivalent conditions (in the form of reduction theorems), which show how different statements about (Weak) Corestriction Principle are related and how useful they are in reducing the problem to a simpler one. In Section 3 we reduce the problem to the quasi-split case. In Section 4 we prove the second main result mentioned above.

### 1. Preliminary results

In this section we present some necessary facts related with the well-known
crossed-diagram construction given by Ono (which was also the notion of*z-exten-*
sions used by Langlands), and we make some preliminary reductions. We will need
the following lemmas.

Lemma 1.1. *Assume that we have the following commutative diagram*

*A*^{0}

*β*

²²

*p** ^{0}* //

*B*

^{0}*γ*

²²

*q** ^{0}* //

*C*

^{0}*A* * ^{p}* //

*B*

*//*

^{q}*C,*

*whereA,B,A*^{0}*,B*^{0}*are groups, the left diagram is a commutative diagram of groups.*

*Lete** ^{0}*=

*q*

*(1),*

^{0}*e*=

*q(1), where*1

*is the identity element of the corresponding groups.*

*Then ifγ(q** ^{0−1}*(e

*))*

^{0}*⊂q*

*(e)*

^{−1}*then*

*β(r** ^{0−1}*(e

*))*

^{0}*⊂r*

*(e),*

^{−1}*withr** ^{0}*=

*q*

^{0}*p*

^{0}*,r*=

*qp.*

The proof is easy so we omit it.

Recall that a connected reductive *k-group* *H* is called (after Langglands) a *z-*
*extension*of a*k-group* *G*if*H* is an extension of *G*by an induced*k-torus* *Z, such*
that the derived subgroup (called also the semisimple part) [H, H] of*H* is simply
connected. For a field extension*K/k* and an element *x∈*H^{1}(K, G), a*z-extension*
*H* of*G*over*k* is called*x-lif ting* if*x∈*Im (H^{1}(K, H)*→*H^{1}(K, G)).

Lemma 1.2 ([T2]). *Let* *Gbe a connected reductive* *k-group,K* *a finite extension*
*of* *k, x an element of*H^{1}(K, G). Then there is a*z-extension*

1*→Z* *→H* *→G→*1,

*of G, where all groups and morphisms are defined over k, which is x-lifting.*

Lemma 1.3 ([T2]). *Let* *α*:*G*1*→G*2 *be a homomorphism of connected reductive*
*groups, all defined over k,* *x∈*H^{1}(K, G1), where K is a finite extension of k. Then
*there exists a x-lifting z-extension* *α** ^{0}* :

*H*1

*→H*2

*of*

*α, i.e.,*

*H*

*i*

*is a z-extension of*

*G*

*i*(i= 1,2), and we have the following commutative diagram

*H*1

²²

*α** ^{0}* //

*H*2

²²*G*1 *α* //*G*2*,*

*with all groups and morphisms defined over k.*

We need the following extension of a lemma of Borovoi ([Bor], p. 45).

Lemma 1.4. *Let*

1*→G*1*→G*2*→G*3*→*1

*be an exact sequence of connected reductive groups over a field k of characteristic 0,*
*k*^{0}*a finite extension of k andx∈*H^{1}(k^{0}*, G*3). Then there exists a z-extension of this
*sequence, which is x-lifting, i.e., an exact sequence of connected reductive k-groups*

1*→H*1*→H*2*→H*3*→*1,

*such that each* *H**i* *is a z-extension of* *G**i**,H*3 *is x-lifting and the following diagram*
*commutes*

1 //*H*1

²² //*H*2

²² //*H*3

²² //1

1 //*G*1 //*G*2 //*G*3 //1

*Proof.* By Lemma 1.2 there exists a *z-extension* *H*3 of *G*3 defined over *k, which*
is *x-lifting. Now we apply the same proof of Lemma 3.10.1 of [Bor] to get the*
result.

Lemma 1.5. *Let A* (resp. A* ^{0}*)

*be a pointed set, B*(resp. B

*)*

^{0}*and C*(resp. C

*)*

^{0}*be groups with homomorphisms*

*N*

*B*:

*B*

^{0}*→*

*B, N*

*C*:

*C*

^{0}*→*

*C. Let*

*f*:

*A*

*→*

*B*

*andh*:

*A→C*(resp. f

*:*

^{0}*A*

^{0}*→B*

^{0}*andh*

*:*

^{0}*A*

^{0}*→C*

*)*

^{0}*be maps of pointed sets and*

*g*:

*B→C, g*

*:*

^{0}*B*

^{0}*→C*

^{0}*be homomorphisms such thath*=

*g◦f, h*

*=*

^{0}*g*

^{0}*◦f*

^{0}*, N*

*C*

*◦g*

*=*

^{0}*g◦N*

*B*

*. If*

*N**B*(Im*f** ^{0}*)

*⊂ hImfi,*

*then*

*N**C*(Im*h** ^{0}*)

*⊂ hImhi.*

The proof is easy and is omitted.

Next we consider a reduction theorem for (Weak) Corestriction Principle for non-
abelian Galois cohomology. Namely we show that it is possible to reduce the problem
of proving the Corestriction Principle (CP) (resp. Weak Corestriction Principle
(WCP)) for connecting maps related with an exact sequence of connected linear
algebraic groups to the same problem where only connected reductive groups are
involved. We refer the readers to [Bo] for basic notions of algebraic groups. For a
linear algebraic group*G*we denote by*G** ^{◦}* its identity component.

Theorem 1.6. *Let* *α* : H* ^{p}*(k, G)

*→*H

*(k, T)*

^{q}*be a connecting map of Galois co-*

*homology, which is induced from an exact sequence of connected linear algebraic*

*groups*

(∗) 1*→A→B→C→*1,

*where all of them are defined over k and T is commutative. Then there is a connect-*
*ing map* *α*1: H* ^{p}*(k, G1)

*→*H

*(k, T1), with*

^{q}*T*1

*commutative, induced from an exact*

*sequence of connected reductive k-groups*1

*→*

*A*1

*→*

*B*1

*→*

*C*1

*→*1

*canonically*

*attached to*(∗), such that if the CP (resp. WCP) holds for the image of

*α*1

*then the*

*same holds for*

*α. The same is true for the connecting map*H

^{1}(k, G)

*→*H

^{1}(k, T),

*induced from a*

*k-morphism of connected*

*k-groups*

*f*:

*G→T, where T is commu-*

*tative.*

*Proof.* The proof consists of a case-by-case consideration, depending on the values
of*p, q.*

*Case* *p*= *q*= 0.Since *p*=*q, we are given an exact sequence of linear connected*
algebraic groups

1*→G*1*→G→*^{π}*T* *→*1.

Here *G*1 is considered as a subgroup of*G. We have* *T* =*T**s**×T**u*, where*T**s* is the
maximal torus of*T* and*T**u* is the unipotent part of*T*, all are defined over*k. Let*
*G*=*L.U* be a Levi decomposition of*G, whereU* =*R**u*(G) the unipotent radical of
*G*and*L* is a maximal connected reductive*k-subgroup ofG. Since the image ofU*
(resp. of*L) viaπ*is a unipotent (resp. reductive) group, it follows that *π(L) =T**s*,

*π(U*) =*T**u*. Let*k*^{0}*/k* be any finite extension of *k,* *g*^{0}*∈* *G(k** ^{0}*). Since

*G*=

*LU*is a semidirect product we have

*G(k*

*) =*

^{0}*L(k*

*)U(k*

^{0}*), so*

^{0}*g*

*=*

^{0}*l*

^{0}*u*

*,*

^{0}*l*

^{0}*∈L(k*

*), u*

^{0}

^{0}*∈U*(k

*).*

^{0}Let *N* = *N**k*^{0}*/k* : *T*(k* ^{0}*)

*→*

*T*(k) be the norm map. Then

*N*induces the norm maps, denoted also by

*N*,

*N*:

*T*

*s*(k

*)*

^{0}*→*

*T*

*s*(k), and

*N*:

*T*

*u*(k

*)*

^{0}*→*

*T*

*u*(k). Let

*π(u*

*) =*

^{0}*v*

^{0}*∈T*

*u*(k

*), N(v*

^{0}*) =*

^{0}*v∈T*

*u*(k). Denote by

*K*

*u*the kernel of the restriction of

*π*to

*U*. Then

*K*

*u*is a connected unipotent normal

*k-subgroup ofU*. It is well-known that the first Galois cohomology of unipotent

*k-groups over perfect fields is trivial*(see [Se], Chap. III), so from this it follows that

*π(U*(k

*)) =*

^{0}*T*

*u*(k

*), π(U(k)) =*

^{0}*T*

*u*(k).

So we have

*N(π(g** ^{0}*)) =

*N*(π(l

^{0}*u*

*))*

^{0}=*N*(π(l* ^{0}*)π(u

*))*

^{0}=*N*(π(l* ^{0}*))N(π(u

*))*

^{0}=*N*(π(l* ^{0}*))π(u)

where *u* *∈* *U*(k). Therefore to prove that *N*(π(g* ^{0}*))

*∈*

*π(G(k)) is equivalent to*proving that

*N(π(l*

*))*

^{0}*∈*

*π(G(k)).*If

*N(π(l*

*)) =*

^{0}*π(lv), with*

*l*

*∈*

*L(k), v*

*∈*

*U*(k), then

*π(v) = 1, since*

*N*(π(l

*))*

^{0}*∈*

*T*

*s*(k), so

*N*(π(l

*)) =*

^{0}*π(l). Thus to prove that*

*N*(π(G(k

*)))*

^{0}*⊂π(G(k)) is equivalent to proving thatN(π(L(k*

*)))*

^{0}*⊂π(L(k)). Now*we consider the following commutative diagram

1 //(G_{1}*∩L)*^{◦}

²² //*L*

=

²²

*π** ^{0}* //

*T*

^{0}*r*

²² //1

1 //(G1*∩L)* //*L* * ^{π}* //

*T*

*//1*

_{s}Here*T** ^{0}* =

*L/(G*1

*∩L)*

*is a torus. For*

^{◦}*x∈L, we havex(G*1

*∩L)*

^{◦}*x*

*= (G1*

^{−1}*∩L)*

*, since*

^{◦}*G*1

*∩L*is a normal subgroup of

*L. Furthermore, ifV*is the unipotent radical of (G1

*∩L)*

*then*

^{◦}*xV x*

*=*

^{−1}*V*as it is not hard to see. Thus

*V*

*⊂R*

*u*(L) =

*{1}, and*(G1

*∩L)*

*is a connected reductive subgroup of*

^{◦}*L. Now the decompositionπ*=

*r◦π*

*shows that the (CP) for the image of*

^{0}*π*follows from that for

*π*

*, hence one may pass further to the case where all groups involved are connected and reductive.*

^{0}*Case p=0, q=1.* We are given an exact sequence 1 *→* *T* *→* *G*1 *→* *G* *→* 1 of
connected groups with*T* commutative. Let*T* =*T**s**×T**u*be the decomposition of*T*
into semisimple and unipotent parts. We have the following commutative diagram

1 //*T**s*

²² //*G*1

²²

*π** ^{0}* //

*G*

^{∗}²² //1

1 //*T**s**×T**u* //*G*1 *π* //*G* //1

where*T* is embedded into*G*1,*G** ^{∗}*=

*G*1

*/T*

*s*. Since

*T*is normal in

*G*1, as in the case

*p*=

*q*= 0 we see that

*T*

*u*is also normal in

*G*1, so

*G'G*

^{∗}*/T*

*u*(T

*u*is embeded into

*G** ^{∗}*). Since the cohomology of

*T*

*u*is trivial, this case is clear.

*Case p = q =1.*We are given an exact sequence 1*→G*1*→G→*^{π}*T* *→*1, with*T*
commutative and all of them are connected linear algebraic*k-groups. As in the case*
*p*=*q*= 0, we let *T* =*T**s**×T**u*,*T**s* is the maximal torus of *T,* *T**u* is the unipotent
part of *T*,*G*=*LU* is a Levi decomposition of*G. Soπ(L) =T**s*, *π(U*) =*T**u*. If*S*
denotes the connected center of*L, thenL*=*S[L, L], thus we have* *π(S) =T**s*. Let
*g∈*H^{1}(k^{0}*, G),t*=*π** ^{∗}*(g)

*∈*H

^{1}(k

^{0}*, T*). We know that since the 1-Galois cohomology of unipotent groups are trivial, there is canonical bijections H

^{1}(k

^{0}*, G)'*H

^{1}(k

^{0}*, L),*H

^{1}(k

^{0}*, T*)

*'*H

^{1}(k

^{0}*, T*

*s*). Hence we may use the same commutative diagram as in the case

*p*=

*q*= 0 to reduce our problem of proving the CP (resp. WCP) for the image of the connecting map H

^{1}(k, G)

*→*H

^{1}(k, T) to that of the connecting map H

^{1}(k, L)

*→*H

^{1}(k, T

*s*). Then, with the same notation used there, it suffices to prove the CP (resp. WCP) for the image of the connecting map H

^{1}(k, L)

*→*H

^{1}(k, T

*), where Ker (L*

^{0}*→T*

*) is now a connected reductive*

^{0}*k-group, and we are done.*

*Case p=1, q=2. We use the same notation as in the case* *p*= 0, q = 1. We are
given an exact sequence 1*→* *T* *→* *G*1 *→G* *→*1 with *T* commutative. We arrive
again at the following commutative diagram as in the case*p*= 0, q= 1 :

1 //*T**s*

²² //*G*1

=

²² //*G*^{∗}

²² //1

1 //*T**s**×T**u* //*G*1 //*G* //1

Since *T**u* is commutative, we have H^{2}(k, T*u*) = 0 by a theorem of Serre ([Se],
Chap. III), so we have canonical isomorphism H^{2}(k, T*s**×T**u*) *'* H^{2}(k, T*s*). Hence
*G*^{∗}*/T**u**'G*(here*T**u*is embedded into*G** ^{∗}*) and we have the following exact sequence
for any extension

*l/k*

H^{1}(l, G* ^{∗}*)

*→*

*H*

^{p}^{1}(l, G)

*→*H

^{2}(l, T

*u*) = (0), and the following commutative diagram

H^{1}(l, G* ^{∗}*)

*p*^{∗}

²²

∆* ^{∗}* //H

^{2}(l, T

*s*)

*j* *'*

²²

H^{1}(l, G) ^{∆}//H^{2}(l, T*s**×T**u*)

We have a similar diagram where*l* is replaced by*k** ^{0}* and

*k, wherek*

*is a finite extension of*

^{0}*k. Since*

*p*

*is surjective for any extension*

^{∗}*l/k, it is clear that to prove*CP for the image of ∆ it suffices to prove the same thing for the image of ∆

*. We claim that the same is true for WCP. Indeed, let*

^{∗}*k*

^{0}*/k*be a finite extension,

*g*

^{0}*∈*H

^{1}(k

^{0}*, G). Let*

*g*

^{∗}*∈*H

^{1}(k

^{0}*, G*

*) such that*

^{∗}*p*

*(g*

^{∗}*) =*

^{∗}*g*

*. Assuming the WCP for*

^{0}∆* ^{∗}*, we have

*Cores*_{k}^{0}* _{/k}*(∆

*(g*

^{∗}*)) =X*

^{∗}*i*

∆* ^{∗}*(g

_{i}*),*

^{∗}where*g*_{i}^{∗}*∈*H^{1}(k, G* ^{∗}*) for all

*i. Due to the functoriality we have*

*Cores*

*k*

^{0}*/k*(∆(g

*)) =*

^{0}*Cores*

*k*

^{0}*/k*(∆p

*(g*

^{∗}*))*

^{∗}=*Cores*_{k}^{0}* _{/k}*(j(∆

*(g*

^{∗}*))*

^{∗}=*j(Cores*_{k}^{0}* _{/k}*(∆

*(g*

^{∗}*)))*

^{∗}=*j(*P

*i*∆* ^{∗}*(g

^{∗}*))*

_{i}=P

*i**j(∆** ^{∗}*(g

^{∗}*))*

_{i}=P

*i*∆p* ^{∗}*(g

^{∗}*)*

_{i}=P

*i*∆(g*i*)
with*g**i*=*p** ^{∗}*(g

_{i}*) as required.*

^{∗}Next we consider the validity of CP (resp. WCP) for the kernel of a connecting map. In a similar way we have the following result.

Theorem 1.7. *Let* *α* : H* ^{p}*(k, T)

*→*H

*(k, G)*

^{q}*be a connecting map of Galois co-*

*homology, which is induced from an exact sequence of connected linear algebraic*

*groups*

(∗) 1*→A→B→C→*1,

*where all of them are defined over k and T is commutative. Then there is a connect-*
*ing map* *α*_{1}: H* ^{p}*(k, T

_{1})

*→*H

*(k, G*

^{q}_{1}), with

*T*

_{1}

*commutative, induced from an exact*

*sequence of connected reductive k-groups*1

*→*

*A*1

*→*

*B*1

*→*

*C*1

*→*1

*canonically*

*attached to*(∗), such that if the CP (resp. WCP) holds for the kernel of

*α*1

*then the*

*same holds for*

*α. The same is true for the connecting map*H

^{1}(k, T)

*→*H

^{1}(k, G),

*induced from a*

*k-morphism of connected*

*k-groups*

*f*:

*T*

*→G, where T is commu-*

*tative.*

*Proof.* Since*α*is a connecting map in the long exact sequence of Galois cohomology
of algebraic groups, there are two possibilities : either Ker (α) = Im (β) for some
other connecting map*β, or there is no such aβ. In the first case we are reduced to*
Theorem 1.1, hence we need only consider the second case. Then we have*p*=*q.*

*Case* *p*=*q* = 0.We are given a*k-morphism of algebraick-groupsT* *→*^{f}*G, where*
*T* is commutative. Let *S*1 = Ker (f),*S*2 = Im (f), so we have the following exact
sequence 1*→S*1*→T* *→S*2*→*1, and this case becomes trivial.

*Casep*=*q*= 1.The above morphism*f*induces the connecting map*f** ^{∗}*: H

^{1}(k, T)

*→*H

^{1}(k, G). Notice that it induces also the following sequence of groups

1*→S*1*→T* *→*^{p}*S*2 *i*

*→G,*
and*f* =*i◦p.*Now the assertion follows from Lemma 1.1.

From Theorems 1.6 and 1.7 it follows that in the study of (weak) corestriction principle we may restrict ourselves to the case of connected reductive groups.

### 2. (Weak) Corestriction Principle : equivalent relations

In this section we will discuss some relations between the validity of (Weak)
Corestriction Principles for connecting maps of various types. For simplicity we
consider only connected reductive groups. If *G* is a connected reductive *k-group,*
there exist canonical maps*ab*^{i}* _{G}* : H

*(k, G)*

^{i}*→*H

^{i}*(k, G) (i= 0,1). Here H*

_{ab}

^{∗}*denotes the abelianized Galois cohomology in the sense of Borovoi - Kottwitz theory and*

_{ab}*ab*

^{∗}*denotes the canonical map between the cohomologies. We refer the reader to [Bor] for basic notions and properties of abelianized Galois cohomology of linear algebraic groups.*

_{G}2.0. Let *k* be a field of characteristic 0 and for a connected reductive *k-group*
*G, we use the following notation. Denote by ˜G*(resp. ¯*G) the simply connected cov-*
ering (resp. the adjoint) group of the semisimple part*G** ^{0}*:= [G, G] of

*G, and denote*by ˜

*F*= Ker ( ˜

*G→G),*¯

*F*

*= Ker (G*

^{0}

^{0}*→G) the corresponding kernels. We consider*¯ the following statements.

*a)The (Weak) Corestriction Principle holds for the image of any connecting map*
*α* : H* ^{p}*(k, G)

*→*H

*(k, T), where G, T are connected reductive k-groups, with T a*

^{q}*torus and with given p,q satisfying*06

*p*61,06

*p*6

*q*6

*p*+ 1.

*b)The (Weak) Corestriction Principle holds for the image of the functorial map*
*ab*^{p}* _{G}* : H

*(k, G)*

^{p}*→*H

^{p}*(k, G), for any connected reductive*

_{ab}*k-group G and given p,*06

*p*61.

*c)The (Weak) Corestriction Principle holds for the image of the coboundary map*
H* ^{p}*(k, G)

*→*H

*(k, T),*

^{p+1}*for any exact sequence*1

*→T*

*→G*1

*→G→*1

*of reductive*

*k-groups, whereG*1

*is connected, G is semisimple, T is a central subgroup, and p is*

*given ,*06

*p*61.

*d)* *The same statement as in* *c), but* *G*1 *and G are supposed to be semisimple*
*groups.*

*e)The (Weak) Corestriction Principle holds for the image of the coboundary map*
H* ^{p}*(k,

*G)*¯

*→*H

*(k, F*

^{p+1}*), for any exact sequence 1*

^{0}*→F*

^{0}*→G*1

*→G*¯

*→*1

*of reduc-*

*tive k-groups, whereG*1

*is semisimple,*

*G*¯

*is adjoint,*

*F*

^{0}*is a finite central subgroup*

*and p is given ,*06

*p*61.

*f*) *The (Weak) Corestriction Principle holds for the image of coboundary map*
H* ^{p}*(k,

*G)*¯

*→*H

*(k,*

^{p+1}*F), for any adjoint group*˜

*G*¯

*with fundamental groupF*˜

*and for*

*given p,*06

*p*61.

For the statements*a) -f*) considered above, let us denote by*x(p, q) (resp.y(p))*
the statement *x) (resp.* *y)) evaluated at (p, q), for 0* 6*p* 6 *q* 62. For example,
*a(1,*2) means the statement *a) withp*= 1, q= 2, or *f*(1) means the statement*f*)
with*p*= 1. We say that the statement *x) holds if for any possible values of (p, q),*
the corresponding statement is true. Note that for any*p,*06*p*61, we have obvious
imlications :*c(p)⇒d(p)⇒e(p)⇒f*(p), i.e., *c)⇒d)⇒e)⇒f*).

The relations between these statements are given in the following results. We will
give the proof only in the case of Weak Corestriction Principle since all proofs hold
true simultaneously for Corestriction and Weak Corestriction Principles, except
possibly Proposition 2.9. (There, in the part *b), we have to restrict ourselves to*
Corestriction Principle only.)

Proposition 2.1. *For a given connected reductive k-group G, and a given p with*
0 6*p*61, if the (Weak) Corestriction Principle holds for the image of *ab*^{p}_{G}*then*
*it also holds for any connecting map (if any)* H* ^{p}*(k, G)

*→*H

*(k, T)*

^{q}*where*

*T*

*is a*

*k-torus and*06

*q*62. In particular, if

*b(p)*

*holds for any p thena(p, q)*

*holds for*

*any pairs (p,q) which make sense.*

*Proof.* The proof follows immediately from the functoriality of the maps *ab*^{p}* _{G}* :
H

*(k, G)*

^{p}*→*H

^{p}*(k, G),*

_{ab}*p*= 0,1, proved in [Bor].

Proposition 2.2. *For a given connected reductive group G, with a z-extension H,*
*if the (Weak) Corestriction Principle holds forab*^{p}_{H}*for some p*(06*p*61)*then the*
*same holds for G. In particular, ifb(p)holds for connected reductive k-groups with*
*simply connected semisimple parts thenb(p)holds itself.*

*Proof.* For any finite extension*k** ^{0}*of

*k*let

*θ∈*H

*(k*

^{p}

^{0}*, G) be any element. We choose*a

*θ-liftingz-extension, all defined overk*: 1

*→Z*

*→H→G→*1,which is possible due to Lemma 1.2. Recall that

*H*is a connected reductive

*k-group with simply*connected semisimple part and

*Z*is an induced

*k-torus. Let denote the induced*(connecting) maps

*φ*: H* ^{p}*(k, H)

*→*H

*(k, G),*

^{p}*ψ*: H

*(k*

^{p}

^{0}*, H)→*H

*(k*

^{p}

^{0}*, G),*

and let*φ** ^{0}* and

*ψ*

*stand for similar maps where H*

^{0}*is replaced by H*

^{p}

^{p}*.*

_{ab}We have the following commutative diagram, where two skew (south - east)
arrows are corestriction maps for abelian Galois cohomology. Here all the vertical
maps are the maps*ab** ^{0}*, where

*ab*

*will denote the same map when we restrict to*

^{0}*k*

*:*

^{0}H* ^{p}*(k, H)

²²

*ψ* //H* ^{p}*(k, G)

²²

H* ^{p}*(k

^{0}*, H)*

²²

*φ* //H* ^{p}*(k

^{0}*, G)*

²²

H^{p}* _{ab}*(k, H)

^{ψ}*//H*

^{0}

^{p}*(k, G)*

_{ab}H^{p}* _{ab}*(k

^{0}*, H)*

33h

hh
hh
hh
hh
hh
hh
hh
hh
hh
hh_{φ}^{0}

//H^{p}* _{ab}*(k

^{0}*, G)*

33h

hh hh hh hh hh hh hh hh hh hh

Let*η∈*H* ^{p}*(k

^{0}*, H*) such that

*φ(η) =θ. Then*

*ab*^{0}* _{G}*(θ) =

*ab*

^{0}*(φ(η)) =*

_{G}*φ*

*(ab*

^{0}

^{0}*(η)).*

_{H}Assuming that*d) holds forH, then there are* *α**i**∈*H* ^{p}*(k, H) such that
X

*i*

*ab**H*(α*i*) =*Cores(ab*^{0}* _{H}*(η)).

Hence

*Cores(ab*^{0}* _{G}*(θ)) =

*Cores(ab*

^{0}*(φ(η)))*

_{G}=*Cores(φ** ^{0}*(ab

^{0}*(η)))*

_{H}=*ψ** ^{0}*(Cores(ab

^{0}*(η)))*

_{H}=*ψ** ^{0}*(P

*i**ab**H*(α*i*))

=P

*i**ab**G*(ψ(α*i*))
as required.

Proposition 2.3. *Assume that for anyk-morphismG→T* *of connected reductive*
*k-groups, with T a torus, the (Weak) Corestriction Principle holds for the image of*
*the induced connecting map*H* ^{p}*(k, G)

*→*H

*(k, T), and for some p, 06*

^{p}*p*61. Then

*the same holds for*

*ab*

^{p}*: H*

_{G}*(k, G)*

^{p}*→*H

^{p}*(k, G), i.e.,*

_{ab}*a(p, p)⇒b(p). In particular,*

*if*

*a)*

*holds, thenb)holds.*

*Proof.* By Proposition 2.2, we may assume that *G** ^{0}* is simply connected. By [Bor]

we have

H^{p}* _{ab}*(k, G) = H

*(k, G/G*

^{p}*),*

^{0}hence *ab*^{p}* _{G}* becomes just connecting map (p = 0,1). Since

*G/G*

*is a torus, the proposition follows.*

^{0}Proposition 2.4. *If the statementa)holds for p=q=0*(resp. p=q=1)*then it also*
*holds for p=0, q=1* (resp. p=1, q=2 ), i.e., we have *a(0,*0) *⇒* *a(0,*1), *a(1,*1) *⇒*
*a(1,*2).

*Proof.* It follows from the equivalence*a)⇔d) above, Proposition 2.2, from the func-*
toriality of the map *ab**G* : H* ^{p}*(k, G)

*→*H

^{p}*(k, G) and the fact that for a connected reductive*

_{ab}*k-groupH*with

*H*

*:= [H, H] simply connected, H*

^{0}

^{p}*(k, H) = H*

_{ab}*(k, H/H*

^{p}*) (by definition) (see [Bor]).*

^{0}Proposition 2.5. *Let* *G*¯ *be an adjoint semisimple k-group with fundamental group*
*F*˜ *and letF* *be a subgroup ofF*˜*. If for some p,*06*p*61, the (Weak) Corestriction
*Principle holds for the image of the coboundary map* *δ* : H* ^{p}*(k,

*G)*¯

*→*H

*(k,*

^{p+1}*F)*˜

*then the same holds for the image ofδ*1: H

*(k,*

^{p}*G)*¯

*→*H

*(k, F). In particular, we*

^{p+1}*havee(p)⇔f*(p).

*Proof.* We need only show that *f*) *⇒* *e). Let* *G* = ˜*G/F*, *F** ^{0}* = Ker (G

*→*

*G).*¯ Consider the following commutative diagram.

H* ^{p}*(k

^{0}*,G)*˜

²²

*q** ^{0}* //H

*(k*

^{p}

^{0}*,G)*¯

=

²²

*δ** ^{0}* //H

*(k*

^{p+1}

^{0}*,F*˜)

*γ*^{0}

²²

H* ^{p}*(k

^{0}*, G*

*)*

^{0}

^{π}*//H*

^{0}*(k*

^{p}

^{0}*,G)*¯

^{δ}*0*1 //H* ^{p+1}*(k

^{0}*, F*

*).*

^{0}Recall that ˜*G*is the simply connected covering for both ¯*G*and*G** ^{0}*.) One sees that

*δ*

^{0}_{1}=

*γ*

^{0}*δ*

*. Thus, if the Weak Corestriction Principle for images holds for*

^{0}*δ, then by*Lemmas 1.1 and 1.5, the same holds for

*δ*1.

Proposition 2.6. *Assume that a(p,q) holds for all G with simply connected semisim-*
*ple part* *G*^{0}*. Then a(p,q) holds itself.*

*Proof.* For *p*= 0 the assertion follows easily by considering any*z-extension ofG.*

For*p*=*q*= 1, it follows from Lemma 1.3 that for any finite extension*k** ^{0}* of

*k*and any element

*x∈*H

^{1}(k

^{0}*, G), there exists a*

*x-lifting*

*z-extension of*

*π*:

*G*

*→*

*T*, all defined over

*k*:

*H*1

²²

*π*1 //*H*2

²²*G* * ^{π}* //

*T*

Here *H*2 is a torus and *H*1 has simply connected semisimple part. By assump-
tion, the Weak Corestriction Principle holds for the image of the induced map
*π*^{∗}_{1} : H^{1}(k, H1) *→* H^{1}(k, H2). Let *α* : H^{1}(k, G) *→* H^{1}(k, T) be the connecting
map. By chasing on suitable diagrams one sees that the image of *x* in H^{1}(k, T)
via*Cores◦α*lies in the subgroup generated by the image of H^{1}(k, G). Hence the
Weak Corestriction Principle for images holds for*α.*

The case*p*= 1, q= 2 is considered in a similar way.

Proposition 2.7. *Let* 1 *→T* *→* *G*1 *π*

*→G→*1 *be an exact sequence of* *k-groups,*
*where* *Gis semisimple and* *T* *is a central subgroup of a reductive* *k-group* *G*1*. Let*
*G*1 =*G*^{0}_{1}*S*1*, where* *G*^{0}_{1} = [G1*, G*1], *S*1 *is a central torus and* *F* :=*S*1*∩G*^{0}_{1}*. If for*
*some p,* 0 6*p*61, the (Weak) Corestriction Principle holds for the image of the
*coboundary map* *δ* : H* ^{p}*(k, G)

*→*H

*(k, F), then the same holds for the image*

^{p+1}*of the coboundary map*H

*(k, G)*

^{p}*→*H

*(k, T). In particular,*

^{p+1}*d(p)*

*⇒*

*c(p), i.e.,*

*c)⇔d).*

*Proof.* It follows from the assumption that *G*1 is also connected and *π(G*^{0}_{1}) =*G.*

It is clear that *T* is a central subgroup of *G* which contains *S*1. We consider the
following commutative diagram

1 //*F*

²² //*G*^{0}_{1}

²² //*G*

=

²² //1

1 //*T* //*G*1 //*G* //1

Now the proposition follows easily from this diagram as in the proof of Proposi- tion 2.5, by combining with Lemmas 1.1 and 1.5.

Proposition 2.8. *Assume that for some p,* 0 6*p*61, the (Weak) Corestriction
*Principle holds for the image of the connecting map* H* ^{p}*(k, H)

*→*H

*(k, T)*

^{p}*for any*

*k-homomorphism*

*H*

*→*

*T*

*of connected reductive k-groups H, T, with T a torus.*

*Then the same holds for the image of the coboundary map* H* ^{p}*(k, G)

*→*H

*(k, F*

^{p+1}*)*

^{0}*deduced from any isogeny*1

*→*

*F*

^{0}*→*

*G*1

*→*

*G*

*→*1

*with kernel F’ and connected*

*reductive*

*k-group G, i.e.,*

*a(p, p)⇒d(p). In particular, ifa)holds thend)holds.*

*Proof.* To prove the assertion, we use the following Ono’s crossed diagram (see
[O] for details) which allows one to embed an exact sequence with finite kernel of
multiplicative type (i.e., isogeny) into another one with induced*k-torus as a kernel.*

We will denote all maps in the following diagrams (over *k* and *k** ^{0}*) by the same
symbols:

1

²²

1

²²1 //*F*

²² //*G*1

²²

*α* //*G*

=

²² //1

1 //*T*1
*γ*

²² //*H*

*γ*

²²

*α* //*G* //1 (∗)

*T*

²²

= *T*

²²1 1

where*T*1is an induced*k-torus. From this diagram we derive the following com-*
mutative diagram

1

²²

1

²²

*T*(k* ^{0}*)

*δ*

²²

1 //_{F}_{(k}²² ^{0}_{)} //_{G}_{1}_{(k}²² ^{0}_{)}

*λ* //_{G(k}^{0}_{)}

=

²²

*β* //_{H}^{1}_{(k}^{0}_{, F}_{)}

*θ*

²²1 //_{T}_{1}_{(k}^{0}_{)}

*γ*

²² //_{H}_{(k}^{0}_{)}

*γ*

²²

*α* //_{G(k}^{0}_{)} //_{1} _{(∗}^{0}_{)}

*T*(k* ^{0}*)

*δ*

²²

= *T*(k* ^{0}*)

*ζ*

²²

*G(k** ^{0}*)

*//*

^{β}_{H}

^{1}

_{(k}

^{0}

_{, F}_{)}

^{i}*//*

^{∗}_{H}

^{1}

_{(k}

^{0}

_{, G}_{1}

_{).}

We need the following lemma due to Merkurjev, which is valid for any cross-
diagram (∗) above, where*T* is not necessarily an induced torus.

Lemma 2.8.1. *[Me2] We have the following anti-commutative diagram*
*H(k** ^{0}*)

*γ*

²²

*α* //*G(k** ^{0}*)

*β*

²²

*T*(k* ^{0}*)

*//H*

^{δ}^{1}(k

^{0}*, F*)

*for any field extension*

*k⊂k*

^{0}*.*

We continue the proof of Proposition 2.8 and we assume first that*p*= 0. Note
that in this case, the Weak Corestriction Principle is just the usual Corestriction
Principle.

Since *T*1 is an induced *k-torus,* *H*^{1}(K, T1) = 0 for any field extension *K/k, so*
we have *α(H*(k* ^{0}*)) =

*G(k*

*). Now for any*

^{0}*g*

^{0}*∈*

*G(k*

*), let*

^{0}*h*

^{0}*∈*

*H*(k

*) such that*

^{0}*α(h*

*) =*

^{0}*g*

*, and denote*

^{0}*t*

*=*

^{0}*γ(h*

*),*

^{0}*f*

*=*

^{0}*β(g*

*). Since the diagram in Lemma 2.8.1 is anti-commutative, we have*

^{0}*δ(t** ^{0}*) =

*δ(γ(h*

*))*

^{0}=*−β(α(h** ^{0}*))

=*−f** ^{0}*.

Then for*f** ^{0}* =

*β(g*

*) we have*

^{0}*θ(f*

*) = 0 hence*

^{0}*y*

*=*

^{0}*δ(z*

*) for some*

^{0}*z*

^{0}*∈*

*T(k*

*). The image*

^{0}*z∈T(k) ofz*

*via*

^{0}*N*

*k*

^{0}*/k*:

*T*(k

*)*

^{0}*→T*(k) is such that

*δ(z) =f*:=

*Cores*

*F*(f

*).*

^{0}Now look at the diagram on the left hand side. By assumption, the Corestriction
Principle holds for the image of*H*(k)*→T*(k), so there are*h**i**∈H*(k) such that

X

*i*

*γ(h**i*) =*t*:=*N*_{k}^{0}* _{/k}*(t

*).*

^{0}Let*g*=*α(h),g**i* =*α(h**i*). Then
*δ(t) =*P

*i**δ(γ(h**i*))

=*−*P

*i**β*(α(h*i*))

=*−*P

*i**β*(g*i*)

=*δ(N**k*^{0}*/k*(t* ^{0}*))

=*Cores**F*(δ(t* ^{0}*)).

Since *δ(t** ^{0}*) =

*−f*

*(see above) and*

^{0}*δ(z*

*) =*

^{0}*f*

*, we have*

^{0}*δ(t*

^{0}*z*

*) = 0 and*

^{0}*Cores*

*F*(δ(t

^{0}*z*

*)) = 0, so*

^{0}*Cores**F*(δ(t* ^{0}*)) =

*−Cores*

*F*(δ(z

*))*

^{0}=*−f.*

Therefore*f* =P

*i**β(g**i*), so*f* *∈ hIm (β)i*and the case*p*= 0 is proved.

Now let *p* = 1. For any finite extension *k** ^{0}* of

*k*and for any element

*g*

*from H*

^{0}^{1}(k

^{0}*, G), by Lemma 1.2 we may choose ag*

*-lifting*

^{0}*z-extension*

1*→T*1*→H* *→G→*1,

defined over*k, such that there is an embedingF ,→T*1. We consider the following
diagram, which is similar to the one we have just considered, with the only difference
that the dimension is shifted.

H^{1}(k^{0}*, T*)

∆

²²

H^{1}(k^{0}*, F*)

²² //_{H}^{1}_{(k}^{0}_{, G}_{1}_{)}²²

*λ* //_{H}^{1}_{(k}^{0}_{, G)}

=

²²

*β* //_{H}^{2}_{(k}^{0}_{, F}_{)}

*θ*

²²

H^{1}(k^{0}*, T*1)

*π*

²² //_{H}^{1}_{(k}^{0}_{, H)}

*γ*

²²

*α* //_{H}^{1}_{(k}^{0}* _{, G)}* //

_{H}

^{2}

_{(k}

^{0}

_{, T}_{1}

_{)}

H^{1}(k^{0}*, T*)

∆

²²

= H^{1}(k^{0}*, T*)

H^{1}(k^{0}*, G)* * ^{β}* //

_{H}

^{2}

_{(k}

^{0}

_{, F}_{)}

We need the following analog of 2.8.1 for higher dimension.

Lemma 2.8.2. *We have the following anti-commutative diagram for any cross-*
*diagram*(∗)

H^{1}(k^{0}*, H*)

*γ*

²²

*α* //H^{1}(k^{0}*, G)*

*β*

²²

H^{1}(k^{0}*, T*) ^{∆} //H^{2}(k^{0}*, F*)

*Proof.* Let *h*= [(h*s*)]*∈* H^{1}(k^{0}*, H), g*= *α(h)∈*H^{1}(k^{0}*, G).*Then *g* = [(g*s*)], where
*g**s* =*α(h**s*). We choose for each *s* an element *g*^{0}_{s}*∈* *G*1(k*s*) such that *g**s* = *α(g*^{0}* _{s}*).

Then

*h**s*=*g*_{s}^{0}*t**s**, t**s**∈T*1(k*s*).

One deduces from this

*f**sr**g*_{sr}* ^{0}* =

*g*

_{s}

^{0}

^{s}*g*

_{r}*(s, r*

^{0}*∈Gal(k*

*s*

*/k*

*))*

^{0}for some*f**sr* *∈F(k**s*) and we know (see [Se], Chap. I) that (f*sr*) is a 2-cocycle which
is a representative of*β([(g**s*)]).From*h**s*=*g*_{s}^{0}*t**s*we deduce that

*γ(h**s*) =*γ(g*_{s}* ^{0}*)γ(t

*s*) =

*γ(t*

*s*), hence for

*t*= [γ(h

*s*)]

*∈*H

^{1}(k

^{0}*, T*) we have

∆(t) = [(t^{−1}_{sr}*t**s**s**t**r*)]*∈*H^{2}(k^{0}*, F*).

Now the product of two 2-cocycles is

(t^{−1}_{sr}*t*_{s}^{s}*t** _{r}*)(g

_{sr}

^{0−1}*g*

^{0}

_{s}

^{s}*g*

^{0}*) =*

_{r}*h*

^{−1}

_{sr}*h*

_{s}

^{s}*h*

*= 1, since (h*

_{r}*s*) is a 1-cocycle. Thus

*β*(α(h)) =*−∆(γ(h)),* (**)

and the lemma follows.

With*g*^{0}*∈*H^{1}(k^{0}*, G) as above, let* *h*^{0}*∈* H^{1}(k^{0}*, H) such thatg** ^{0}* =

*α(h*

*). (Recall that*

^{0}*H*is a

*g*

*-lifting*

^{0}*z-extension.) Take a cocycle representative (g*

*s*) of

*g*

*and let*

^{0}*g*

*s*=

*α(g*1,s), g1,s

*∈G*1(k

*s*). Let (h

^{0}*)*

_{s}*s*be a representative of

*h*

*,*

^{0}*h*

^{0}

_{s}*∈H(k*

*s*). Then

*β(g** ^{0}*) = [(g

^{−1}_{1,st}

*g*1,s

*s*

*g*1,t)]

=*−∆(γ(h** ^{0}*))
by the lemma above. Therefore

*Cores**F*(β(g* ^{0}*)) =

*−Cores*

*F*(∆(γ(h

*)))*

^{0}=*−∆(Cores**T*(γ(h* ^{0}*))).

By assumption, we have *Cores**T*(γ(h* ^{0}*)) = Σ

*i*

*γ(h*

*i*) for some

*h*

*i*

*∈*H

^{1}(k, H). Let

*g*

*i*=

*α(h*

*i*)

*∈*H

^{1}(k, G). Then from above we have

*Cores**F*(β(g* ^{0}*)) =

*−∆(Cores*

*T*(γ(h

*)))*

^{0}=*−∆(Σ**i**γ(h**i*))

=*−Σ**i*∆(γ(h*i*))

= Σ*i**β*(α(h*i*)) (by (∗∗))

= Σ_{i}*β*(g* _{i}*)
as required.

For a connected reductive group*G*we denote by*Ad(G) the adjoint group ofG,*
*Ad(G) =G/Cent(G).*

Proposition 2.9. *Let* *G*¯ *be an adjoint semisimple k-group with fundamental group*
*F*˜*.*

*a) (p* = 0) *Assume that Weak Corestriction Principle holds for the image of the*
*coboundary map* *δ*: H^{0}(k,*G)*¯ *→*H^{1}(k,*F*˜). Then the same holds for the connecting
*map* *α*: H^{0}(k, G)*→* H^{0}(k, T) *for all connected reductive k-groups* *G, T, with T a*
*torus such thatAd(G) = ¯G. In particular, iff*(0)*holds then* *a(0,*0) *holds.*

*b)(p*= 1) Assume that Corestriction Principle holds for the image of the cobound-
*ary map* ∆ : H^{1}(k,*G)*¯ *→* H^{2}(k,*F*˜). Then the same holds for the connecting map
H^{1}(k, G) *→*H^{1}(k, T) *for all connected reductive groups G, T with a torus T such*
*that* *Ad(G) = ¯G.*

*Proof.* *a) Notice that in the case* *p*= 0, the Weak Corestriction Principle is just
the Coretsriction Principle. Assume that we are given an exact sequence

1*→G*1*→G→T* *→*1,

of connected reductive*k-groups with* *T* a torus. Let*G** ^{0}*= [G, G],

*G*=

*G*

^{0}*.S, where*

*S*is a central torus of

*G. Denote*

*F*

*=*

^{0}*Cent(G*

*),*

^{0}*F*=

*G*

^{0}*∩S, which are finite*central subgroup of

*G*

*. From Proposition 2.5 and its proof it follows that the Weak Corestriction Principle holds for the connecting map*

^{0}*δ*: H

^{0}(k,

*G)*¯

*→*H

^{1}(k, F

*).*

^{0}Consider the following commutative diagram
1 //*F*^{0}

²² //*G*^{0}*.S*

=

²² // ¯*G×S/F*

²² //1

1 //*G** ^{0}* //

*G*

^{0}*.S*//

*S/F*//1 and also the following commutative diagram

*G(k** ^{0}*)

=

²²

*β*//^{0}*G(k*¯ * ^{0}*)

*×*(S/F)(k

*)*

^{0}*p*^{0}

²²

*δ** ^{0}* //H

^{1}(k

^{0}*, F*

*)*

^{0}*q*^{0}

²²

*G(k** ^{0}*)

^{α}*//(S/F)(k*

^{0}*)*

^{0}

^{δ}*//H*

^{0}^{1}(k

^{0}*, G*

*)*

^{0}