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Symd(A, σ)— /∼ ’ ’ σ F — / Nrd(A— ),
’’ ’
(where Nrpσ is the pfa¬an norm). Hence the invariant
i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )
2

is given by the formula
i(G)(u/∼) = Nrpσ (u) ∪ [A].
The exact sequence
µ
1 ’ Sp(A, σ) ’ GSp(A, σ) ’ Gm ’ 1,

where µ is the multiplier map, induces the following exact sequence in cohomology:
H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ) ’ 1
since H 1 (F, Gm ) = 1 by Hilbert™s Theorem 90. If deg A is divisible by 4, it turns
out that the invariant i(G) induces an invariant
i GSp(A, σ) : H 1 F, GSp(A, σ) ’ H 3 (F, µ—2 ).
2

Indeed, viewing H 1 F, GSp(A, σ) as the set of conjugacy classes of symplectic
involutions on A (see (??)), the canonical map
Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, GSp(A, σ)
takes u/∼ to the conjugacy class of Int(u) —¦ σ. For z ∈ F — and u ∈ Symd(A, σ)—
we have Nrpσ (zu) = z deg A/2 Nrpσ (u), hence Nrpσ (zu) = Nrpσ (u) in H 1 (F, µ2 )
if deg A is divisible by 4. Therefore, in this case we may set
i GSp(A, σ) Int(u) —¦ σ = i Sp(A, σ) (u/∼) = Nrpσ (u) ∪ [A].
(31.45) Example. Consider the particular case where deg A = 4. Since the quad-
ratic form Nrpσ is an Albert form of A by (??), its Hasse-Witt invariant is [A].
Therefore,
Nrpσ (u) ∪ [A] = e3 Nrpσ (u) · Nrpσ
and it follows by (??) that
i GSp(A, σ) („ ) = e3 jσ („ )
for every symplectic involution „ on A.
§31. COHOMOLOGICAL INVARIANTS 441


(31.46) Example. Let A = EndQ (V ) where V is a vector space of even dimen-
sion over a quaternion division algebra Q, and let σ be a hyperbolic involution
on A. For every nonsingular hermitian form h on V (with respect to the conju-
gation involution on Q), the invariant i GSp(A, σ) (σh ) of the adjoint involution
σh is the cohomological version of the Jacobson discriminant of h, see the notes to
Chapter ??. Indeed, if h has a diagonalization ±1 , . . . , ±n , then we may assume
σh = Int(u) —¦ σ where u is the diagonal matrix
u = diag(±1 , ’±2 , . . . , ±n’1 , ’±n ).
Then Nrpσ (u) = (’1)n/2 ±1 . . . ±n , hence
i GSp(A, σ) (σh ) = (’1)n/2 ±1 . . . ±n ∪ [Q].
Type Dn . Assume that char F = 2. Let G be an absolutely simple simply
connected group of type Dn (n ≥ 5) over F , so that G = Spin(A, σ) where A
is a central simple algebra of degree 2n over F with an orthogonal involution σ.
The case where A is split, i.e., G = Spin(q) for some quadratic form q, has been
considered in (??).
Assume that the algebra A is not split. In this case nG = 4. The exact sequence
similar to (??) yields a map
i1 : F — /F —2 = H 1 (F, µ2 ) ’ H 1 F, Spin(A, σ) .
The image i1 (a · F —2 ) for a ∈ F — corresponds to the torsor Xa given in the Cli¬ord
group “(A, σ) by the equation σ(x)x = a. The Rost invariant i(G) on Xa is given
by the formula
i(G)(Xa ) = (a) ∪ [A]
and therefore it is in general nontrivial. Hence the invariant does not factor through
the image of
H 1 F, Spin(A, σ) ’ H 1 F, O+ (A, σ)
as is the case when A is split.
Exceptional types.
G2 . Let G be an absolutely simple simply connected group of type G2 over F ,
so that G = Aut(C) where C is a Cayley algebra over F . The set H 1 (F, G) classi¬es
Cayley algebras over F . One has nG = 2 and the Rost invariant
i(G) : H 1 (F, G) ’ H 3 (F, µ—2 )
2

is given by the formula
i(G)(C ) = e3 (nC ) + e3 (nC )
where nC is the norm form of the Cayley algebra C (which is a 3-fold P¬ster form)
and e3 is the Arason invariant.
D4 . An absolutely simple simply connected algebraic group of type D4 over F
is isomorphic to Spin(T ) where T = (E, L, σ, ±) is a trialitarian algebra (see §??).
Here E is a central simple algebra with an orthogonal involution σ over a cubic
´tale extension L of F .
e
Assume ¬rst that L splits completely, i.e., L = F — F — F . Then E = A1 —
A2 — A3 where the Ai are central simple algebras of degree 8 over F . In this case
nG = 2 or 4. The ¬rst case occurs if and only if at least one of the algebras Ai is
split.
442 VII. GALOIS COHOMOLOGY


Assume now that L is not a ¬eld but does not split completely, i.e., L = F — K
where K is a quadratic ¬eld extension of F , hence E = A — C where A and C are
central simple algebras of degree 8 over F and K respectively (see §??). In this
case also nG = 2 or 4 and the ¬rst case takes place if and only if A is split.
Finally assume that L is a ¬eld (this is the trialitarian case). In this case
nG = 6 or 12. The ¬rst case occurs if and only if E is split.
F4 . nG = 6. The set H 1 (F, G) classi¬es absolutely simple groups of type F4
and also exceptional Jordan algebras. The cohomological invariant is discussed in
Chapter ??.
E6 . nG = 6 (when G is split).
Isn™t the
statement for E6 E7 . nG = 12 (when G is split).
and E7 true E8 . nG = 60.
whenever the Tits
algebras of G
are split and G
is inner ? (Skip Exercises
G.)
1. Let G be a pro¬nite group and let A be a (continuous) G-group. Show that
there is a natural bijection between the pointed set H 1 (G, A) and the direct
limit of H 1 (G/U, AU ) where U ranges over all open normal subgroups in G.
ˆ ˆ
2. Let Z be the inverse limit of Z/nZ, n ∈ N, and A be a Z-group such that any
element of A has a ¬nite order. Show that there is a natural bijection between
ˆ
the pointed set H 1 (Z, A) and the set of equivalence classes of A where the
equivalence relation is given by a ∼ a if there is b ∈ A such that a = b’1 ·a·σ(b)
ˆ
(σ is the canonical topological generator of Z).
3. Show that Aut(GL2 ) = Aut(SL2 ) — Z/2Z. Describe the twisted forms of GL2 .
4. Let Sn act on (Z/2Z)n through permutations and let G = (Z/2Z)n Sn . Let F
be an arbitrary ¬eld. Show that H 1 (F, G) classi¬es towers F ‚ L ‚ E with
L/F ´tale of dimension n and E/L quadratic ´tale.
e e
5. Let G = GLn /µ2 . Show that there is a natural bijection between H 1 (F, G)
and the set of isomorphism classes of triples (A, V, ρ) where A is a central
simple F -algebra of degree n, V is an F -vector space of dimension n2 and
ρ : A —F A ’ EndF (V ) is an isomorphism of F -algebras.
Hint: For an n-dimensional F -vector space U there is an associated triple
(AU , VU , ρU ) where AU = EndF (U ), VU = U —2 and where
ρ : EndF (U ) —F EndF (U ) ’ EndF (U —2 )
is the natural map. If F is separably closed, then any triple (A, V, ρ) is isomor-
phic to (AU , VU , ρU ). Moreover the homomorphism
GL(U ) ’ { (±, β) ∈ AutF (AU ) — GL(VU ) | ρ —¦ (± — ±) = Ad(β) —¦ µ }
given by γ ’ Ad(γ), γ —2 is surjective with kernel µ2 .
6. Let G be as in Exercise ??. Show that the sequence
2δ 1
»
H 1 (F, G) ’ H 1 (F, PGLn ) ’ ’ H 2 (F, Gm )
’ ’
is exact. Here » is induced from the natural map GLn ’ PGLn and δ 1 is the
connecting homomorphism for (??).
Using this result one may restate Albert™s theorem on the existence of
involutions of the ¬rst kind (Theorem (??)) by saying that the natural inclusion
EXERCISES 443


PGOn ’ G induces a surjection
H 1 (F, PGOn ) ’ H 1 (F, G).
The construction of Exercise ?? in Chapter ?? can be interpreted in terms
of Galois cohomology via the natural homomorphism GL(U ) ’ PGO H(U )
where U is an n-dimensional vector space and H(U ) is the hyperbolic quadratic
space de¬ned in §??.
7. Let K/F be separable quadratic extension of ¬elds. Taking transfers, the exact
sequence (??) induces an exact sequence
1 ’ RK/F (Gm ) ’ RK/F (GLn ) ’ RK/F (PGLn ) ’ 1.
Let N : RK/F (Gm ) ’ Gm be the transfer map and set
G = RK/F (GLn )/ ker N.
Show that there is a natural bijection between H 1 (F, G) and the set of iso-
morphism classes of triples (A, V, ρ) where A is a central simple K-algebra of
degree n, V is an F -vector space of dimension n2 and
ρ : NK/F (A) ’ EndF (V )
is an isomorphism of F -algebras. Moreover show that the sequence
corK/F —¦δ 1
»
H (F, G) ’ H (K, PGLn ) ’ ’ ’ ’ H 2 (F, Gm )
1 1
’ ’ ’ ’’
is exact. Here δ 1 is the connecting homomorphism for the sequence (??) and
» is given by
H 1 (F, G) ’ H 1 F, RK/F (PGLn ) = H 1 (K, PGLn ).
Using this result one may restate the theorem on the existence of involutions of
the second kind (Theorem (??)) by saying that the natural inclusion PGU n =
SUn / ker N ’ G induces a surjection
H 1 (F, PGUn ) ’ H 1 (F, G).
The construction of Exercise ?? in Chapter ?? can be interpreted in terms of
Galois cohomology via the natural homomorphism GL(UK ) ’ PGU H1 (UK )
where U is an n-dimensional F -vector space and H1 (UK ) is the hyperbolic
hermitian space de¬ned in §??.
8. Let (A, σ, f ) be a central simple F -algebra with quadratic pair. Let GL1 (A)
act on the vector space Symd(A, σ) • Sym(A, σ)— by
ρ(a)(x, g) = axσ(a), ag ,
where ag(y) = g σ(a)ya for y ∈ Sym(A, σ). Show that the stabilizer of (1, f )
is O(A, σ, f ) and that the twisted ρ-forms of (1, f ) are the pairs (x, g) such that
x ∈ A— and g y + σ(y) = TrdA (y) for all y ∈ A. Use these results to give
an alternate description of H 1 F, O(A, σ, f ) , and describe the canonical map
induced by the inclusion O(A, σ, f ) ’ GO(A, σ, f ).
9. Let L be a Galois Z/nZ-algebra over a ¬eld F of arbitrary characteristic. Using
the exact sequence 0 ’ Z ’ Q ’ Q/Z ’ 0, associate to L a cohomology class
[L] in H 2 (F, Z) and show that the class (L, a) ∈ H 2 (F, Gm ) corresponding
to the cyclic algebra (L, a) under the crossed product construction is the cup
product [L] ∪ a, for a ∈ F — = H 0 (F, Gm ).
444 VII. GALOIS COHOMOLOGY


10. Let K/F be a separable quadratic extension of ¬elds with nontrivial auto-
morphism ι, and let n be an integer which is not divisible by char F . Use
Proposition (??) to identify H 1 (F, µn[K] ) to the factor group
{ (x, y) ∈ F — — K — | xn = NK/F (y) }
.
{ (NK/F (z), z n ) | z ∈ K — }
For (x, y) ∈ F — — K — such that xn = NK/F (y), let [x, y] ∈ H 1 (F, µ[K] ) be the
corresponding cohomology class.
(a) Suppose n = 2. Since µ2[K] = µ2 , there is a canonical isomorphism
H 1 (F, µ2[K] ) F — /F —2 . Show that this isomorphism takes [x, y] to
NK/F (z) · F —2 , where z ∈ K — is such that x’1 y = zι(z)’1 .
(b) Suppose n = rs for some integers r, s. Consider the exact sequence
j
i
1 ’ µr[K] ’ µn[K] ’ µs[K] ’ 1.
’ ’
Show that the induced maps
j1
i1
H 1 (F, µr[K] ) ’ H 1 (F, µn[K] ) ’ H 1 (F, µs[K] )
’ ’
can be described as follows:
i1 [x, y] = [x, y s ] j 1 [x, y] = [xr , y].
and
(Compare with (??).)
(c) Show that the restriction map
res: H 1 (F, µn[K] ) ’ H 1 (K, µn ) = K — /K —n
takes [x, y] to y · K —n and the corestriction map
cor : H 1 (K, µn ) ’ H 1 (F, µn[K] )
takes z · K —n to [1, zι(z)’1 ].
11. Show that for n dividing 24, µn — µn and Z/nZ are isomorphic as Galois
modules.
12. Let (A, σ) be a central simple algebra over F with a symplectic involution σ.
Show that the map
Symd(A, σ)— /∼ = H 1 F, Sp(A, σ) ’ H 1 F, SL1 (A) = F — / Nrd(A— )
induced by the inclusion Sp(A, σ) ’ SL1 (A) takes a ∈ Sym(A, σ)— to its
pfa¬an norm NrpA (a) modulo Nrd(A— ).
13. Let A be a central simple algebra over F . For any c ∈ F — write Xc for the set
of all x ∈ A— such that Nrd(x) = c. Prove that
sep
(a) Xc is a SL1 (Asep )-torsor.
(b) Any SL1 (Asep )-torsor is isomorphic to Xc for some c.
(c) Xc Xd if and only if cd’1 ∈ Nrd(A— ).
14. Describe H 1 F, Spin(V, q) in terms of twisted forms of tensors.
15. Let (A, σ, f ) be a central simple F -algebra with quadratic pair of even degree 2n
over an arbitrary ¬eld F . Let Z be the center of the Cli¬ord algebra C(A, σ, f )
and let „¦(A, σ, f ) be the extended Cli¬ord group.
(a) Show that the connecting map
δ 1 : H 1 F, PGO+ (A, σ, f ) ’ H 2 F, RZ/F (Gm,Z ) = Br(Z)
EXERCISES 445


in the cohomology sequence associated to
χ
1 ’ RZ/F (Gm,Z ) ’ „¦(A, σ, f ) ’ PGO+ (A, σ, f ) ’ 1

’1
maps the 4-tuple (A , σ , f , •) to C(A , σ , f )—Z Z C(A, σ, f ) , where
the tensor product is taken with respect to •.
(b) Show that the multiplication homomorphism
Spin(A, σ, f ) — RZ/F (Gm,Z ) ’ „¦(A, σ, f )
induces an isomorphism
„¦(A, σ, f ) Spin(A, σ, f ) — RZ/F (Gm,Z ) /C
where C is isomorphic to the center of Spin(A, σ, f ). Similarly, show that
GO+ (A, σ, f ) O+ (A, σ, f ) — Gm /µ2
where µ2 is embedded diagonally in the product.
(c) Assume that n is even. Let ± : „¦(A, σ, f ) ’ GO+ (A, σ, f ) be the ho-
momorphism which, under the isomorphism in (??), is the vector rep-
resentation χ on Spin(A, σ, f ) and the norm map on RZ/F (Gm,Z ). By
relating via ± the exact sequence in (??) to a similar exact sequence for
GO+ (A, σ, f ), show that for all 4-tuple (A , σ , f , •) representing an ele-
ment of H 1 F, PGO+ (A, σ, f ) ,
’1
= [A ][A]’1
NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) in Br(F ).
In particular, NZ/F C(A, σ, f ) = [A].
Similarly, using the homomorphism „¦(A, σ, f ) ’ RZ/F (Gm,Z ) which is
trivial on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that
’1 2
C(A , σ , f ) —Z Z C(A, σ, f ) = 1.
2
In particular, C(A, σ, f ) = 1. (Compare with (??).)
(d) Assume that n is odd. Let G = O+ (A, σ, f ) — RZ/F (Gm,Z ) /µ2 . Using
the homomorphism ± : „¦(A, σ, f ) ’ G which is the vector representation
χ on Spin(A, σ, f ) and the squaring map on RZ/F (Gm,Z ), show that for all
4-tuple (A , σ , f , •) representing an element of H 1 F, PGO+ (A, σ, f ) ,
’1 2
= [AZ ][AZ ]’1
C(A , σ , f ) —Z Z C(A, σ, f ) in Br(Z).
2
In particular, C(A, σ, f ) = [AZ ].
Using the character of „¦(A, σ, f ) which is trivial on Spin(A, σ, f ) and is
the norm on RZ/F (Gm,Z ), show that
’1
NZ/F C(A , σ , f ) —Z Z C(A, σ, f ) = 1.
In particular, NZ/F C(A, σ, f ) = 1. (Compare with (??).)
16. (Qu´guiner [?]) Let (B, „ ) be a central simple F -algebra with unitary involution
e
of degree n. Let K be the center of B and let „ = Int(u) —¦ „ for some unit u ∈
Sym(B, „ ). Assume that char F does not divide n. Show that the Tits classes
t(B, „ ) and t(B, „ ) in H 2 (F, µn[K] ) are related by t(B, „ ) = t(B, „ ) + ζK ∪
NrdB (u) where ζK is the nontrivial element of H 1 (F, Z[K] ) and NrdB (u) =
NrdB (u) · F —n ∈ F — /F —n = H 1 (F, µn ). (Compare with (??).)
446 VII. GALOIS COHOMOLOGY


Notes
§??. The concept of a nonabelian cohomology set H 1 (“, A) has its origin in the
theory of principal homogeneous spaces (or torsors) due to Grothendieck [?], see
also Frenkel [?] and Serre [?]. The ¬rst steps in the theory of principal homogeneous
spaces attached to an algebraic group (in fact a commutative group variety) are
found in Weil [?].
Galois descent was implicitly used by Chˆtelet [?], in the case where A is an
a
elliptic curve (see also [?]). An explicit formulation (and proof) of Galois descent
in algebraic geometry was ¬rst given by Weil [?]. The idea of twisting the action
of the Galois group using automorphisms appears also in this paper, see Weil™s
commentaries in [?, pp. 543“544].
No Galois cohomology appears in the paper [?] on principal homogeneous spaces
mentioned above. The fact that Weil™s group of classes of principal homogeneous
spaces for a commutative group variety A over a ¬eld F stands in bijection with
the Galois 1-cohomology set H 1 (F, A) was noticed by Serre; details are given in
Lang and Tate [?], see also Tate™s Bourbaki talk [?].
The ¬rst systematic treatment of Galois descent, including nonabelian cases
(linear groups, in particular PGLn with application to the Brauer group), appeared
in Serre™s book “Corps locaux” [?], which was based on a course at the Coll`ge de
e
France in 1958/59. Twisted forms of algebraic structures viewed as tensors are
mentioned as examples. Applications to quadratic forms are given in Springer [?].
Another early application is the realization by Weil [?], following an observation of
“un amateur de cocycles tr`s connu”33 , of Siegel™s idea that classical groups can be
e
described as automorphism groups of algebras with involution (Weil [?, pp. 548“
549]).
Since then this simple but very useful formalism found many applications. See
the latest revised and completed edition of the Lecture Notes of Serre [?] and his
Bourbaki talk [?] for more information and numerous references. A far-reaching
generalization of nonabelian Galois cohomology, which goes beyond Galois exten-
sions and applies in the setting of schemes, was given by Grothendieck [?].
Our presentation in this section owes much to Serre™s Lecture Notes [?] and to
the paper [?] of Borel and Serre. The technique of changing base points by twisting
coe¬cients in cohomology, which we use systematically, was ¬rst developed there.
Note that the term “co-induced module” is used by Serre [?] and by Brown [?] for
the modules which we call “induced”, following Serre [?].
§??. Lemma (??), the so-called “Shapiro lemma”, was independently proved
by Eckmann [?, Theorem 4], D. K. Faddeev [?], and Arnold Shapiro. Shapiro™s
proof appears in Hochschild-Nakayama [?, Lemma 1.1].
Besides algebras and quadratic forms, Severi-Brauer varieties also have a nice
interpretation in terms of Galois cohomology: the group scheme PGL n occurs not
only as the automorphism group of a split central simple algebra of degree n, but
also as the automorphism group of the projective space Pn’1 . The Severi-Brauer
variety SB(A) attached to a central simple algebra A is a twisted form of the
projective space, given by the cocycle of A (see Artin [?]).
For any quadratic space (V, q) of even dimension 2n, the Cli¬ord functor de¬nes
a homomorphism C : PGO(V, q) ’ Autalg C0 (V, q) (see (??)). The induced map
in cohomology C 1 : H 1 F, PGO(V, q) ’ H 1 F, Autalg C0 (V, q) associates to
33 also referred to as “Mr. P. (the famous winner of many cocycle races)”
NOTES 447


every central simple F -algebra with quadratic pair of degree 2n a separable F -
algebra of dimension 22n’1 ; this is the de¬nition of the Cli¬ord algebra of a central
simple algebra with quadratic pair by Galois descent.
§??. Although the cyclic algebra construction is classical, the case considered
here, where L is an arbitrary Galois Z/nZ-algebra, is not so common in the lit-
erature. It can be found however in Albert [?, Chapter VII]. Note that if L is a
¬eld, its Galois Z/nZ-algebra structure designates a generator of the Galois group
Gal(L/F ).
The exact sequence (??) was observed by Arason-Elman [?, Appendix] and by
Serre [?, Chapter I, §2, Exercise 2]. (This exercise is not in the 1973 edition.) The
special case where M = µ2 (Fsep ) (Corollary (??)) plays a crucial rˆle in Arason [?].
o
The cohomological invariants f1 , g2 , f3 for central simple F -algebras with
unitary involution of degree 3 are discussed in Haile-Knus-Rost-Tignol [?, Corol-
lary 32]. It is also shown in [?] that these invariants are not independent and that
the invariant g2 (B, „ ) gives information on the ´tale F -subalgebras of B. To state
e
precise results, recall from (??) that cubic ´tale F -algebras with discriminant ∆
e
1
are classi¬ed by the orbit set H (F, A3[∆] )/S2 . Suppose char F = 2, 3 and let
F (ω) = F [X]/(X 2 + X + 1), so that µ3 = A3[F (ω)] . Let (B, „ ) be a central simple
F -algebra with unitary involution of degree 3 and let L be a cubic ´tale F -algebra
e
1
with discriminant ∆. Let K be the center of B and let cL ∈ H (F, A3[∆] ) be a co-
homology class representing L. The algebra B contains a subalgebra isomorphic to
L if and only if g2 (B, „ ) = cL ∪d for some d ∈ H 1 (F, A3[K—F (ω)—∆] ). (Compare with
Proposition (??).) If this condition holds, then B also contains an ´tale subalgebra
e
L with associated cohomology class d (hence with discriminant K — F (ω) — ∆).
Moreover, there exists an involution „ such that Sym(B, „ ) contains L and L .
See [?, Proposition 31].
§??. Let (A, σ) be a central simple algebra with orthogonal involution of even
degree over a ¬eld F of characteristic di¬erent from 2. The connecting homomor-
phism

δ 1 : H 1 F, O+ (A, σ) ’ H 2 (F, µ2 ) = 2 Br(F )

in the cohomology sequence associated to the exact sequence

1 ’ µ2 ’ Spin(A, σ) ’ O+ (A, σ) ’ 1

is described in Garibaldi-Tignol-Wadsworth [?]. Recall from (??) the bijection

H 1 F, O+ (A, σ) SSym(A, σ)— /≈.

For (s, z) ∈ SSym(A, σ)— , consider the algebra A = M2 (A) EndA (A2 ) with the
involution σ adjoint to the hermitian form 1, ’s’1 , i.e.,

’σ(c)s’1
a b σ(a)
σ = for a, b, c, d ∈ A.
sσ(d)s’1
c d ’sσ(b)

01
∈ A . We have s ∈ Skew(A , σ ) and NrdA (s ) = NrdA (s) = z 2 .
Let s =
s0

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