3 — Etale Technology
3.2 Central simple algebras
LetK be a field of characteristic zero, choose an algebraic closure K withabsolute Galois group GK =Gal(K/K).
Lemma 3.2 The following are equivalent 1. K - Ais ´etale
2. A⊗KK'K×. . .×K 3. A=Q
LiwhereLi/Kis a finite field extension
Proof. Assume (1), thenA=K[x1, . . . , xn]/(f1, . . . , fn) wherefihave invertible Jacobian matrix.
ThenA⊗Kis a smooth commutative algebra (hence reduced) of dimension 0 so (2) holds.
Assume (2), then
HomK−alg(A,K)'HomK−alg(A⊗K,K)
3.2. Central simple algebras 121
hasdimK(A⊗K) elements. On the other hand we have by theChinese remainder theoremthat A/J ac A=Y
i
Li
withLia finite field extension ofK. However, dimK(A⊗K) =X
i
dimK(Li) =dimK(A/J ac A)≤dimK(A) and as both ends are equalAis reduced and henceA=Q
iLiwhence (3).
Assume (3), then each Li =K[xi]/(fi) with∂fi/∂xi invertible inLi. But then A =Q Li is
´
etale overKwhence (1).
To every finite ´etale extension A = Q
Li we can associate the finite set rts(A) = HomK−alg(A,K) on which the Galois group GK acts via a finite quotient group. If we write A=K[t]/(f), thenrts(A) is theset of rootsinKof the polynomialf with obvious action byGK. Galois theory, in the interpretation of Grothendieck, can now be stated as
Proposition 3.4 The functor Ket
rts(−)
- finite GK−sets is an anti-equivalence of categories.
We will now give a similar interpretation of the Abelian sheaves onKet. LetGbe a presheaf on Ket. Define
MG= lim - G(L)
where the limit is taken over all subfieldsL⊂- Kwhich are finite overK. The Galois groupGK
acts onG(L) on the left through its action onLwheneverL/Kis Galois. Hence,GK acts anMG andMG=∪MGH whereH runs through theopen subgroups(that is, containing a normal subgroup having a finite quotient) ofGK. That is,MGis acontinuousGK-module.
Conversely, given a continuousGK-moduleM we can define a presheafGM onKetsuch that
• GM(L) =MH whereH =GL=Gal(K/L).
• GM(Q
Li) =Q
GM(Li).
One verifies thatGM is a sheaf of Abelian groups onKet. Theorem 3.1 There is an equivalence of categories
S(Ket)
GK−mod
induced by the correspondences G 7→ MG and M 7→ GM. Here, GK −mod is the category of continuousGK-modules.
Proof. AGK-morphismM - M0 induces a morphism of sheavesGM - GM0. Conversely, if H is an open subgroup of GK with L = KH, then if G φ- G0 is a sheafmorphism, φ(L) : G(L) - G0(L) commutes with the action ofGK by functoriality ofφ. Therefore, lim
- φ(L) is aGK-morphismMG - MG0.
One verifies easily thatHomGK(M, M0) - Hom(GM,GM0) is an isomorphism and that the
canonical mapG - GMG is an isomorphism.
In particular, we have that G(K) =G(K)GK for every sheaf G of Abelian groups on Ket and whereG(K) =MG. Hence, the right derived functors of Γ and (−)Gcoincide for Abelian sheaves.
The categoryGK−modof continuousGK-modules is Abelian having enough injectives. There-fore, the left exact functor
(−)G:GK−mod - abelian
admits right derived functors. They are called theGalois cohomology groups and denoted Ri MG=Hi(GK, M)
Therefore, we have.
Proposition 3.5 For any sheaf of Abelian groupsGonKetwe have a group isomorphism Heti (K,G)'Hi(GK,G(K))
Therefore, ´etale cohomology is a natural extension of Galois cohomology to arbitrary commu-tative algebras. The following definition-characterization of central simple algebras is classical, see for example [66].
Proposition 3.6 LetAbe a finite dimensionalK-algebra. The following are equivalent : 1. Ahas no proper twosided ideals and the center of AisK.
2. AK=A⊗KK'Mn(K) for somen.
3. AL=A⊗KL'Mn(L)for somen and some finite Galois extensionL/K.
4. A'Mk(D) for somekwhereD is a division algebra of dimensionl2 with centerK.
The last part of this result suggests the following definition. Call two central simple algebrasA andA0equivalent if and only ifA'Mk(∆) andA0'Ml(∆) with ∆ a division algebra. From the second characterization it follows that the tensorproduct of two central simpleK-algebras is again central simple. Therefore, we can equip the set of equivalence classes of central simple algebras with a product induced from the tensorproduct. This product has the class [K] as unit element and [∆]−1 = [∆opp], the opposite algebra as ∆⊗K∆opp'EndK(∆) =Ml2(K). This group is called
3.2. Central simple algebras 123
theBrauer group and is denotedBr(K). We will quickly recall its cohomological description, all of which is classical.
GLr is an affine smooth algebraic group defined overK and is the automorphism group of a vectorspace of dimensionr. It defines a sheaf of groups onKetthat we will denote byGLr. Using the fact that the first cohomology classifies twisted forms of vectorspaces of dimensionrwe have Lemma 3.3
Het1(K,GLr) =H1(GK, GLr(K)) = 0 In particular, we have’Hilbert’s theorem 90’
Het1(K,Gm) =H1(GK,K∗) = 0
Proof. The cohomology group classifies K-module isomorphism classes of twisted forms of
r-dimensional vectorspaces overK. There is just one such class.
P GLnis an affine smooth algebraic group defined overK and it is the automorphism group of theK-algebraMn(K). It defines a sheaf of groups onKetdenoted byPGLn. By proposition 3.6 we know that any central simpleK-algebra ∆ of dimensionn2 is a twisted form ofMn(K). Therefore, Lemma 3.4 The pointed set ofK-algebra isomorphism classes of central simple algebras of dimen-sionn2 overKcoincides with the cohomology set
Het1(K,PGLn) =H1(GK, P GLn(K)) Theorem 3.2 There is a natural inclusion
Het1(K,PGLn)⊂ - Het2(K, µn) =Brn(K) whereBrn(K)is then-torsion part of the Brauer group ofK. Moreover,
Br(K) =Het2(K,Gm) is a torsion group.
Proof. Consider the exact commutative diagram of sheaves of groups onKetof figure 3.2. Taking cohomology of the second exact sequence we obtain
GLn(K) det- K∗ - Het1(K,SLn) - Het1(K,GLn) where the first map is surjective and the last term is zero, whence
Het1(K,SLn) = 0
1 1
1 - µn
?
- Gm
? (−)n
- Gm - 1
||
1 - SLn
?
- GLn
? det
- Gm - 1
PGLn
?
= PGLn
?
1?
1?
Figure 3.2: Brauer group diagram.
Taking cohomology of the first vertical exact sequence we get
Het1(K,SLn) - Het1(K,PGLn) - Het2(K, µn) from which the first claim follows.
As for the second assertion, taking cohomology of the first exact sequence we get Het1(K,Gm) - Het2(K, µn) - Het2(K,Gm) n.- Het2(K,Gm)
By Hilbert 90, the first term vanishes and henceHet2(K, µn) is equal to then-torsion of the group Het2(K,Gm) =H2(GK,K∗) =Br(K)
where the last equality follows from the crossed product result, see for example [66].