Theorem 5

There exists a group isomorphism

Ψ : Ex$_{\tau}(G,L^*)\longrightarrow$ Br(L$/K$)

that sends the class of a $\tau$-extension $\mathscr{E}$ of G by $L^*$ to the class in Br(L$/K$) of the algebra $\mathbf{A}[\mathscr{E}; L]$.

To define Ψ and verify that it is a bijection, we must establish the following points:

a) If $\mathscr{E}$ and $\mathscr{E}'$ are isomorphic $\tau$-extensions of G by $L^*$, then the algebras $\mathbf{A}[\mathscr{E}; L]$ and $\mathbf{A}[\mathscr{E}'; L]$ are isomorphic.

b) Conversely, if the algebras $\mathbf{A}[\mathscr{E}; L]$ and $\mathbf{A}[\mathscr{E}'; L]$ are isomorphic, then the $\tau$-extensions $\mathscr{E}$ and $\mathscr{E}'$ of G by $L^*$ are isomorphic.

c) In every class in Br(L$/K$), there is an algebra E containing L as a maximal commutative subalgebra.

d) If E is a central simple algebra of finite degree over K containing L as a maximal commutative subalgebra, then there exists a $\tau$-extension $\mathscr{E}$ of G by $L^*$ such that E is isomorphic to $\mathbf{A}[\mathscr{E}; L]$.

Assertion a) follows from the construction of the cross product; b) follows from VIII, p. 263, Corollary. Assertion c) follows from Proposition 5 of VIII, p. 281, and d) is simply Theorem 4 above.

It remains to verify that Ψ is a group homomorphism; for this, it suffices to prove that if $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ are $\tau$-extensions, then the algebra $\mathbf{A}[\mathscr{E}_1\mathscr{E}_2; L]$ is equivalent to the algebra $\mathbf{A}[\mathscr{E}_1; L]\otimes \mathbf{A}[\mathscr{E}_2; L]$. We denote the product extension $\mathscr{E}_1\mathscr{E}_2$ by $\mathscr{E}= (\Gamma , \iota , \pi )$. The group Γ is isomorphic to the cokernel of the homomorphism $\rho$ from $L^*$ to the fiber product $\Gamma_1\times_G\Gamma_2$ that sends $\mu$ to $(\iota_1(\mu), \iota_2(\mu)^{-1})$. For $i\in \{1,2\}$, write $A_i=\mathbf{A}[\mathscr{E}_i; L]$. Let $u_i: L\rightarrow A_i$ and $v_i: \Gamma_i\rightarrow A^*_i$ be the canonical homomorphisms. We identify L with its images by the homomorphisms $u_i$, which make L into maximal commutative subalgebras of the $A_i$. We denote by $V_i$ the L-vector space defined by left multiplication in $A_i$. The ring $L\otimes_KA^o_i$ acts on $V_i$. Since it is simple and has dimension $n^2$ over L, we obtain an isomorphism from $L\otimes_KA^o_i$ to End$_L(V_i)$. Hence the ring $L\otimes_KA^o_1\otimes_KA^o_2$, which we can identify with $(L\otimes_KA^o_1)\otimes_L(L\otimes_KA^o_2)$, is isomorphic to End$_L(V_1\otimes_LV_2)$. Set C = End$_{A^o_1\otimes_KA^o_2}(V_1\otimes_LV_2)$. By Lemma 3 of VIII, p. 282, the ring C is similar to $A_1\otimes_KA_2$, and $L\otimes 1\otimes 1$ is a maximal commutative subalgebra of C. For every pair $(\gamma_1, \gamma_2)\in \Gamma_1\times \Gamma_2$ satisfying $\pi_1(\gamma_1) =\pi_2(\gamma_2)$, we denote by $w(\gamma_1, \gamma_2)$ the unique $\lambda (\pi_1(\gamma_1$))-semilinear (II, §1, No. 13, p. 223) endomorphism such that

for $x_1\in V_1$ and $x_2\in V_2$. We have $w(\gamma_1, \gamma_2)\in C^*$, and $w$ is a group homomorphism from the fiber product $\Gamma_1\times_G\Gamma_2$ to $C^*$. This homomorphism is trivial on the image of $\rho$ and induces a homomorphism $v$ from Γ to $C^*$. Denote by $u: L\rightarrow C$ the morphism given by $u:\ell \mapsto \ell \otimes 1\otimes 1$. We can verify the relations

$u(\alpha ) =v(\iota (\alpha ))$ and $u(^{\gamma}\beta ) =v(\gamma )u(\beta )v(\gamma )^{-1}$

for $\alpha \in L^*,\beta \in L$, and $\gamma \in \Gamma$. Proposition 12 of VIII, p. 315 gives a homomorphism $f$ from the algebra $\mathbf{A}[\mathscr{E}; L]$ to C. Since the algebra $\mathbf{A}[\mathscr{E}; L]$ is simple, the homomorphism $f$ is injective. The algebras C and $\mathbf{A}[\mathscr{E}; L]$ have the same dimension over K, so $f$ is an isomorphism.