§ 16. Other Descriptions of the Brauer Group

§ 16. Other descriptions of the brauer group

In this section, if F is an abelian group and $g$ an automorphism of F, then we write $g\cdot f$ for $g(f)$ .

1. $\tau$ -Extensions of Groups

In this subsection, we fix a group G, an abelian group F written multiplicatively, and a group homomorphism $\tau$ from G to the automorphism group Aut(F) of F. We denote the identity element of G by $e$ and the identity element of F by 1.

Recall (I, §6, No. 1, p. 65) that an extension $\mathscr{E}$ of G by F is a triple $(\Gamma , \iota , \pi )$ , where Γ is a group, $\pi : \Gamma \rightarrow G$ is a surjective homomorphism, and $\iota$ is an injective homomorphism from F to Γ such that Im( $\iota$ ) $=$ Ker( $\pi$ ). Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be such an extension. For every $\gamma \in \Gamma$ , the mapping $\varphi_{\gamma}: F\rightarrow F$ defined by

\iota (\varphi_{\gamma}(f)) =\gamma \iota (f)\gamma^{-1}

for $f\in F$ is an automorphism of F. Since F is commutative, for every $f\in F$ , the automorphism defined by $\iota (f)$ is the identity mapping on F. By passing to the quotient, we obtain a homomorphism Int $_{\mathscr{E}}$ from G to Aut(F) characterized by

$\gamma \iota (f)\gamma^{-1}=\iota$ (Int $_{\mathscr{E}}(\pi (\gamma ))\cdot f$ )

for $\gamma \in \Gamma$ and $f\in F$ .

$A\tau$ -extension of G by F is an extension $\mathscr{E}= (\Gamma , \iota , \pi )$ such that Int $_{\mathscr{E}}$ is equal to $\tau$ , in other words, that satisfies the relation

\gamma \iota (f)\gamma^{-1}=\iota (\tau (\pi (\gamma ))\cdot f) \tag{1}

for $\gamma \in \Gamma$ and $f\in F$ . If $\mathscr{E}= (\Gamma , \iota , \pi )$ and $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ are $\tau$ -extensions, then a morphism of $\tau$ -extensions from $\mathscr{E}$ to $\mathscr{E}'$ is a morphism of extensions from $\mathscr{E}$ to $\mathscr{E}'$ (I, §6, No. 1, p. 65), that is, a group homomorphism $u: \Gamma \rightarrow \Gamma '$ such that $\pi '\circ u=\pi$ and $\iota '=u\circ \iota$ . Note that the $\tau$ -extensions form a species of structures for which every morphism is an isomorphism (Set Theory, IV, §1, No. 5, p. 264 and I, §6, No. 1, p. 66, Proposition 1). Denote by Iso $_{\tau}(\mathscr{E},\mathscr{F})$ the relation

“ $\mathscr{E}$ and $\mathscr{F}$ are isomorphic $\tau$ -extensions.”

This is an equivalence relation; the class of the extension $\mathscr{E}$ is the class of objects equivalent to $\mathscr{E}$ for Iso $_{\tau}($ Set Theory, II, §6, No. 6, p. 122).

Lemma 1 00I9

The relation

“ $\alpha$ is the class of a $\tau$ -extension for Iso $_{\tau}$ ”

is collectivizing in $\alpha$ .

Set $E_0= F\times G$ , and consider the mappings $\iota_0: F\rightarrow E_0$ and $\pi_0: E_0\rightarrow G$ defined by $\iota_0(f) = (f, e)$ for $f\in F$ and $\pi_0(f, g) =g$ for $(f, g)\in F\times G$ . Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension of G by F. The mapping $\pi$ is surjective, so has a section $\sigma : G\rightarrow \Gamma$ such that $\sigma (e) =e$ . The mapping $u: (f, g)\mapsto \iota (f)\sigma (g)$ from $F\times G$ to Γ is bijective. We endow $F\times G$ with the group law obtained by transfer of structure. The triple $(E_0, \iota_0, \pi_0)$ is then a $\tau$ -extension isomorphic to $\mathscr{E}$ . Lemma 1 now follows from Set Theory, II, §6, No. 6, p. 122.

We denote by Ex $_{\tau}(G,F)$ the set of classes of $\tau$ -extensions for the relation Iso $_{\tau}$ .

Example 1 00IA

The external semidirect product $(F\times_{\tau}G, i, p)$ (I, §6, No. 1, p. 66, Proposition 3) is a $\tau$ -extension; we denote it by $\mathscr{I}_{\tau}$ . We say that a $\tau$ -extension is semitrivial if it is isomorphic to the $\tau$ -extension $\mathscr{I}_{\tau}$ .

Example 2 00IB

For $i\in \{1,2\}$ , take a group $G_i$ , an abelian group $F_i$ , and a group homomorphism $\tau_i$ from $G_i$ to the automorphism group of $F_i$ . We denote by $\tau_1\times \tau_2: G_1\times G_2\rightarrow$ Aut(F $_1\times F_2$ ) the homomorphism defined by

(\tau_1\times \tau_2)(g_1, g_2)\cdot (f_1, f_2) = (\tau_1(g_1)\cdot f_1, \tau_2(g_2)\cdot f_2)

for $g_1\in G_1,g_2\in G_2,f_1\in F_1$ , and $f_2\in F_2$ . Let $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ be a $\tau_1$ -extension of $G_1$ by $F_1$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ a $\tau_2$ -extension of $G_2$ by $F_2$ . Then the triple $(\Gamma_1\times \Gamma_2, \iota_1\times \iota_2, \pi_1\times \pi_2)$ is a $\tau_1\times \tau_2$ -extension of $G_1\times G_2$ by $F_1\times F_2$ ; it is called the exterior product of the extensions $\mathscr{E}_1$ and $\mathscr{E}_2$ and denoted by $\mathscr{E}_1\times \mathscr{E}_2$ .

2. Inverse Image of a $\tau$ -Extension

Let G and $G'$ be groups. Let F be an abelian group, and let $\tau : G\rightarrow$ Aut(F) and $u: G'\rightarrow G$ be group homomorphisms. Consider the group homomorphism $\tau '=\tau \circ u: G'\rightarrow$ Aut(F), and denote by $\Gamma '$ the fiber product $\Gamma \times_GG'$ with respect to $\pi$ and $u$ (I, §4, No. 8, p. 46). It is the subgroup of $\Gamma \times G'$ consisting of the pairs $(\gamma , g')$ such that $\pi (\gamma ) =u(g')$ . Let $\iota ': F\rightarrow \Gamma '$ be the group homomorphism given by $\iota '(\alpha ) = (\iota (\alpha ), e)$ for $\alpha \in F$ , and denote the second projection by $\pi ': \Gamma '\rightarrow G'$ . Then the morphism $\iota '$ is injective, the morphism $\pi '$ is surjective because $\pi$ is, and the image of $\iota '$ coincides with the kernel of $\pi '$ . Moreover, for every $\alpha \in F$ and $\gamma '= (\gamma , g)\in \Gamma '$ , we have the relations

\gamma '\iota '(\alpha )\gamma^{'-1}= (\gamma , g)(\iota (\alpha ), e)(\gamma , g)^{-1}= (\iota (\tau (\pi (\gamma )).\alpha ), e) =\iota '(\tau '(\pi '(\gamma ')).\alpha )

Consequently, $(\Gamma ', \iota ', \pi ')$ is a $\tau '$ -extension of $G'$ by F; we call it the inverse image of $\mathscr{E}$ by $u$ and denote it by $u^*(\mathscr{E})$ . The first projection is a group homomorphism $\varphi : \Gamma '\rightarrow \Gamma$ that we call canonical.

Proposition 1 00IC

The following diagram commutes:

F $^{\iota'}$ // ${\Gamma'}^{\pi'}$ // $G'$

(2)

\varphi u

F $^{\iota}$ // Γ $^{\pi}$ // G.

Moreover, if $\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1)$ is a $\tau '$ -extension and $\varphi_1: \Gamma '_1\rightarrow \Gamma$ is a group homomorphism such that the diagram

F $^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'$

\varphi_1u

F $^{\iota}$ // Γ $^{\pi}$ // G

commutes, then there exists a unique morphism $\psi$ of $\tau '$ -extensions from $\mathscr{E}_1'$ to $u^*(\mathscr{E})$ such that we have $\varphi_1=\varphi \circ \psi$ .

The commutativity of the first diagram follows from the definition of $\varphi$ . The existence and uniqueness of $\psi$ follow from Lemma 2 below.

Lemma 2 00ID

Let $F'_1$ be an abelian group, and let $w: F'_1\rightarrow$ F and $\tau_1: G'\rightarrow$ Aut(F $'_1$ ) be group homomorphisms such that

w(\tau_1(g)(f)) =\tau (u(g))(w(f))

for every $f\in F'_1$ and $g\in G'$ . Let $\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1)$ be a $\tau_1$ -extension of $G'$ by $F'_1$ and $\varphi_1: \Gamma '_1\rightarrow \Gamma$ a group homomorphism such that the following diagram commutes:

${F'_1}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'$

w\varphi_1u

F $^{\iota}$ // Γ $^{\pi}$ // G.

Then there exists a unique group homomorphism $\psi : \Gamma '_1\rightarrow \Gamma '$ such that the diagram

${F'_1}^{\iota'_1}$ // ${\Gamma'_1}^{\pi_1'}$ // $G'$

w\psi

F $^{\iota'}$ // ${\Gamma'}^{\pi'}$ // $G'$

commutes and $\varphi_1=\varphi \circ \psi$ .

If the group homomorphism $\psi : \Gamma '_1\rightarrow \Gamma '$ has the desired properties, then it satisfies the relations

\psi (\gamma ) = (\varphi \circ \psi (\gamma ), \pi '\circ \psi (\gamma )) = (\varphi_1(\gamma ), \pi '_1(\gamma ))

for every $\gamma \in \Gamma '_1$ . Conversely, the group homomorphism from $\Gamma '_1$ to $\Gamma \times G'$ defined by $\gamma \mapsto (\varphi_1(\gamma ), \pi_1'(\gamma ))$ has values in the fiber product $\Gamma \times_GG'$ because $\pi \circ \varphi_1(\gamma ) =u\circ \pi_1'(\gamma )$ for every $\gamma \in \Gamma '_1$ .

Corollary 1 00IE

Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be $\tau$ -extensions of G by F, and let $\psi$ be a morphism of $\tau$ -extensions from $\mathscr{E}_1$ to $\mathscr{E}_2$ . Denote by $\varphi_1($ resp. $\varphi_2)$ the canonical homomorphism for $u^*(\mathscr{E}_1) ($ resp. $u^*(\mathscr{E}_2))$ . Then there exists a unique morphism of $\tau '$ -extensions from $u^*(\mathscr{E}_1)$ to $u^*(\mathscr{E}_2)$ , denoted by $u^*(\psi )$ , such that $\varphi_2\circ u^*(\psi ) =\psi \circ \varphi_1$ .

It suffices to apply Proposition 1 to $\psi \circ \varphi_1$ .

The class of the $\tau '$ -extension $u^*(\mathscr{E})$ therefore depends only on the class of $\mathscr{E}$ . We also denote by $u^*:$ Ex $_{\tau}(G,F)\rightarrow$ Ex $_{\tau'}(G',F)$ the mapping that sends the class of a $\tau$ -extension $\mathscr{E}$ to the class of the $\tau '$ -extension $u^*(\mathscr{E})$ .

Corollary 2 00IF

Let $u': G''\rightarrow G'$ be a group homomorphism, and let $\mathscr{E}$ be a $\tau$ -extension of G by F. Set $\tau ''=\tau '\circ u'$ , and denote by $\varphi ($ resp. $\varphi ',\varphi '')$ the canonical homomorphism associated with the $\tau '$ -extension $u^*(\mathscr{E}) ($ resp. the $\tau ''$ -extension $u^{'*}(u^*(\mathscr{E}))$ , the $\tau ''$ -extension $(u\circ u')^*(\mathscr{E}))$ . Then there exists a unique morphism $\psi$ from the $\tau ''$ -extension $u^{'*}(u^*(\mathscr{E}))$ to the $\tau ''$ -extension $(u\circ u')^*(\mathscr{E})$ such that $\varphi ''\circ \psi =\varphi \circ \varphi '$ .

Example 00RI

Let H be a subgroup of G and $j: H\rightarrow G$ the canonical injection. Then for every $\tau$ -extension $\mathscr{E}= (\Gamma , \iota , \pi )$ , the $\tau \circ j$ -extension $j^*(\mathscr{E})$ is isomorphic to $(\overset{-1}{\pi}(H), \iota ', \pi ')$ , where $\iota ': F\rightarrow \overset{-1}{\pi}(H)$ (resp. $\pi ':\overset{-1}{\pi}(H)\rightarrow H$ ) is the group homomorphism $f\mapsto \iota (f)$ (resp. $\gamma \mapsto \pi (\gamma )$ ). More generally, if the group homomorphism $u: G'\rightarrow G$ is injective, then the canonical homomorphism $\varphi$ is injective with image $\overset{-1}{\pi}(u(G'))$ .

3. Direct Image of a $\tau$ -Extension

Let G be a group, let F and $F'$ be abelian groups, let $\tau$ (resp. $\tau '$ ) be a group homomorphism from G to the automorphism group of F (resp. $F'$ ), and let $v: F\rightarrow F'$ be a group homomorphism such that

v(\tau (g)\cdot f) =\tau '(g)\cdot v(f) \tag{3}

for every $g\in G$ and $f\in F$ . Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension of G by F. Let $\widetilde{\Gamma}$ be the external semidirect product $F'\times_{\tau'\circ\pi}\Gamma$ . We denote by $\widetilde{\imath}: F'\rightarrow \widetilde{\Gamma}$ the homomorphism $f\mapsto (f, e)$ and by $\widetilde{p}:\widetilde{\Gamma}\rightarrow \Gamma$ the first projection. Let $j: F\rightarrow \widetilde{\Gamma}$ be the mapping defined by $f\mapsto (v(f), \iota (f)^{-1})$ . Since the image of $\iota$ is contained in the kernel of $\tau '\circ \pi$ , the mapping $j$ is a group homomorphism; it is injective because $\iota$ is injective. We have the relations

(f', \gamma )j(f)(f', \gamma )^{-1}= (f', \gamma )(v(f), \iota (f)^{-1})(f', \gamma )^{-1}

= (\tau '(\pi (\gamma ))\cdot v(f), \iota (\tau (\pi (\gamma ))\cdot f)^{-1}) \tag{4}

=j(\tau (\pi (\gamma ))\cdot f)

for $f\in F,\gamma \in \Gamma$ , and $f'\in F'$ . Consequently, the image of $j$ is a normal subgroup of $\widetilde{\Gamma}$ . We denote by $\Gamma '$ the quotient of $\widetilde{\Gamma}$ by the image of $j$ . We denote by $\iota '$ the composition of the canonical surjection from $\widetilde{\Gamma}$ to $\Gamma '$ and $\widetilde{\imath}$ . The kernel of the homomorphism $\pi \circ \widetilde{p}$ is the product $F'\times \iota$ (F), which contains the image of $j$ . We define $\pi ': \Gamma '\rightarrow G$ as the group homomorphism deduced from $\pi \circ \widetilde{p}$ by passing to the quotient. Since $\iota$ is injective, the intersection of $j(F)$ and $\widetilde{\imath}(F')$ is reduced to the identity element of $\widetilde{\Gamma}$ ; it follows that $\iota '$ is injective. The mapping from $F'\times F$ to $F'\times \iota (F) =$ Ker( $\pi \circ \widetilde{p}$ ) given by

(f', f)\longmapsto (f'v(f), \iota (f)^{-1})

is a group isomorphism. The image of $\iota '$ therefore coincides with the kernel of $\pi '$ . Since $\pi$ and $\widetilde{p}$ are surjective, so is $\pi '$ . This proves that $\mathscr{E}'= (\Gamma ', \iota , \pi ')$ is a $\tau '$ -extension of G by $F'$ ; we call it the direct image of $\mathscr{E}$ by $v$ and denote it by $v_*(\mathscr{E})$ . The composition of the canonical surjection from $\widetilde{\Gamma}$ to $\Gamma '$ and the group homomorphism from Γ to $\widetilde{\Gamma}$ given by $\gamma \mapsto (1, \gamma )$ is a group homomorphism $\varphi : \Gamma \rightarrow \Gamma '$ that we call canonical.

Proposition 2 00IG

In the notation above, the following diagram commutes:

F $^{\iota}$ // Γ $^{\pi}$ // G

v\varphi \tag{5}

${F'}^{\iota'}$ // ${\Gamma'}^{\pi'}$ // G .

Let $\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi_1')$ be a $\tau '$ -extension of G by $F'$ , and let $\varphi_1: \Gamma \rightarrow \Gamma '_1$ be a group homomorphism such that the following diagram commutes:

F $^{\iota}$ // Γ $^{\pi}$ // G

v\varphi_1

${F'}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // G.

Then there exists a unique morphism $\psi$ of $\tau '$ -extensions from $v_*(\mathscr{E})$ to $\mathscr{E}_1'$ such that we have $\varphi_1=\psi \circ \varphi$ .

The commutativity of the first diagram follows by construction. The existence and uniqueness of $\psi$ follow from Lemma 3 below.

Lemma 3 00IH

Let $G'_1$ be a group, and let $w: G\rightarrow G'_1$ and $\tau_1: G'_1\rightarrow$ Aut(F $'$ ) be group homomorphisms such that $\tau '=\tau_1\circ w$ . Let $\mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1)$ be a $\tau_1$ -extension of $G'_1$ by $F'$ , and let $\varphi_1: \Gamma \rightarrow \Gamma '_1$ be a group homomorphism such that the following diagram commutes:

F $^{\iota}$ // Γ $^{\pi}$ // G

v\varphi_1w

${F'}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'_1$ .

Then there exists a unique group homomorphism $\psi : \Gamma '\rightarrow \Gamma '_1$ such that the diagram

${F'}^{\iota'}$ // ${\Gamma'}^{\pi'}$ // G

\psi w

${F'}^{\iota'_1}$ // ${\Gamma'_1}^{\pi'_1}$ // $G'_1$ commutes and $\varphi_1=\psi \circ \varphi$ .

For any $(f, \gamma )\in F'\times \Gamma$ , denote the class of $(f, \gamma )$ in $\Gamma '$ by $(f, \gamma )$ . If the group homomorphism $\psi : \Gamma '\rightarrow \Gamma '_1$ has the desired properties, then it satisfies the relations

\psi (f', \gamma )=\psi (\iota '(f')\varphi (\gamma )) =\iota '_1(f')\varphi_1(\gamma )

for every $f'\in F'$ and $\gamma \in \Gamma$ . Conversely, the mapping $\widetilde{\psi}$ from $F'\times_{\tau'\circ\pi}\Gamma$ to $\Gamma '_1$ given by $(f, \gamma )\mapsto \iota '_1(f)\varphi_1(\gamma )$ is a group homomorphism. Indeed, we have the relations

\iota '_1(f)\varphi_1(\gamma )\iota '_1(f')\varphi_1(\gamma ') =\iota '_1(f \tau_1(\pi '_1(\varphi_1(\gamma )))\cdot f')\varphi_1(\gamma \gamma ')

=\iota '_1(f \tau '(\pi (\gamma ))\cdot f')\varphi_1(\gamma \gamma ')

for $f, f'\in F'$ and $\gamma , \gamma '\in \Gamma$ . The kernel of $\widetilde{\psi}$ contains the image of $j$ because $\iota '_1(v(f)) =\varphi_1(\iota (f))$ for $f\in F$ , and the morphism $\psi$ deduced from $\widetilde{\psi}$ by passing to the quotient has the desired properties.

Remark 00II

Denote by Σ the species of structures of $\tau '$ -extensions, and define $\alpha$ -mappings to be mappings from Γ to a group $\Gamma '$ underlying a $\tau '$ -extension that are group homomorphisms and that make the following diagram commute:

F $^{\iota}$ // Γ $^{\pi}$ // G

v\varphi \tag{6}

${F'}^{\iota'}$ // ${\Gamma'}^{\pi'}$ // G .

Proposition 2 expresses that $v_*(\mathscr{E})$ is a solution of the corresponding universal mapping problem (Set Theory, IV, §3, No. 1, p. 284).

Corollary 1 00IJ

Let $\mathscr{E}_1$ and $\mathscr{E}_2$ be $\tau$ -extensions of G by F, and let $\psi$ be a morphism of $\tau$ -extensions from $\mathscr{E}_1$ to $\mathscr{E}_2$ . Denote by $\varphi_1($ resp. $\varphi_2)$ the canonical homomorphism for $v_*(\mathscr{E}_1) ($ resp. $v_*(\mathscr{E}_2))$ . Then there exists a unique morphism of $\tau '$ -extensions from $v_*(\mathscr{E}_1)$ to $v_*(\mathscr{E}_2)$ , denoted by $v_*(\psi )$ , such that $\varphi_2\circ \psi =v_*(\psi )\circ \varphi_1$ .

It suffices to apply Proposition 2 to $\varphi_2\circ \psi$ .

The class of the $\tau '$ -extension $v_*(\mathscr{E})$ therefore depends only on the class of $\mathscr{E}$ . We also denote by $v_*:$ Ex $_{\tau}(G,F)\rightarrow$ Ex $_{\tau'}(G,F')$ the mapping that sends the class of a $\tau$ -extension $\mathscr{E}$ to the class of the $\tau '$ -extension $v_*(\mathscr{E})$ .

Corollary 2 00RJ

We keep the notation of the proposition. Let $F''$ be an abelian group, and let $\tau '': G\rightarrow$ Aut(F $''$ ) and $v': F'\rightarrow F''$ be group homomorphisms such that

\tau ''(g)\cdot v'(f) =v'(\tau '(g)\cdot f)

for $g\in G$ and $f\in F'$ . Let $\mathscr{E}$ be a $\tau$ -extension of G by F, and denote by $\varphi ($ resp. $\varphi ',\varphi '')$ the canonical homomorphism associated with $v_*(\mathscr{E}) ($ resp. $v'_*(v_*(\mathscr{E}))$ , $(v'\circ v)_*(\mathscr{E}))$ . Then there exists a unique morphism $\psi$ from the $\tau ''$ -extension $v'_*(v_*(\mathscr{E}))$ of G by $F''$ to the $\tau ''$ -extension $(v'\circ v)_*(\mathscr{E})$ such that $\varphi ''=\psi \circ \varphi '\circ \varphi$ .

Example 1 00IK

Let $j:\{1\} \rightarrow F$ be the canonical injection. The semitrivial extension $\mathscr{I}_{\tau}$ is isomorphic to $j_*((G, i$ , Id $_G))$ , where $i:\{e\} \rightarrow G$ is the canonical injection. Let $c: F\rightarrow F$ be the constant homomorphism $f\mapsto 1$ , and let $\mathscr{E}$ be a $\tau$ -extension. The $\tau$ -extension $c_*(\mathscr{E})$ is also isomorphic to $\mathscr{I}_{\tau}$ .

Example 2 00IL

Let E be a subgroup of F stable under the action of G. Denote by $F'$ the quotient of F by E, by $v: F\rightarrow F'$ the canonical homomorphism, and by $\tau ': G\rightarrow$ Aut(F $'$ ) the action of G on $F'$ characterized by

\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)

for $g\in G$ and $f\in F$ . Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension of G by F. Then $\iota (E)$ is a normal subgroup of Γ, and the $\tau '$ -extension $v_*(\mathscr{E})$ of G by $F'$ is isomorphic to the extension $(\Gamma /\iota (E), \iota , \pi )$ , where $\overline{\iota}$ and $\overline{\pi}$ are the group homomorphisms deduced from $\iota$ and $\pi$ by passing to the quotients. By this isomorphism, the canonical homomorphism $\varphi$ associated with $v_*(\mathscr{E})$ corresponds to the canonical homomorphism from Γ to $\Gamma /\iota (E)$ .

Proposition 3 00IM

Let G and $G'$ be groups, and let F and $F'$ be abelian groups. Let $\tau : G\rightarrow$ Aut(F) $,\tau ': G\rightarrow$ Aut(F $'$ ) $,u: G'\rightarrow G$ , and $v: F\rightarrow F'$ be group homomorphisms such that

\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)

for $g\in G$ and $f\in F$ . We write $\tau ''=\tau '\circ u$ . Let $\mathscr{E}$ be a $\tau$ -extension of G by F. We denote by $\varphi_u($ resp. $\varphi_v,\varphi '_u,\varphi '_v)$ the canonical homomorphism corresponding to the $\tau \circ u$ -extension $u^*(\mathscr{E}) ($ resp. to the $\tau '$ -extension $v_*(\mathscr{E})$ , to the $\tau ''$ -extensions $u^*(v_*(\mathscr{E}))$ and $v_*(u^*(\mathscr{E})))$ . Then there exists a unique morphism $\psi$ of $\tau ''$ -extensions from $v_*(u^*(\mathscr{E}))$ to $u^*(v_*(\mathscr{E}))$ such that $\varphi_v\circ \varphi_u=$ $\varphi '_u\circ \psi \circ \varphi '_v$ .

We denote the $\tau \circ u$ -extension $u^*(\mathscr{E})$ (resp. the $\tau ''$ -extension $u^*(v_*(\mathscr{E}))$ ) by $(\Gamma_u, \iota_u, \pi_u)$ (resp. $(\Gamma '_u, \iota '_u, \pi '_u)$ ). Applying Lemma 2 of VIII, p. 287 to $\varphi_v\circ \varphi_u$ , we find that there exists a group homomorphism $\psi_1: \Gamma_u\rightarrow \Gamma '_u$ such that the diagram

F $^{\iota_u}$ // $\Gamma_u^{\pi_u}$ // $G'$

v\psi_1

${F'}^{\iota'_u}$ // ${\Gamma'_u}^{\pi'_u}$ // $G'$

commutes and $\varphi '_u\circ \psi_1=\varphi_v\circ \varphi_u$ . The existence of $\psi$ follows from Proposition 2 applied to $\psi_1$ . Conversely, if $\psi '$ also has the desired properties, then we have $\psi '\circ \varphi '_v=\psi_1$ by Lemma 2, so $\psi '=\psi$ (Proposition 2).

4. Group Law on the Classes of $\tau$ -Extensions

Let G be a group, F be an abelian group, and $\tau : G\rightarrow$ Aut(F) be a group homomorphism. We denote by $\delta : G\rightarrow G\times G$ the diagonal mapping $g\mapsto$ $(g, g)$ and by $m: F\times F\rightarrow F$ the multiplication homomorphism $(f_1, f_2)\mapsto$ $f_1f_2$ . We denote by $s: F\rightarrow F$ the group homomorphism given by $f\mapsto f^{-1}$ . Let $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ be $\tau$ -extensions of G by F. Since we have the relation

m(((\tau \times \tau )\circ \delta )(g)\cdot (f_1, f_2)) =\tau (g)\cdot m(f_1, f_2)

for every $g\in G$ and all $f_1, f_2\in F$ , the extension $m_*(\delta^*(\mathscr{E}_1\times \mathscr{E}_2))$ is a $\tau$ -extension; we call it the product of the $\tau$ -extensions $\mathscr{E}_1$ and $\mathscr{E}_2$ and denote it by $\mathscr{E}_1\mathscr{E}_2$ . The class of this extension depends only on the classes of the extensions $\mathscr{E}_1$ and $\mathscr{E}_2$ (VIII, p. 288, Corollary 1 and VIII, p. 291, Corollary 1). So this gives a law of composition on Ex $_{\tau}(G,F)$ .

Remark 00IN

Let $\mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1)$ and $\mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2)$ be $\tau$ -extensions of G by F. Let $\mathscr{E}_1\mathscr{E}_2= (\Gamma , \iota , \pi )$ be the product of these extensions. Following the example of VIII, p. 289, the construction gives a surjective group homomorphism from the fiber product $\Gamma_1\times_G\Gamma_2$ to Γ whose kernel is the image of the group homomorphism $f\mapsto (\iota_1(f), \iota_2(f)^{-1})$ from F to $\Gamma_1\times_G\Gamma_2$ .

Proposition 4 00IO

The product of $\tau$ -extensions endows the set Ex $_{\tau}(G,F)$ with the structure of an abelian group. Its identity element is the class of the semitrivial extension $\mathscr{I}_{\tau}$ . The inverse of the class of a $\tau$ -extension $\mathscr{E}$ is the class of $s_*(\mathscr{E})$ .

The associativity of the law follows from the commutativity of the diagrams

\begin{array}{ccc} G & \xrightarrow{\delta} & G \times G \\ \downarrow \delta & & \downarrow \operatorname{Id} \times \delta \\ G \times G & \xrightarrow{\delta \times \operatorname{Id}} & G \times G \times G \end{array}

and

\begin{array}{ccc} F \times F \times F & \xrightarrow{m \times \operatorname{Id}} & F \times F \\ \downarrow \operatorname{Id} \times m & & \downarrow m \\ F \times F & \xrightarrow{m} & F \end{array}

and from Corollary 2 of VIII, p. 288; Corollary 2 of VIII, p. 291; and Proposition 3 of VIII, p. 292.

Let $\Delta : F \to F \times F$ be the diagonal mapping $f \mapsto (f, f)$ . Let $\mathscr{E} = (\Gamma, \iota, \pi)$ be a $\tau$ -extension. Let $\tilde{\Delta} : \Gamma \to \Gamma \times_G \Gamma$ be the group homomorphism given by $\gamma \mapsto (\gamma, \gamma)$ . The following diagram commutes:

\begin{array}{ccccc} F & \xrightarrow{\iota} & \Gamma & \xrightarrow{\pi} & G \\ \downarrow \Delta & & \downarrow \tilde{\Delta} & & \parallel \\ F \times F & \longrightarrow & \Gamma \times_G \Gamma & \longrightarrow & G. \end{array}

By Proposition 2 of VIII, p. 290, it follows that the $(\tau \times \tau) \circ \delta$ -extension $\delta^*(\mathscr{E} \times \mathscr{E})$ is isomorphic to $\Delta_*(\mathscr{E})$ .

Denote by $c : F \to F$ the constant homomorphism $f \mapsto 1$ . by Example 1 of VIII, p. 292, the fact that $\mathscr{I}_\tau$ is an identity element for this law of composition follows from the isomorphism from $\delta^*(\mathscr{E} \times \mathscr{E})$ to $\Delta_*(\mathscr{E})$ and the commutative diagram

\begin{array}{ccc} F & \xrightarrow{(\operatorname{Id}_F \times c) \circ \Delta} & F \times F \\ (c \times \operatorname{Id}_F) \circ \Delta & & \downarrow \operatorname{Id}_F \\ & & \downarrow m \\ F \times F & \xrightarrow{m} & F. \end{array}

The last assertion follows from the commutative diagram

\begin{array}{ccc} F & \xrightarrow{(\operatorname{Id}_F \times s) \circ \Delta} & F \times F \\ (s \times \operatorname{Id}_F) \circ \Delta & & \downarrow c \\ & & \downarrow m \\ F \times F & \xrightarrow{m} & F. \end{array}

Let $\mathscr{E}_1 = (\Gamma_1, \iota_1, \pi_1)$ and $\mathscr{E}_2 = (\Gamma_2, \iota_2, \pi_2)$ be $\tau$ -extensions. The group isomorphism $\Gamma_1 \times \Gamma_2 \to \Gamma_2 \times \Gamma_1$ given by $(\gamma_1, \gamma_2) \mapsto (\gamma_2, \gamma_1)$ restricts to a group isomorphism $\sigma : \Gamma_1 \times_G \Gamma_2 \to \Gamma_2 \times_G \Gamma_1$ . Because of the relations

\sigma(\iota_1(f), \iota_2(f)^{-1}) = (\iota_2(f^{-1}), \iota_1(f^{-1})^{-1})

for $f\in F$ , the group homomorphism $\sigma$ induces, by passing to the quotients, a morphism of $\tau$ -extensions from $\mathscr{E}_1\mathscr{E}_2$ to $\mathscr{E}_2\mathscr{E}_1$ . So the law of composition is commutative.

Proposition 5 00IP

a) Let G and $G'$ be groups, and let F be an abelian group. Let $\tau : G\rightarrow$ Aut(F) and $u: G'\rightarrow G$ be group homomorphisms. The mapping $u^*:$ Ex $_{\tau}(G,F)\rightarrow$ Ex $_{\tau\circ u}(G',F)$ is a group homomorphism.

b) Let G be a group, and let F and $F'$ be abelian groups. Let $\tau : G\rightarrow$ Aut(F) $,\tau ': G\rightarrow$ Aut(F $'$ ), and $v: F\rightarrow F'$ be group homomorphisms such that

\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)

for $g\in G$ and $f\in F$ . The mapping $v_*:$ Ex $_{\tau}(G,F)\rightarrow$ Ex $_{\tau'}(G,F')$ is a group homomorphism.

This follows from the commutativity of the diagrams

${G'}^{\delta}$ // $G'\times G'F\times F^m$ // F

$_uu_{\times u}$ and $v_{\times v}v$

G $^{\delta}$ // $G\times GF'\times {F'}^m$ // $F'$ .

5. Cohomological Description

Let G be a group, let F be an abelian group, and let $\tau : G\rightarrow$ Aut(F) be a group homomorphism. For any $g\in G$ and $f\in F$ , we also write $^gf$ for $\tau (g)\cdot f$ . A 2-cocycle of G with values in F is a mapping $c: G\times G\rightarrow F$ such that for every $(g_1, g_2, g_3)\in G\times G\times G$ , we have

g_{_1}c(g_2, g_3)c(g_1, g_2g_3) =c(g_1, g_2)c(g_1g_2, g_3) \tag{7}

Since F is commutative, the set of 2-cocycles is a subgroup of the group of mappings from $G\times G$ to F; we denote it by $Z^2(G,F)$ . We denote the group of mappings from G to F by $C^1(G,F)$ . For every $h\in C^1(G,F)$ and $(g_1, g_2)\in G\times G$ , set

(\partial h)(g_1, g_2) =^{g_1}h(g_2)h(g_1g_2)^{-1}h(g_1) \tag{8}

A simple calculation shows that the mapping $\partial h: G\times G\rightarrow F$ is a 2-cocycle. The mapping $\partial : C^1(G,F)\rightarrow Z^2(G,F)$ is a group homomorphism. For any $h\in C^1(G,F)$ , the 2-cocycle $\partial h$ is called a 2-coboundary. The group $Z^2(G,F)/\partial (C^1(G,F))$ is denoted by $H^2(G,F)$ and is called the second cohomology group of G with coefficients in F.

$*$ The notation above agrees with that of X, §6, n $^o8$ , p. 112 concerning group cohomology. $*$

Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension. Let $\sigma$ be a section of the surjective mapping $\pi ($ Set Theory, II, §3, No. 8, p. 86), that is, a mapping from G to Γ such that $\pi (\sigma (g)) =g$ for every $g$ in G. For every $g_1, g_2\in G$ , $\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1}$ belongs to Ker( $\pi$ ), so that there exists a unique mapping $c_{\sigma}: G\times G\rightarrow F$ such that

\iota (c_{\sigma}(g_1, g_2)) =\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1} \tag{9}

for all $g_1, g_2\in G$ . The mapping $c_{\sigma}$ is constant with value 1 if and only if $\sigma$ is a group homomorphism.

Proposition 6 00IQ

The mapping $c_{\sigma}$ is an element of the group $Z^2(G,F)$ , and its class in the cohomology group $H^2(G,F)$ depends only on the class of the $\tau$ -extension $\mathscr{E}$ .

We say that the mapping $c_{\sigma}$ is the 2-cocycle associated with $\sigma$ and that its cohomology class in $H^2(G,F)$ is the cohomology class of the $\tau$ -extension $\mathscr{E}$ .

Let us first verify that $c_{\sigma}$ satisfies the cocycle condition. Let $g_1,g_2$ , and $g_3$ be elements of G. Using formula (1) of VIII, p. 285 and (9), we obtain the relations

\iota (^{g_1}c_{\sigma}(g_2, g_3)c_{\sigma}(g_1, g_2g_3))

=\sigma (g_1)\sigma (g_2)\sigma (g_3)\sigma (g_2g_3)^{-1}\sigma (g_1)^{-1}\sigma (g_1)\sigma (g_2g_3)\sigma (g_1g_2g_3)^{-1}

=\sigma (g_1)\sigma (g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}

and

\iota (c_{\sigma}(g_1, g_2)c_{\sigma}(g_1g_2, g_3))

=\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1}\sigma (g_1g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}

=\sigma (g_1)\sigma (g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}

The mapping $c_{\sigma}$ is therefore an element of $Z^2(G,F)$ .

Let us now prove that the image of $c_{\sigma}$ in $H^2(G,F)$ is independent of the chosen section $\sigma$ . Let $\sigma$ and $\sigma '$ be such sections. For every element $g$ of G, there exists a unique element $a_g$ of F such that $\sigma '(g) =\iota (a_g)\sigma (g)$ . Let $g_1$ and $g_2$ be elements of G. Using the definition (9), we obtain the equalities

\iota (c_{\sigma'}(g_1, g_2)) =\sigma '(g_1)\sigma '(g_2)\sigma '(g_1g_2)^{-1}

=\iota (a_{g_1})\sigma (g_1)\iota (a_{g_2})\sigma (g_2)\sigma (g_1g_2)^{-1}\iota (a_{g_1g_2})^{-1} \tag{10}

=\iota (a_{g_1})\iota (^{g_1}a_{g_2})\iota (c_{\sigma}(g_1, g_2))\iota (a_{g_1g_2})^{-1}

Since the group F is commutative, we have the relations

c_{\sigma'}(g_1, g_2)c_{\sigma}(g_1, g_2)^{-1}= (^{g_1}a_{g_2})a^{-1}_{g_1g_2}a_{g_1}= (\partial a)(g_1, g_2) \tag{11}

as follows from (8). This proves that the classes of $c_{\sigma'}$ and $c_{\sigma}$ in $H^2(G,F)$ coincide.

Finally, let $\mathscr{E}= (\Gamma , \iota , \pi )$ and $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ be isomorphic $\tau$ -extensions, let $u:\mathscr{E}\rightarrow \mathscr{E}'$ be a morphism of $\tau$ -extensions, and let $\sigma$ be a section of the mapping $\pi$ . The mapping $u\circ \sigma$ is a section of the mapping $\pi '$ , and we have $c_{\sigma}=c_{u\circ\sigma}$ . The class of $c_{\sigma}$ in $H^2(G,F)$ therefore depends only on the class of $\mathscr{E}$ in Ex $_{\tau}(G,F)$ .

We denote by $\Theta_{\tau}$ : Ex $_{\tau}(G,F)\rightarrow H^2(G,F)$ , or more simply Θ, the mapping that sends the class of a $\tau$ -extension $(\Gamma , \iota , \pi )$ to the class of the 2-cocycle $c_{\sigma}$ , where $\sigma$ is a section of the surjection $\pi$ .

Theorem 1 00IR

The mapping Θ is a group isomorphism from Ex $_{\tau}(G,F)$ to $H^2(G,F)$ .

Let us first prove that Θ is a group homomorphism. Let $\mathscr{E}= (\Gamma , \iota , \pi )$ and $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ be $\tau$ -extensions, and let $\sigma$ (resp. $\sigma '$ ) be a section of the mapping $\pi$ (resp. $\pi '$ ). We denote by $\mathscr{E} \mathscr{E}'= (\Gamma '', \iota '', \pi '')$ the product of the $\tau$ -extensions $\mathscr{E}$ and $\mathscr{E}'$ . We denote by $[\gamma , \gamma ']$ the image in $\Gamma ''$ of an element $(\gamma , \gamma ')$ of $\Gamma \times_G\Gamma '$ by the surjective homomorphism from the remark of VIII, p. 293. The mapping from G to $\Gamma ''$ that sends an element $g$ to $[\sigma (g), \sigma '(g)]$ is a section $\sigma ''$ of the mapping $\pi ''$ . Let $g_1$ and $g_2$ be elements of G. We have the relations

\iota ''(c_{\sigma''}(g_1, g_2)) =\sigma ''(g_1)\sigma ''(g_2)\sigma ''(g_1g_2)^{-1}

= [\iota (c_{\sigma}(g_1, g_2)), \iota '(c_{\sigma'}(g_1, g_2)]

=\iota ''(c_{\sigma}(g_1, g_2)c_{\sigma'}(g_1, g_2))

We therefore have $c_{\sigma''}(g_1, g_2) =c_{\sigma}(g_1, g_2)c_{\sigma'}(g_1, g_2)$ .

Let us prove that the mapping Θ is injective. Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension, and let $\sigma$ be a section of the mapping $\pi$ such that the image of $c_{\sigma}$ in $H^2(G,F)$ is trivial. There exists a mapping $a: G\rightarrow F$ such that

c_{\sigma}(g_1, g_2) = (^{g_1}a(g_2))a(g_1g_2)^{-1}a(g_1)

for all $g_1, g_2\in G$ . We then define $\sigma ': G\rightarrow \Gamma$ by

\sigma '(g) =\iota (a(g)^{-1})\sigma (g)

for $g\in G$ . Using (11), we obtain that $c_{\sigma'}$ is constant with value 1; consequently, $\sigma '$ is a group homomorphism, which proves that the $\tau$ -extension $\mathscr{E}$ is semitrivial (I, §6, No. 1, p. 67, Proposition 4).

Let us prove that the mapping Θ is surjective. Let $c$ be an element of $Z^2(G,F)$ . We endow the set $\Gamma = F\times G$ with the following law of composition: (12) $(f_1, g_1)(f_2, g_2) =(f_1(^{g_1}f_2)c(g_1, g_2), g_1g_2)$

for all $f_1, f_2\in F$ and $g_1, g_2\in G$ . We have the relations

(f_1, g_1) ((f_2, g_2)(f_3, g_3)) = (f_1, g_1)(f_2(^{g_2}f_3)c(g_2, g_3), g_2g_3)

=(f_1(^{g_1}f_2)(^{g_1g_2}f_3)(^{g_1}c(g_2, g_3))c(g_1, g_2g_3), g_1g_2g_3)

and

((f_1, g_1)(f_2, g_2)) (f_3, g_3) =(f_1(^{g_1}f_2)c(g_1, g_2), g_1g_2)(f_3, g_3)

=(f_1(^{g_1}f_2)(^{g_1g_2}f_3)c(g_1, g_2)c(g_1g_2, g_3), g_1g_2g_3)

for all $f_1, f_2, f_3\in F$ and $g_1,g_2, g_3\in G$ . It follows that this law is associative because $c$ is a 2-cocycle. For every $(f, g)\in \Gamma$ , we have

(f, g)(c(e, e)^{-1}, e) = (f(^gc(e, e)^{-1})c(g, e), g)

Now, it follows from the definition of a 2-cocycle that $c(g, e) =^gc(e, e)$ , from which we deduce that $(f, g)(c(e, e)^{-1}, e) = (f, g)$ . We prove likewise that

(c(e, e)^{-1}, e)(f, g) = (f c(e, e)^{-1}c(e, g), g) = (f, g)

The law of composition defined by (12) therefore has identity element $(c(e, e)^{-1}, e)$ , and every element $(f, g)$ of Γ is invertible, with inverse

(^{g^{-1}}(f^{-1}c(e, e)^{-1}c(g, g^{-1})^{-1}), g^{-1})

So we have endowed Γ with a group structure. Let us denote by $\iota : F\rightarrow G$ the mapping that sends $f$ to $(c(e, e)^{-1}f, e)$ , by $\pi : \Gamma \rightarrow G$ the second projection, and by $\sigma$ the mapping $g\mapsto (1, g)$ . The mappings $\iota$ and $\pi$ are group homomorphisms, so the triple $\mathscr{E}= (\Gamma , \iota , \pi )$ is a $\tau$ -extension, $\sigma$ is a section of the mapping $\pi$ , and the associated cocycle $c_{\sigma}$ is equal to $c$ because

\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1}= (1, g_1)(1, g_2)(1, g_1g_2)^{-1}= (c(g_1, g_2), g_1g_2)(1, g_1g_2)^{-1}

= (c(g_1, g_2)c(e, g_1g_2)^{-1}, e) = (c(e, e)^{-1}c(g_1, g_2), e)

for $g_1, g_2\in G$ .

Remark 00RK

v(\tau (g)\cdot f) =\tau '(g)\cdot v(f) \tag{13}

for $g\in G$ and $f\in F$ . Let $\alpha$ be an element of Ex $_{\tau}(G,F)$ . If the cocycle $c$ represents $\Theta (\alpha )$ , then $v\circ c$ represents $\Theta (v_*(\alpha ))\in H^2(G,F')$ .

6. Restriction and Corestriction

Let G and $G'$ be groups, F be an abelian group, and $\tau$ be a homomorphism from G to the automorphism group of F. Let $u: G'\rightarrow G$ be a group homomorphism. We write $\tau '=\tau \circ u$ .

If $\psi : G\times G\rightarrow F$ is a 2-cocycle of G with values in F, then the mapping $\psi \circ (u\times u)$ from $G'\times G'$ to F given by $(g_1, g_2)\mapsto \psi (u(g_1), u(g_2))$ is a 2-cocycle of $G'$ with values in F, and the mapping $\psi \mapsto \psi \circ (u\times u)$ from $Z^2(G,F)$ to $Z^2(G',F)$ induces a group homomorphism $u^*: H^2(G,F)\rightarrow H^2(G',F)$ . For any $\lambda \in H^2(G,F)$ , the element $u^*(\lambda )$ is called the inverse image of $\lambda$ by $u$ . When $G'$ is a subgroup of G and $u: G'\rightarrow G$ is the canonical injection, the homomorphism $u^*$ is called the restriction homomorphism from G to $G'$ ; we denote it by Res $^G_{G'}$ . When G is a quotient of $G'$ and $u: G'\rightarrow G$ is the canonical surjection, the homomorphism $u^*$ is called the inflation homomorphism from G to $G'$ .

Proposition 7 00IS

In the notation from above, the following diagram commutes:

Ex $_{\tau}(G,F)^{u^*}$ // Ex $_{\tau}(G',F)$

\Theta_{\tau}\Theta_{\tau'}

$H_2(G,F)^{u^*}$ // $H_2(G',F)$ .

Let H be a subgroup of G of finite index. Let $s$ be a section of the canonical surjection from G to $H\backslash G$ . We denote by $(g, x)\mapsto x\cdot g$ the right action of G on $H\backslash G$ induced by the right action of G on itself by multiplication. Note that for every $x\in H\backslash G$ and $g\in G$ , the element $s(x)gs(x\cdot g)^{-1}$ belongs to H. For any mapping $c: H\times H\rightarrow F$ , we can therefore define a mapping $\widetilde{c}_s: G\times G\rightarrow F$ by the relation

\widetilde{c}_s(g_1, g_2) =\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

for $g_1,g_2\in G$ . The mapping $c\mapsto \widetilde{c}_s$ is a group homomorphism from $F^{H\times H}$ to $F^{G\times G}$ .

Lemma 4 00IT

If $c: H\times H\rightarrow F$ is a 2-cocycle of H with values in F, then $\widetilde{c}_s$ is a 2-cocycle of G with values in F.

Let $g_1,g_2$ , and $g_3$ be elements of G. For every $i\in \{1,2,3\}$ , we define a mapping $h_i: H\backslash G\rightarrow H$ by the relation

h_i(x) =s(x\cdot g_1\cdots g_{i-1})g_is(x\cdot g_1\cdots g_i)^{-1}

for $x\in H\backslash G$ . Note that

h_1(x)h_2(x) =s(x)g_1g_2s(x\cdot g_1g_2)^{-1}

h_2(x)h_3(x) =s(x\cdot g_1)g_2g_3s(x\cdot g_1g_2g_3)^{-1}

for $x\in H\backslash G$ . We then have the relations

^{g_1}\widetilde{c}_s(g_2, g_3)\widetilde{c}_s(g_1g_2, g_3)^{-1}\widetilde{c}_s(g_1, g_2g_3)\widetilde{c}_s(g_1, g_2)^{-1}

=\prod_{x\in H\backslash G}^{g_1s(x)^{-1}}c(s(x)g_2s(x\cdot g_2)^{-1}, s(x\cdot g_2)g_3s(x\cdot g_2g_3)^{-1})

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(c(h_1(x)h_2(x), h_3(x))^{-1}

\times c(h_1(x), h_2(x)h_3(x))c(h_1(x), h_2(x))^{-1})

=\prod_{x\in H\backslash G}^{s(x\cdot g_1^{-1})^{-1}h_1(x\cdot g^{-1}_1)}c(h_2(x\cdot g_1^{-1}), h_3(x\cdot g_1^{-1}))

\times \prod_{x\in H\backslash G}^{s(x)^{-1}h_1(x)}c(h_2(x), h_3(x))^{-1}

= 1

where the second equality follows from the fact that $c$ is a 2-cocycle.

Lemma 5 00IU

If $c$ is a 2-coboundary, then so is $\widetilde{c}_s$ .

Let $t: H\rightarrow F$ be a mapping such that $c=\partial t$ . Let $\widetilde{t}_s: G\rightarrow F$ be the mapping defined by

\widetilde{t}_s(g) =\prod_{x\in H\backslash G}^{s(x)^{-1}}t(s(x)gs(x\cdot g)^{-1})

for $g\in G$ . It suffices to prove that $\widetilde{c}_s=\partial \widetilde{t}_s$ , which follows from the relations $\widetilde{c}_s(g_1, g_2) =\prod_{x\in H\backslash G}^{s(x)^{-1}s(x)g_1s(x\cdot g_1)^{-1}}t(s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})$

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(t(s(x)g_1g_2s(x\cdot g_1g_2)^{-1})^{-1}t(s(x)g_1s(x\cdot g_1)^{-1}))

=\partial \widetilde{t}_s(g_1, g_2)

for $g_1, g_2\in G$ .

Lemma 6 00IV

Let $c$ be a 2-cocycle of H with values in F. The image of $\widetilde{c}_s$ in the group $H^2(G,F)$ does not depend on the choice of the section $s$ .

Let $s'$ be a section of the canonical surjection from G to $H\backslash G$ . Let $h: H\backslash G\rightarrow H$ be the mapping characterized by the relation

s'(x) =h(x)s(x)

for $x\in H\backslash G$ . The 2-cocycle $\widetilde{c}_{s'}$ is then given by the relations

\widetilde{c}_{s'}(g_1, g_2)

=\prod_{x\in H\backslash G}^{s(x)^{-1}h(x)^{-1}}c(h(x)s(x)g_1s'(x\cdot g_1)^{-1}, s'(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})

=\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}h(x\cdot g_1)^{-1}, h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})

s(x)^{-1}(-1''-1-1

\times \prod_{x\in H\backslash G}c(h(x), s(x)g_1g_2s(x\cdot g_1g_2))

\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))

=\prod_{x\in H\backslash G}^{g_1s(x\cdot g_1)^{-1}}c(h(x\cdot g_1)^{-1}, h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})

s(x)^{-1}-1-1-1

\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(c(h(x)^{-1}, s'(x)g_1g_2s'(x\cdot g_1g_2)^{-1})^{-1}

\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))

=\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1})

s(x)^{-1}-1-1-1

\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))

\times \prod_{x\in H\backslash G}^{g_1s(x)^{-1}}c(h(x)^{-1}, h(x)s(x)g_2s'(x\cdot g_2)^{-1})

s(x)^{-1}(-1''-1-1

\times \prod_{x\in H\backslash G}c(h(x), s(x)g_1g_2s(x\cdot g_1g_2))

\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))

for all $g_1, g_2\in G$ . The first equality comes from the cocycle relation (VIII, p. 295, formula (7)) applied to the elements

$h(x)^{-1},h(x)s(x)g_1s'(x\cdot g_1)^{-1}$ , and $s'(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1}$ ;

the second is obtained by applying the cocycle relation to the elements

$s(x)g_1s(x\cdot g_1)^{-1},h(x\cdot g_1)^{-1}$ , and $h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1}$ ; and the last simply uses the fact that the mapping $x\mapsto x\cdot g_1$ is a permutation of $H\backslash G$ .

The last two lines of the obtained expression correspond to a 2-coboundary. We find that $\widetilde{c}_{s'}$ has the same class in $H^2(G,F)$ as the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}h(x\cdot g_1g_2)^{-1})

s(x)^{-1}-1-1-1

\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))

=\prod_{x\in H\backslash G}^{g_1s(x\cdot g_1)^{-1}}c(s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}, h(x\cdot g_1g_2)^{-1})^{-1}

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(c(s(x)g_1g_2s(x\cdot g_1g_2)^{-1}, h(x\cdot g_1g_2)^{-1})

-1-1-1)

\times c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))

=\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

gs(x)^{-1}-1-1-1

\times \prod_{x\in H\backslash G^1}c(s(x)g_2s(x\cdot g_2), h(x\cdot g_2))

\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(c(s(x)g_1g_2s(x\cdot g_1g_2)^{-1}, h(x\cdot g_1g_2)^{-1})

\times c(s(x)g_1s(x\cdot g_1)^{-1}, h(x\cdot g_1)^{-1})^{-1})

where the first equality follows from the cocycle relation applied to the elements

$s(x)g_1s(x\cdot g_1)^{-1},s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}$ , and $h(x\cdot g_1g_2)^{-1}$ .

The last two lines of the obtained expression correspond to a 2-coboundary. We find that the class of $\widetilde{c}_{s'}$ coincides with that of $\widetilde{c}_s$ .

We have constructed a homomorphism from the group $H^2(H,F)$ to the group $H^2(G,F)$ , which we call the corestriction homomorphism from H to G and denote by Cor $^G_H$ .

Proposition 8 00IW

Let H be a subgroup of G of finite index. The endomorphism Cor $^G_H\circ$ Res $^G_H$ of the group $H^2(G,F)$ coincides with multiplication by the index (G : H).

Let $\alpha$ be an element of $H^2(G,F)$ , and let $c$ be an element of $Z^2(G,F)$ representing $\alpha$ . The element Cor $^G_H\circ$ Res $^G_H(\alpha )$ is the class of the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

\prod_{x\in H\backslash G}^{s(x)^{-1}}c(s(x)g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

=\prod_{x\in H\backslash G}c(g_1s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

\times \prod_{x\in H\backslash G}(c(s(x)^{-1}, s(x)g_1g_2s(x\cdot g_1g_2)^{-1})^{-1}c(s(x)^{-1}, s(x)g_1s(x\cdot g_1)^{-1}))

=\prod_{x\in H\backslash G}^{g_1}c(s(x\cdot g_1)^{-1}, s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1})

\times \prod_{x\in H\backslash G}c(g_1, g_2s(x\cdot g_1g_2)^{-1})c(g_1, s(x\cdot g_1)^{-1})^{-1}

\times \prod_{x\in H\backslash G}(c(s(x)^{-1}, s(x)g_1g_2s(x\cdot g_1g_2)^{-1})^{-1}c(s(x)^{-1}, s(x)g_1s(x\cdot g_1)^{-1}))

=\prod_{x\in H\backslash G}c(g_1, g_2s(x\cdot g_1g_2)^{-1})c(g_1, s(x\cdot g_1)^{-1})^{-1}

\times \prod_{x\in H\backslash G}^{g_1}c(s(x)^{-1}, s(x)g_2s(xg_2)^{-1})

\times \prod_{x\in H\backslash G}(c(s(x)^{-1}, s(x)g_1g_2s(x\cdot g_1g_2)^{-1})^{-1}c(s(x)^{-1}, s(x)g_1s(x\cdot g_1)^{-1}))

The first equality comes from the cocycle relation (VIII, p. 295, formula (7)) applied to the elements

$s(x)^{-1},s(x)g_1s(x\cdot g_1)^{-1}$ , and $s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}$ ;

the second is obtained by applying the cocycle relation to the elements

$g_1,s(x\cdot g_1)^{-1}$ , and $s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1}$ .

By removing a coboundary, we obtain that Cor $^G_H\circ$ Res $^G_H(\alpha )$ is the class of the cocycle whose value in $(g_1, g_2)\in G^2$ is given by the expression

\prod_{x\in H\backslash G}c(g_1, g_2s(x\cdot g_1g_2)^{-1})c(g_1, s(x\cdot g_1)^{-1})^{-1}

=\prod_{x\in H\backslash G}^{g_1}c(g_2, s(x\cdot g_1g_2)^{-1})^{-1}c(g_1g_2, s(x\cdot g_1g_2)^{-1})c(g_1, g_2)

\times \prod_{x\in H\backslash G}c(g_1, s(x\cdot g_1)^{-1})^{-1}

=\prod_{x\in H\backslash G}^{g_1}c(g_2, s(x\cdot g_2)^{-1})^{-1}c(g_1g_2, s(x\cdot g_1g_2)^{-1})c(g_1, s(x\cdot g_1)^{-1})^{-1}

\times c(g_1, g_2)^{(G:H)}

which proves the result.

7. Galois Algebras

Let K be a commutative field. If E is a K-algebra, then we denote its automorphism group by Aut $_K(E)$ . If E is a Galois extension of the field K, then the group Aut $_K(E)$ is simply the Galois group Gal(E $/K$ ) (V, §10, No. 2, p. 58).

Let G be a group. $A (K,G)$ -algebra is a K-algebra E endowed with a group homomorphism $\lambda : G\rightarrow$ Aut $_K(E)$ . The homomorphism $\lambda$ then endows E with the structure of a group with operators in G as well as the structure of a left K[G]-module with external law given by

((\sum_{g\in G}\mu_gg)x=\sum_{g\in G}\mu_g\lambda (g).x \tag{14}

for every $x\in L$ and every element $(\mu_g)_{g\in G}$ of K[G]. A morphism of $(K,G)$ -algebras is a morphism of algebras that is also a morphism of groups with operators.

For every family $(E_i)_{i\in I}$ of $(K$ , G)-algebras, the product K-algebra $\prod_{i\in I}E_i$ endowed with its structure of a product group with operators is a $(K$ , G)-algebra.

If E is a $(K$ , G)-algebra, then the set $E^G$ of elements of E invariant under G is a subalgebra of E.

Let E be a $(K$ , G)-algebra, where G acts by $\lambda : G\rightarrow$ Aut $_K(E)$ . If $K'$ is an extension of K, then for any $g\in G$ , let $\lambda '(g)$ be the automorphism Id $_{K'}\otimes \lambda (g)$ of the $K'$ -algebra $L_{(K')}$ . Then $\lambda ':g\mapsto \lambda '(g)$ endows $L_{(K')}$ with the structure of a $(K'$ , G)-algebra.

Given a group H and H-sets X and Y, we denote by $\mathscr{F}_H(X,Y)$ the set of morphisms of H-sets from X to Y. It is the set of mappings $f: X\rightarrow Y$ such that $f(hx) =hf(x)$ for every $h\in H$ and $x\in X$ .

Let G be a group with identity element $e$ , let H be a subgroup of G, and let E be a $(K$ , H)-algebra. The $(K,G)$ -algebra deduced from E by coinduction from H to G, denoted by Coind $^G_H$ (E), is the K-algebra $\mathscr{F}_H(G,E)$ endowed with the action of G given by the homomorphism $\lambda$ from G to Aut $_K$ (Coind $^G_H(E)$ ) defined by

(\lambda (g)\cdot f)(g') =f(g'g) \tag{15}

for $f\in$ Coind $^G_H(E)$ and $g, g'\in G$ .

Lemma 7 00IX

a) Let S be a subset of G such that every element of G can be written uniquely as $hs$ with $h\in H$ and $s\in S$ . Then the mapping that sends $f$ to $(f(s))_{s\in S}$ is an isomorphism from the K-algebra Coind $^G_H(E)$ to the K-algebra $\mathscr{F}(S,E)$ of mappings from S to E.

b) The algebra Coind $^G_H(E)$ has finite degree over K if and only if E has finite degree over K and the index of H in G is finite. In this case, we have the formula

[Coind $^G_H(E) : K] = (G : H)[E : K]$ .

c) Let $E^H$ be the algebra of invariants of the group H in E and Coind $^G_H(E)^G$ that of the group G in Coind $^G_H(E)$ . The mapping $f\mapsto f(e)$ from Coind $^G_H(E)$ to E restricts to an algebra isomorphism from Coind $^G_H(E)^G$ to $E^H$ .

Assertion a) follows from the definitions and implies b). By (15), a mapping from G to E is an element of Coind $^G_H(E)$ invariant under G if and only if it is constant with value an element of E invariant under H.

Remark 1 00IY

Let G be a group, H a subgroup of G, and N a subgroup of H. Let E be a $(K$ , N)-algebra. Let $\alpha$ be an element of $\mathscr{F}_H(G,\mathscr{F}_N(H,E))$ . We have the relation

\alpha (g)(nh) =n(\alpha (g)(h)) \tag{16}

for every $g\in G,h\in H$ , and $n\in N$ and the relations

\alpha (hg)(h') = (h\alpha (g))(h') =\alpha (g)(h'h) \tag{17}

for all $h, h'\in H$ and every $g\in G$ . Consequently, $\alpha (ng)(e) =\alpha (g)(n) =$ $n(\alpha (g)(e))$ for every $g\in G$ and $n\in N$ . We can therefore consider the mapping

\psi :\mathscr{F}_H(G,\mathscr{F}_N(H,E))\rightarrow \mathscr{F}_N(G,E)

defined by the relation $\psi (\alpha )(g) =\alpha (g)(e)$ for $\alpha$ in $\mathscr{F}_H(G,\mathscr{F}_N(H,E))$ and $g$ in G. The mapping $\psi$ is an algebra isomorphism from Coind $^G_H$ (Coind $^H_N(E)$ ) to Coind $^G_N(E)$ whose inverse sends an element $\beta$ from $\mathscr{F}_N(G,E)$ to the mapping $\alpha$ defined by the relation $\alpha (g)(h) =\beta (hg)$ for $g\in G$ and $h\in H$ .

We now suppose given a finite group G and a reduced (V, §6, No. 6, p. 32) commutative K-algebra L of finite degree endowed with an action of G given by a homomorphism $\lambda$ from G to Aut $_K(L)$ . For $x\in L$ and $g\in G$ , we denote by $g\cdot x$ the transform of $x$ under the automorphism $\lambda (g)$ of L. Let $\mathscr{S}$ be the set of maximal ideals of L; we denote by $g\cdot \mathfrak{m}$ the transform of an element $\mathfrak{m}$ of $\mathscr{S}$ under the automorphism $\lambda (g)$ of L. It is an element of $\mathscr{S}$ . For every $\mathfrak{m}$ in $\mathscr{S}$ , the field $L/\mathfrak{m}$ is a finite extension of K. We write $\pi_{\mathfrak{m}}: L\rightarrow L/\mathfrak{m}$ for the projection and denote by $G_{\mathfrak{m}}$ the stabilizer of $\mathfrak{m}$ in G, that is, the set of $g\in G$ such that $g\cdot \mathfrak{m}=\mathfrak{m}$ . The K-algebra $L/\mathfrak{m}$ is endowed with an action of $G_{\mathfrak{m}}$ through the homomorphism $\lambda_{\mathfrak{m}}$ from $G_{\mathfrak{m}}$ to Aut $_K(L/\mathfrak{m})$ that sends an element $h$ of $G_{\mathfrak{m}}$ to the automorphism of $L/\mathfrak{m}$ deduced from $\lambda (h)$ when passing to the quotients.

Let $\mathscr{O}$ be the set of orbits of G in $\mathscr{S}$ . Given an orbit $\sigma \in \mathscr{O}$ , set $\mathfrak{a}_{\sigma}=\bigcap_{\mathfrak{m}\in\sigma}\mathfrak{m}$ and $L_{\sigma}= L/\mathfrak{a}_{\sigma}$ . Since $\mathfrak{a}_{\sigma}$ is invariant under G, by passing to the quotients, the action of G on L defines a homomorphism $\lambda_{\sigma}$ from G to Aut $_K(L_{\sigma})$ . Finally, denote by $\pi_{\sigma}$ the canonical mapping from L to $L_{\sigma}$ .

Lemma 8 00RL

a) For every $g\in G,\sigma \in \mathscr{O}$ , and $\mathfrak{m}\in \sigma$ , we have

$[L_{\sigma}: K] =$ Card( $\sigma$ ) $[L/\mathfrak{m}: K]$ .

b) The mapping $\pi :x\mapsto (\pi_{\sigma}(x))_{\sigma\in\mathscr{O}}$ is an isomorphism of $(K,G)$ -algebras from L to $\prod_{\sigma\in\mathscr{O}}L_{\sigma}$ .

c) Denote by $L^G($ resp. $L^G_{\sigma})$ the subalgebra of L (resp. $L_{\sigma}$ ) of elements invariant under the action of G. Then $\pi$ induces an isomorphism from $L^G$ to $\prod_{\sigma\in\mathscr{O}}L^G_{\sigma}$ .

Since the algebra L is reduced and of finite degree, the intersection of the maximal ideals of L is reduced to 0 (VIII, p. 173, Corollary 2). Moreover, if $\mathfrak{m}$ and $\mathfrak{m}'$ are two distinct maximal ideals of L, then we have $\mathfrak{m}+\mathfrak{m}'= L$ . By Proposition 10 of I, §8, No. 11, p. 110, the canonical mapping from L to $\prod_{\mathfrak{m}\in\mathscr{S}}L/\mathfrak{m}$ is an isomorphism, as is the canonical mapping from $L/\mathfrak{a}_{\sigma}$ to $\prod_{\mathfrak{m}\in\sigma}L/\mathfrak{m}$ for every $\sigma \in \mathscr{O}$ . Assertion a) follows. Since $\mathscr{O}$ is a partition of $\mathscr{S}$ , assertion b) follows; assertion c) is an immediate consequence of b).

Let us now fix an orbit $\sigma \in \mathscr{O}$ and an element $\mathfrak{m}$ of $\sigma$ . Set $F_{\mathfrak{m}}$ = Coind $^G_{G_{\mathfrak{m}}}(L/\mathfrak{m})$ , and denote by $\lambda_{F_{\mathfrak{m}}}$ the action of G on $F_{\mathfrak{m}}$ . For any $x\in L$ , denote by $\overline{x}$ the mapping from G to $L/\mathfrak{m}$ that sends $g$ to $\pi_{\mathfrak{m}}(gx)$ . By formula (15) of VIII, p. 305 and the definition of the action of $G_{\mathfrak{m}}$ on $L/\mathfrak{m}$ , it is immediate that $\overline{x}$ belongs to $F_{\mathfrak{m}}$ , and the mapping $u:x\mapsto \overline{x}$ from L to $F_{\mathfrak{m}}$ satisfies $\lambda_{F_{\mathfrak{m}}}(g)\circ u=u\circ \lambda (g)$ for $g\in G$ . In other words, $u$ is a morphism of $(K$ , G)-algebras.

Lemma 9 00IZ

The morphism $u$ is surjective with kernel $\mathfrak{a}_{\sigma}$ . The mapping deduced from $u$ by passing to the quotient is an isomorphism of $(K,G)$ -algebras from $L_{\sigma}$ to $F_{\mathfrak{m}}$ .

Since the kernel of $\pi_{\mathfrak{m}}$ is equal to $\mathfrak{m}$ , that of $u$ is equal to $\bigcap_{g\in G}g^{-1}.\mathfrak{m}=\mathfrak{a}_{\sigma}$ . To prove that $u$ is surjective, it suffices to prove that the vector spaces $L_{\sigma}=$ $L/\mathfrak{a}_{\sigma}$ and $F_{\mathfrak{m}}$ have the same dimension over K. Now, all ideals $g\cdot \mathfrak{m}$ have the same codimension in L, and, by Lemma 8, a),

$[L_{\sigma}: K] =$ Card( $\sigma$ ) $.[L/\mathfrak{m}: K]$ .

Moreover, by Lemma 7, b), we have

[F_{\mathfrak{m}}: K] = (G : G_{\mathfrak{m}})[L/\mathfrak{m}: K]

Since we have Card( $\sigma$ ) $= (G : G_{\mathfrak{m}})$ , we have proved the equality $[L_{\sigma}: K] =$ $[F_{\mathfrak{m}}: K]$ .

We now suppose, furthermore, that the homomorphism $\lambda$ from G to Aut $_K(L)$ is injective. We identify K with its image in L through the mapping $\xi \mapsto \xi \cdot 1$ . Let Ω be an algebraically closed extension of K. The set $\mathscr{F}(G,\Omega )$ of mappings from G to Ω coincides with the coinduced $(\Omega$ , G)-algebra Coind $^G_{\{e\}}(\Omega )$ . It is a free Ω[G]-module of rank 1. Let $\mathscr{H}$ be the set of K-algebra homomorphisms from L to Ω. We define a right action of G on $\mathscr{H}$ by $(g, \chi )\mapsto \chi \circ \lambda (g)$ . The Ω-algebra $\mathscr{F}(\mathscr{H},\Omega )$ is then endowed with the structure of an $(\Omega$ , G)-algebra deduced from the right action of G on $\mathscr{H}$ .

Lemma 10 00J0

Let L be a $(K,G)$ -algebra that is étale over K. The mapping $\psi$ from $L_{(\Omega )}$ to $\mathscr{F}(\mathscr{H},\Omega )$ characterized by the relation

\psi (\xi \otimes x) = (\xi \chi (x))_{\chi\in\mathscr{H}}

is an isomorphism of $(K,G)$ -algebras.

Since L is étale, the mapping $\psi$ is an isomorphism of Ω-algebras (V, §6, No. 3, p. 30, Proposition 2 and V, §6, No. 3, p. 29, Proposition 1, c)). We have the relations

$\psi$ ((Id $\otimes \lambda (g)$ ) $(\xi \otimes x)$ ) $= (\xi (\chi \circ \lambda (g))(x))_{\chi\in\mathscr{H}}$

for $\xi \in \Omega ,x\in L$ , and $g\in G$ . So $\psi$ is a morphism of $(\Omega$ , G)-algebras.

Theorem 2 00J1

Let G be a finite group, and let L be a commutative K-algebra of finite degree endowed with an action of G given by an injective homomorphism $\lambda$ from G to Aut $_K(L)$ . The following properties are then equivalent:

(i) There exist a subgroup H of G, a Galois extension E of K of finite degree, an isomorphism from H to Gal(E $/K$ ), and an isomorphism of $(K,G)$ -algebras from L to Coind $^G_H(E)$ .

(ii) The algebra L is étale, and $\mathscr{H}$ is a homogeneous principal G-set (I, §5, No. 6, p. 60, Definition 7).

(iii) There exists an isomorphism of $(\Omega ,G)$ -algebras $\psi : L_{(\Omega )}\rightarrow \mathscr{F}(G,\Omega )$ ; in other words, for every $g\in G$ , the automorphism $\psi \circ \lambda (g)_{(\Omega )}\circ \psi^{-1}$ of $\Omega^G$ is equal to the automorphism

(x_h)_{h\in G}\longmapsto (x_{hg})_{h\in G}

(iv) The algebra L is reduced, and L is a free K[G]-module of rank 1.

(v) The algebra L is reduced, we have Card(G) = [L : K], and K is the subring of L of elements invariant under the action of G.

(vi) The algebra L is reduced, the group G acts transitively on the set of maximal ideals of L, and, for every maximal ideal $\mathfrak{m}$ of L, the stabilizer $G_{\mathfrak{m}}$ of $\mathfrak{m}$ in G acts faithfully on $L/\mathfrak{m}$ and admits K as subfield of invariants.

(i) $\Rightarrow$ (ii): Let E be a Galois extension of K of finite degree and $\tau$ an isomorphism from H to Aut $_K(E)$ . Let S be a system of representatives of the right cosets of G modulo H. The K-algebra F = Coind $^G_H(E)$ is isomorphic to $\mathscr{F}(S,E)$ (Lemma 7, a)); it is therefore étale. We denote by $\lambda_F$ the action of G on F. Furthermore, let $\psi$ be a K-algebra homomorphism from E to Ω, and let $\chi_0$ be the homomorphism $f\mapsto \psi (f(e))$ from F to Ω. Let $g\in G$ be such that we have $\chi_0\circ \lambda_F(g) =\chi_0$ ; since $\psi$ is injective, we then have $f(g) =f(e)$ for every $f\in F$ . In view of Lemma 7, a), we have $g\in H$ , and by the formula

f(h) =\tau (h)\cdot f(e) \tag{18}

which holds for every $h\in H$ , this can only happen if $g=e$ . On the other hand, by Lemma 7, b) and Theorem 3 of V, §10, No. 6, p. 66, we have

[F : K] = (G : H)[E : K] = (G : H) Card(H) = Card(G).

The set $\mathscr{K}$ of K-homomorphisms from F to Ω has cardinal [F : K] because F is étale (V, §6, No. 5, p. 32, Proposition 4), so Card( $\mathscr{K}$ ) $=$ Card(G). Since the stabilizer of $\chi_0$ in G is equal to $\{e\}$ by the above, $\mathscr{K}$ is a homogenous principal G-set.

(ii) $\Rightarrow$ (iii): Suppose that L is étale and that $\mathscr{H}$ is a homogenous principal G-set. By Lemma 10, the $(\Omega$ , G)-algebras $L_{(\Omega )}$ and $\mathscr{F}(\mathscr{H},\Omega )$ are isomorphic. Since $\mathscr{H}$ is a homogenous principal G-set, the $(\Omega$ , G)-algebras $\mathscr{F}(\mathscr{H},\Omega )$ and $\mathscr{F}(G,\Omega )$ are isomorphic.

(iii) $\Rightarrow$ (iv): Suppose that property (iii) holds. Then $L_{(\Omega )}$ is a free module of rank 1 over the algebra Ω[G]; the latter can be canonically identified with $K[G]_{(\Omega )}$ . We then apply Theorem 3 of VIII, p. 37.

The implication (iv) $\Rightarrow$ (v) is immediate.

Let us prove the implication (v) $\Rightarrow$ (vi). The algebra L is reduced. By Lemma 8, c), the group G acts transitively on the set $\mathscr{S}$ of maximal ideals of L. Let $\mathfrak{m}$ be an element of $\mathscr{S}$ . By Lemma 9, since $\bigcap_{\mathfrak{n}\in\mathscr{S}}\mathfrak{n}=\{0\}$ , the algebra L is isomorphic to the algebra Coind $^G_{G_{\mathfrak{m}}}(L/\mathfrak{m})$ . The algebra of invariants of $G_{\mathfrak{m}}$ in $L/\mathfrak{m}$ therefore coincides with K by Lemma 7, c). Hence the homomorphism $\lambda_{\mathfrak{m}}$ from $G_{\mathfrak{m}}$ to Gal((L $/\mathfrak{m}$ ) $/K$ ) is surjective. By Lemma 7, we moreover have

Card(G) $= [L : K] = (G : G_{\mathfrak{m}})[L/\mathfrak{m}: K]$ . So Card(G $_{\mathfrak{m}}$ ) $= [L/\mathfrak{m}: K]$ , and the homomorphism $\lambda_{\mathfrak{m}}$ is injective.

It remains to prove the implication (vi) $\Rightarrow$ (i). Let $\mathfrak{m}$ be a maximal ideal of L. By Lemma 9, the algebra L is isomorphic to the algebra Coind $^G_{G_{\mathfrak{m}}}(L/\mathfrak{m})$ as a $(K$ , G)-algebra. Since $G_{\mathfrak{m}}$ acts faithfully in $L/\mathfrak{m}$ and admits K as subfield of invariants, the group homomorphism $\lambda_{\mathfrak{m}}$ defines an isomorphism from $G_{\mathfrak{m}}$ to Gal((L $/\mathfrak{m}$ ) $/K$ ).

Remark 2 00J2

In Theorem 2, we can replace the assumption that Ω is algebraically closed with the assumption that Ω is separably closed. Indeed, if L is étale, then the image of every K-algebra homomorphism from L to Ω is a separable extension of K.

Remark 3 00J3

The normal basis theorem (V, §10, No. 9, p. 73, Theorem 6) is a specific case of the implication (i) $\Rightarrow$ (iv) in Theorem 2.

Definition 1 00J4

Let G be a finite group, and let L be a nonzero commutative algebra of finite degree over K endowed with the structure of a $(K,G)$ -algebra, where the action of G is given by an injective homomorphism from G to the group Aut $_K(L)$ . We say that L is a Galois algebra with group G if it has the equivalent properties (i) through (vi) of Theorem 2.

Remark 4 00J5

Suppose that L is an extension of K endowed with an action $\lambda$ of G. Then L is a Galois algebra over K if and only if L is a Galois extension of K and $\lambda$ is an isomorphism from G to the Galois group of L over K.

Remark 5 00J6

Suppose that the group G is abelian. If G acts faithfully and transitively on a set X, then the stabilizer of any point of X is reduced to the identity (because the stabilizers of the points of X are all equal and their intersection is reduced to the identity element of G). Consequently, properties (i) through (vi) of Theorem 2 are also equivalent to the following:

(vii) The algebra L is étale, and G acts faithfully and transitively on $\mathscr{H}$ .

Remark 6 00J7

By V, §10, No. 1, p. 75, Lemma 5; V, §10, No. 1, p. 76, Proposition 12; and VIII, p. 137, Proposition 3, a Galois algebra is commutative and semisimple.

Example 1 00J8

Let $n$ be a strictly positive integer, prime to the characteristic exponent of K. Suppose that the group $\mu_n$ of $n$ -th roots of unity in K has order $n$ . Then for every divisor $d$ of $n$ , the group $\mu_d$ of $d$ -th roots of unity has order $d$ . Let $a$ be a nonzero element of K, let L be the algebra $K[X]/(X^n-a)$ , and let $x$ be the class of X in L. The sequence $(1, x, . . . , x^{n-1})$ is a basis of the vector space L over the field K, and we have $x^n=a$ . Moreover, the polynomial $X^n-a$ is prime to its derivative $nX^{n-1}$ , so the algebra L is étale (V, §7, No. 2, p. 37, Proposition 3). For every $\zeta$ in $\mu_n$ , the automorphism $P(X)\mapsto P(\zeta X)$ of the ring K[X] defines, by passing to the quotients, an endomorphism $\lambda (\zeta )$ of L because we have $(\zeta X)^n-a= X^n-a$ ; it is an automorphism. We have

\lambda (\zeta )x^i=\zeta^ix^i \tag{19}

for $0\leqslant i < n$ . The mapping $\lambda :\zeta \mapsto \lambda (\zeta )$ is an injective homomorphism from $\mu_n$ to Aut $_K$ (L), and the ring of invariants of the group $\lambda (\mu_n)$ in L is equal to $K\cdot 1$ . Since the cardinal of $\mu_n$ is equal to $n= [L : K]$ , the algebra L endowed with the action of $\lambda$ is a Galois algebra (VIII, p. 308, Theorem 2, (v)).

Let $r$ be the least strictly positive integer such that $a^r$ belongs to $K^{*n}$ ; it divides $n$ , and there exists an element $b$ of $K^*$ such that $a=b^{n/r}$ . Then (V, §11, No. 8, p. 91, Remark) the polynomial $X^r-b$ is irreducible, and we have $X^n-a=\prod_{\zeta\in\mu_{n/r}}(X^r-\zeta b)$ . Let E be the field $K[Y]/(Y^r-b)$ , and let $y$ be the class of Y in E. There exists an isomorphism $\theta$ from $\mu_{n/r}$ to Gal(E $/K$ ), characterized by the relation $\theta (\xi )(y) =\xi y$ (V, §11, No. 8, p. 91, Example 3). We then verify that the Galois algebra L is isomorphic to the $(K, \mu_n$ )-algebra Coind $^{\mu_n}_{\mu_{n/r}}(E)$ .

Example 2 00J9

Now suppose that the field K has characteristic $p\not= 0$ . Let $c$ be an element of K. The polynomial $f= X^p-X-c$ is prime to its derivative $f'=-1$ , so the algebra $L = K[X]/(f)$ is étale (V, §7, No. 2, p. 37, Proposition 3). We denote the image of X in L by $x$ ; we have the relation $x^p=x+c$ . The sequence $(1, x, . . . , x^{p-1})$ is a basis of L viewed as a vector space over K.

Let P be the additive group of the prime subfield of K; it is a cyclic group of order $p$ , generated by the unit element 1 of K. For every $j$ in P, we have $j^p=j$ (V, §1, No. 3, p. 4, formula (4)) and therefore $f(X +j) =f(X)$ . Hence there exists an automorphism $\gamma (j)$ of the algebra L characterized by the relation $\gamma (j)(x) =x+j$ ; moreover, the resulting mapping $\gamma$ is an injective homomorphism from P to Aut $_K(L)$ .

Let Ω be an algebraically closed extension of K, and let $\xi$ be a root of the polynomial $f$ in Ω. We have $\xi^p=\xi +c$ , hence

X^p-X-c= (X^p-\xi^p)-(X-\xi ) = (X-\xi )^p-(X-\xi ) =\prod_{j\in P}(X-\xi -j)

by V, §12, No. 1, p. 94, formula (1). For every $j$ in P, there exists a unique algebra homomorphism $\chi_j: L\rightarrow \Omega$ that sends $x$ to $\xi +j$ ; moreover, every homomorphism from L to Ω is one of the $\chi_j$ , and we have the relation $\chi_j=\chi_0\circ \gamma (j)$ . The algebra L endowed with $\gamma$ has property (ii) of Theorem 2 of VIII, p. 308 and is therefore a Galois algebra over K.

To describe the structure of L, we must distinguish between two cases:

a) We have $\xi \notin K$ . Then the polynomial $f(X)$ is irreducible in K[X] (V, §11, No. 9, p. 93, Example 3). In this case, L is a cyclic extension of K of degree $p$ , and $\gamma$ is an isomorphism from P to Gal(L $/K$ ).

b) We have $\xi \in K$ . Then the mapping $\psi :y\mapsto (\chi_j(y))_{j\in P}$ is an isomorphism from the algebra L to the product algebra $K^P$ ; moreover, $\psi \circ \gamma (k)\circ \psi^{-1}$ is the automorphism $(x_j)_{j\in P}\mapsto (x_{j+k})_{j\in P}$ of $K^P$ for every $k\in P$ .

8. Actions on Galois Algebras

Proposition 9 00JA

Let G be a finite group, H be a subgroup of G, and E be a Galois algebra over the field K with group H. Then the $(K,G)$ -algebra Coind $^G_H(E)$ deduced from E by coinduction from H to G is a Galois algebra over the field K with group G.

Since E is a Galois algebra over K, by property (v) of Theorem 2 of VIII, p. 308, it is reduced, we have Card(H) = [E : K], and K is the ring of invariants of H in E. But by Lemma 7 of VIII, p. 305, the algebra F = Coind $^G_H(E)$ is reduced, and we have

[F : K] = (G : H)[E : K] = (G : H) Card(H) = Card(G),

and K is the ring of invariants of G in F. So F is a Galois algebra by the criterion given in Theorem 2, (v).

Proposition 10 00JB

Let G be a finite group. Let L be a Galois algebra over the field K with group G, and let $K'$ be an extension of K. Then the $(K',G)$ -algebra $L_{(K')}$ is a Galois algebra over $K'$ .

We use property (v) in Theorem 2 by observing that if the K-algebra L is étale, then the $K'$ -algebra $L_{(K')}$ is too (V, §6, No. 5, p. 32, Corollary 2), that we have the equality $[L_{(K')}: K'] = [L : K]$ , and that the ring of invariants of G in $L_{(K')}$ is $(L^G)_{(K')}$ , where $L^G$ is the ring of invariants of G in L.

Proposition 11 00JC

Let $G_1$ and $G_2$ be groups. Let $L_1$ and $L_2$ be Galois algebras over K with respective actions $\lambda_1: G_1\rightarrow$ Aut $_K(L_1)$ and $\lambda_2: G_2\rightarrow$ Aut $_K(L_2)$ . Set $L = L_1\otimes_KL_2,G = G_1\times G_2$ , and $\lambda (g_1, g_2) =\lambda_1(g_1)\otimes \lambda_2(g_2)$ for $(g_1, g_2)\in G$ . Then the K-algebra L endowed with the action $\lambda$ is a Galois algebra over K.

We reason as before, taking the following into account: If $L_1$ and $L_2$ are étale, then so is the algebra $L = L_1\otimes_KL_2$ (V, §6, No. 5, p. 32, Corollary 1), and we have the equalities

$[L : K] = [L_1: K][L_2: K]$ and Card(G) = Card(G $_1$ ) Card(G $_2$ ).

Moreover, if $L^{G_i}_i$ denotes the ring of invariants of $G_i$ in $L_i$ , then it follows from Lemma 11 below that $L^{G_1}_1\otimes_KL^{G_2}_2$ is the ring of invariants of $G_1\times G_2$ in $L_1\otimes_KL_2$ .

Lemma 11 00JD

Let $G_1$ and $G_2$ be groups, and let $W_1$ and $W_2$ be K-vector spaces. We endow $W_1($ resp. $W_2)$ with an action of $G_1($ resp. $G_2)$ given by a group homomorphism $\rho_1: G_1\rightarrow$ Aut $_K(W_1) ($ resp. $\rho_2: G_2\rightarrow$ Aut $_K(W_2))$ . We consider the group homomorphism

$\rho_1\otimes \rho_2: G_1\times G_2\longrightarrow$ Aut $_K(W_1\otimes_KW_2)$

defined by the relation $(\rho_1\otimes \rho_2)(g_1, g_2)(w_1\otimes w_2) =\rho_1(g_1)(w_1)\otimes \rho_2(g_2)(w_2)$ for $g_1\in G_1,g_2\in G_2,w_1\in W_1$ , and $w_2\in W_2$ . Then the linear mapping from $W^{G_1}_1\otimes_KW^{G_2}_2$ to $W_1\otimes_KW_2$ given by the tensor product of the canonical injections induces an isomorphism of K-vector spaces from $W_1^{G_1}\otimes_KW^{G_2}_2$ to $(W_1\otimes_KW_2)^{G_1\times G_2}$ .

This follows from Lemma 1 of VIII, p. 213 applied to the K[G]-modules $M_1= M_2= K$ endowed with the trivial action of $G, N_1= W_1$ , and $N_2= W_2$ .

Remark 00JE

Let L be a Galois extension of the field K of finite degree, and let G be its Galois group; we denote the identity mapping on G by $\lambda$ . Then L endowed with $\lambda$ is a Galois algebra over K with group G. Let $K'$ be an extension of K. By Proposition 10, the algebra $L_{(K')}$ is Galois over $K'$ , but it is not in general an extension of $K'$ . Analogously, by Proposition 11, the tensor product of Galois extensions E and F of K of finite degree can be viewed as a Galois algebra; in general, it is not a Galois extension of K. It is one, however, if E and F are moreover linearly disjoint subextensions of an extension of K (V, §2, No. 5, p. 13 and V, §10, No. 1, p. 57, Proposition 1).

9. Cross Products

Let K be a field, and let G be a group whose identity element we denote by $e$ . Let L be a commutative K-algebra, and let $\lambda$ be a homomorphism from G to the automorphism group of the K-algebra L. For any $g$ in G, let $\tau (g)$ be the automorphism of the multiplicative group $L^*$ of L induced by $\lambda (g)$ .

Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension of the group G by $L^*$ . Define a right action of the group $L^*$ on the set $L\times \Gamma$ by

(\beta , \gamma ).\alpha = (\beta \alpha , \iota (\alpha )^{-1}\gamma ) \tag{20}

for $\alpha \in L^*,\beta \in L$ , and $\gamma \in \Gamma$ . We denote by E the set of orbits of $L^*$ in $L\times \Gamma$ and by $[\beta ;\gamma ]$ the orbit of the pair $(\beta , \gamma )$ . We therefore have, by construction, the relation

[\beta \alpha ;\gamma ] = [\beta ;\iota (\alpha )\gamma ] \tag{21}

for $\alpha \in L^*,\beta \in L$ , and $\gamma \in \Gamma$ .

Given $\beta$ in L and $\gamma$ in Γ, we denote by $^{\gamma}\beta$ the transform of $\beta$ under the automorphism $\lambda \circ \pi (\gamma )$ of L; we have the relations

(22) $^{\gamma}(^{\gamma'}\beta ) =^{\gamma \gamma'}\beta ,^{\gamma}(\beta +\beta ') =^{\gamma}\beta +^{\gamma}\beta '$ , and $^{\gamma}(\beta \beta ') =^{\gamma}\beta^{\gamma}\beta '$

for $\gamma , \gamma '$ in Γ and $\beta , \beta '$ in L. There exists a law of composition on E characterized by the relation

[\beta ;\gamma ][\beta ';\gamma '] = [\beta^{\gamma}\beta ';\gamma \gamma '] \tag{23}

Indeed, it suffices to verify that the right-hand side does not change if we replace, respectively, $\beta ,\gamma ,\beta '$ , and $\gamma '$ with $\beta \alpha ,\iota (\alpha )^{-1}\gamma ,\beta '\alpha '$ , and $\iota (\alpha ')^{-1}\gamma '$ for all $\alpha ,\alpha '$ in $L^*$ . This follows immediately by applying to $\mathscr{E}$ formula (1) of VIII, p. 285, which can also be written as $\gamma \iota (\alpha ) =\iota (^{\gamma}\alpha )\gamma$ for $\gamma \in \Gamma$ and $\alpha \in L^*$ . By the formulas in (22), the set E endowed with this law is a monoid with identity element $[1;e]$ .

Since $\pi \circ \iota$ is constant with value $e$ , there exists a mapping $\widetilde{\pi}$ from E to G such that we have

\widetilde{\pi}([\beta ;\gamma ]) =\pi (\gamma ) \tag{24}

for $\beta \in L$ and $\gamma \in \Gamma$ .

Let $g$ be an element of G, and let $E_g=\widetilde{\pi}^{-1}(g)$ . If $\gamma_0$ is a fixed element of $\pi^{-1}(g)$ , then the mapping $\beta \mapsto [\beta ;\gamma_0]$ is a bijection from L to $E_g$ , thanks to which we will transfer to $E_g$ the K-vector space structure on L. Since $\pi^{-1}(g)$ is composed of elements of the form $\iota (\alpha )\gamma_0$ , where $\alpha$ runs through $L^*$ , and we have

[\beta ;\iota (\alpha )\gamma_0] = [\beta \alpha ;\gamma_0]

the vector space structure on $E_g$ does not depend on the choice of $\gamma_0$ .

Let $g$ and $g'$ be elements of G; by formulas (22) and (23), the law of composition on E induces by restriction a K-bilinear mapping from $E_g\times E_{g'}$ to $E_{gg'}$ . Consequently, the vector space $P =\bigoplus_{g\in G}E_g$ is endowed with the structure of an associative and unital algebra, whose multiplication induces the previous bilinear mapping from $E_g\times E_{g'}$ to $E_{gg'}$ for all $g$ and $g'$ in G. The algebra P is called the cross product of L and $\mathscr{E}$ and is denoted by $\mathbf{A}[\mathscr{E}; L]$ ; its unit element is the element $[1;e]$ of $E_e$ .

Set

u(\beta ) = [\beta ;e] \tag{25}

for $\beta$ in L. Then $u: L\rightarrow \mathbf{A}[\mathscr{E}; L]$ is an injective homomorphism of K-algebras. By (23), for every $\gamma \in \Gamma$ , the element $[1;\gamma ]$ is invertible in $\mathbf{A}[\mathscr{E},L]$ , and the mapping $v: \Gamma \rightarrow \mathbf{A}[\mathscr{E},L]^*$ that sends $\gamma$ to $[1;\gamma ]$ is an injective group homomorphism. The homomorphisms $u$ and $v$ are called canonical. We have the relations

u(\alpha ) =v(\iota (\alpha )) \tag{26}

u(^{\gamma}\beta ) =v(\gamma )u(\beta )v(\gamma )^{-1} \tag{27}

[\beta ;\gamma ] =u(\beta )v(\gamma ) \tag{28}

for $\alpha \in L^*,\beta \in L$ , and $\gamma \in \Gamma$ .

Conversely, we have the following universal property of the algebra $\mathbf{A}[\mathscr{E}; L]$ .

Proposition 12 00JF

Let B be a K-algebra, $u': L\rightarrow B$ a K-algebra homomorphism, and $v': \Gamma \rightarrow B^*$ a group homomorphism. We assume satisfied the relations

u'(\alpha ) =v'(\iota (\alpha )) \tag{29}

u'(^{\gamma}\beta ) =v'(\gamma )u'(\beta )v'(\gamma )^{-1} \tag{30}

for $\alpha \in L^*,\beta \in L$ , and $\gamma \in \Gamma$ . Then there exists a unique algebra homomorphism $f$ from $\mathbf{A}[\mathscr{E}; L]$ to B such that we have $u'=f\circ u$ and $v'=f\circ v$ .

To prove the uniqueness of $f$ , observe that the vector space $\mathbf{A}[\mathscr{E}; L]$ over the field K is generated by the set of elements of the form $[\beta ;\gamma ] =u(\beta )v(\gamma )$ . If the homomorphism $f':\mathbf{A}[\mathscr{E}; L]\rightarrow B$ satisfies $f'\circ u=u'$ and $f'\circ v=v'$ , then it sends $[\beta ;\gamma ]$ to $u'(\beta )v'(\gamma )$ and therefore coincides with $f$ .

By formula (29), we have

u'(\beta \alpha )v'(\iota (\alpha )^{-1}\gamma ) =u'(\beta )v'(\gamma ) \tag{31}

for $\alpha$ in $L^*,\beta$ in L, and $\gamma$ in Γ. By the definition of E, there consequently exists a mapping $f_0: E\rightarrow B$ such that $f_0([\beta ;\gamma ]) =u'(\beta )v'(\gamma )$ . By formulas (23) and (30), we have $f_0(xx') =f_0(x)f_0(x')$ for $x$ and $x'$ in E. The restriction of $f_0$ to $E_g$ is K-linear for every element $g$ of G; consequently, there exists a unique K-linear mapping $f$ from $\mathbf{A}[\mathscr{E}; L]$ to B that coincides with $f_0$ on E. The mapping $f$ is an algebra morphism that satisfies $u'=f\circ u$ and $v'=f\circ v$ .

Remark 00JG

Let $\sigma : G\rightarrow \Gamma$ be a section of the mapping $\pi$ . Set $\varepsilon_g=v(\sigma (g))$ for every $g$ in G, and denote by $c_{\sigma}$ the 2-cocycle associated with $\sigma$ (VIII, p. 296). In particular, we have

\varepsilon_g\varepsilon_{g'}=u(c_{\sigma}(g, g'))\varepsilon_{gg'} \tag{32}

for all $g, g'\in G$ . Let us, moreover, identify L with a subalgebra of $\mathbf{A}[\mathscr{E}; L]$ via the homomorphism $u$ . Then every element of $\mathbf{A}[\mathscr{E}; L]$ can be written uniquely as $\sum_{g\in G}a_g\varepsilon_g$ , where $(a_g)$ is a family of elements of L with finite support. The multiplication in $\mathbf{A}[\mathscr{E}; L]$ can be expressed by the formula

(\sum_ga_g\varepsilon_g)(\sum_gb_g\varepsilon_g)=\sum_gd_g\varepsilon_g \tag{33}

with

d_g=\sum_{hh'=g}a_h(\lambda (h)\cdot b_{h'})c_{\sigma}(h, h') \tag{34}

If the extension $\mathscr{E}$ is semitrivial, then we can choose a section $\sigma$ of $\pi$ that is a group morphism from G to Γ; the cocycle $c_{\sigma}$ is therefore constant with value 1, and formula (34) simplifies to

d_g=\sum_{hh'=g}a_h\lambda (h)\cdot b_{h'} \tag{35}

Let $K'$ be an extension of the field K. Denote by $L'$ the $K'$ -algebra $L_{(K')}$ , and for any $g$ in G, denote by $\lambda '(g)$ the automorphism $\lambda (g)_{(K')}$ induced by $\lambda (g)$ on $L_{(K')}$ . Moreover, let us denote by $\tau '(g)$ the automorphism of $L^{'*}$ induced by $\lambda '(g)$ . Finally, let $h$ be the homomorphism from $L^*$ to $L^{'*}$ that sends $x$ to $x\otimes 1$ . Let $\mathscr{E}'= (\Gamma ', \iota ', \pi ')$ be the direct image of the extension of $\mathscr{E}$ by $h$ (VIII, p. 289). Let $\mathbf{A}[\mathscr{E}',L']$ be the cross product $K'$ -algebra of $\mathscr{E}'$ and $L'$ , and let $u': L'\rightarrow \mathbf{A}[\mathscr{E}',L']$ and $v': \Gamma '\rightarrow \mathbf{A}[\mathscr{E}',L']^*$ be the canonical homomorphisms.

Proposition 13 00JH

There exists a unique isomorphism of $K'$ -algebras $\varphi$ from $\mathbf{A}[\mathscr{E}; L]_{(K')}$ to $\mathbf{A}[\mathscr{E}'; L']$ that satisfies the relations

u'(h(\beta )) =\varphi (1\otimes u(\beta )) \tag{36}

for $\beta \in L$ and

v'(h(\gamma )) =\varphi (1\otimes v(\gamma )) \tag{37}

for $\gamma \in \Gamma$ .

The proof follows immediately from the constructions.

The isomorphism $\varphi$ is called canonical.

10. Application to the Brauer Group

In this subsection, K is a field, and L is a Galois algebra over K with action $\lambda : G\rightarrow$ Aut $_K(L)$ . We denote the degree of L over K by $n$ . We have $n=$ Card(G).

Theorem 3 00JI

Let $\mathscr{E}= (\Gamma , \iota , \pi )$ be a $\tau$ -extension of G by $L^*$ . The algebra $\mathbf{A}[\mathscr{E}; L]$ is central simple and of degree $n^2$ over K. Moreover, the canonical homomorphism $u$ from L to $\mathbf{A}[\mathscr{E}; L]$ is an isomorphism from L to a maximal commutative subalgebra of $\mathbf{A}[\mathscr{E}; L]$ .

Lemma 12 00JJ

a) There does not exist any ideal of L, other than $\{0\}$ and L that is invariant under G.

b) Let $g$ be an element of G distinct from the identity element, and let $\mathfrak{a}_g$ be the ideal of L generated by the elements of the form $x-\lambda (g)x$ , where $x$ runs through L. We have $\mathfrak{a}_g= L$ .

Let $\mathscr{S}$ be the set of maximal ideals of L; for any subset S of $\mathscr{S}$ , set $\mathfrak{a}(S) =\bigcap_{\mathfrak{m}\in S}\mathfrak{m}$ . Since the ring L is commutative and semisimple (VIII, p. 310, Remark 6), the map $W\mapsto \mathfrak{a}(W)$ is a bijection from the set of subsets of $\mathscr{S}$ to the set of ideals of L (VIII, p. 142, Proposition 9). The ideal $\mathfrak{a}(S)$ is invariant under G if and only if S is. Since G acts transitively on $\mathscr{S}$ (VIII, p. 308, Theorem 2, vi)), the only subsets of $\mathscr{S}$ invariant under G are $\emptyset$ and $\mathscr{S}$ . Since $\mathfrak{a}(\emptyset ) = L$ and $\mathfrak{a}(\mathscr{S}) =\{0\}$ , we have proved assertion a).

Let us prove b) by contradiction; suppose that we have $\mathfrak{a}_g\not= L$ . Let $\mathfrak{m}$ be a maximal ideal of L containing $\mathfrak{a}_g$ . We then have $\lambda (g)x\equiv x$ mod $\mathfrak{m}$ for every $x$ in L. In particular, $\mathfrak{m}$ is invariant under $g$ , and $\lambda (g)$ is induced by the identity on the field $L/\mathfrak{m}$ . By property (vi) in Theorem 2 of VIII, p. 308, the element $g$ of $G_{\mathfrak{m}}$ is trivial, which contradicts the assumption of b).

Let us now prove Theorem 3. The vector space $\mathbf{A}[\mathscr{E}; L]$ has finite dimension Card(G)[L $: K] =n^2$ over K, by VIII, p. 316 and the assumed equality Card(G) $= [L : K] =n$ .

Let $\mathfrak{a}$ be a nonzero two-sided ideal of the algebra $\mathbf{A}[\mathscr{E}; L]$ . We use the notation of VIII, p. 316. Every element $a$ of $\mathbf{A}[\mathscr{E}; L]$ can be written uniquely as $a=\sum_{g\in G}a_g\varepsilon_g$ with $a_g\in L$ for every $g\in G$ ; we denote by $\Phi (a)$ the set of elements $g$ of G such that $a_g\not= 0$ . By formula (32) of VIII, p. 316, we have the relation

\varepsilon_g\varepsilon_{g'}=c_{\sigma}(g, g')\varepsilon_{gg'} \tag{38}

for all $g, g'\in G$ and consequently

\Phi (a\varepsilon_g) = \Phi (a).g \tag{39}

for every $g\in G$ and $a\in \mathbf{A}[\mathscr{E}; L]$ .

Let $a$ be a nonzero element of $\mathfrak{a}$ for which $\Phi (a)$ is minimal for the inclusion; by formula (39), if necessary replacing $a$ with an element of the form $a\varepsilon_{g^{-1}}$ with $g\in \Phi (a)$ , we may assume that we have $e\in \Phi (a)$ . Let $s$ be an element of $\Phi (a)$ distinct from $e$ . By Lemma 12, b), there exists an element $x$ of L such that $a_s(x-\lambda (s)\cdot x)\not= 0$ . But we have the relation

xa-ax=\sum_ga_g(x-\lambda (g)\cdot x)\varepsilon_g \tag{40}

where the sum is taken over the elements $g$ of $\Phi (a)$ distinct from $e$ . Because of the minimality of $\Phi (a)$ , we then have $xa-ax= 0$ , but this contradicts the assumption on $x$ . We have, therefore, proved by contradiction that $\Phi (a)$ contains only the identity element $e$ of G, so we have $a\in L$ .

Hence $L\cap \mathfrak{a}$ is an ideal of L not reduced to $\{0\}$ . Moreover, for every $x$ in L, we have $\varepsilon_gx\varepsilon^{-1}_g=\lambda (g)\cdot x$ , so $L\cap \mathfrak{a}$ is invariant under G. By Lemma 12, we therefore have $L\cap \mathfrak{a}= L$ , that is, $L\subset \mathfrak{a}$ . Since L contains the identity element of $\mathbf{A}[\mathscr{E}; L]$ , we have $\mathfrak{a}=\mathbf{A}[\mathscr{E}; L]$ .

Since the algebra $\mathbf{A}[\mathscr{E}; L]$ has nonzero finite dimension over K and its only two-sided ideals are $\{0\}$ and $\mathbf{A}[\mathscr{E}; L]$ , it is simple.

Let us prove that every element $a$ of $\mathbf{A}[\mathscr{E}; L]$ that commutes with L belongs to L. We write $a$ as $\sum_{g\in G}a_g\varepsilon_g$ with coefficients $a_g$ in L. For every $x$ in L, we have $xa-ax= 0$ , and relation (40) shows that we have

a_g(x-\lambda (g)\cdot x) = 0 \tag{41}

for every $g\in G$ and $x\in L$ . By Lemma 12, we therefore have $a_g= 0$ for $g\not=e$ ; consequently, $a\in L$ .

Let us, finally, determine the center of $\mathbf{A}[\mathscr{E}; L]$ . If $z$ belongs to the center, then it commutes with L, so belongs to L. But we then have

0 =\varepsilon_gz-z\varepsilon_g= (\lambda (g)\cdot z-z)\varepsilon_g

for every $g$ of G. Hence $z$ is invariant under the automorphism group $\lambda (G)$ of L. We therefore have $z\in K$ by assumption.

Theorem 4 00JK

Let A be a central simple algebra of finite degree over K, and let L be a maximal commutative subalgebra of A. Then there exists a $\tau$ -extension $\mathscr{E}$ of G by $L^*$ such that A is isomorphic to $\mathbf{A}[\mathscr{E}; L]$ .

Without loss of generality, we may assume that L is a maximal commutative subalgebra of A. Let Γ be the multiplicative group consisting of the invertible elements $\gamma$ of A such that there exists a $g$ in G with

\gamma x\gamma^{-1}=\lambda (g).x \tag{42}

for every element $x$ of L. If $\gamma \in \Gamma$ is given, then the element $g$ satisfying this relation is unique; we denote it by $\pi (\gamma )$ . It is immediate that $\pi$ is a homomorphism from Γ to G and that its kernel is equal to $L^*$ . The surjectivity of $\pi$ follows from the Skolem–Noether theorem (VIII, p. 263, Corollary).

If we denote the canonical injection of $L^*$ into Γ by $\iota$ , then it follows from the constructions that $\mathscr{E}= (\Gamma , \iota , \pi )$ is a $\tau$ -extension of G by $L^*$ . Let

$u: L\rightarrow \mathbf{A}[\mathscr{E}; L]$ and $v: \Gamma \rightarrow \mathbf{A}[\mathscr{E}; L]$

be the canonical homomorphisms. By the universal property of $\mathbf{A}[\mathscr{E}; L]$ (VIII, p. 315, Proposition 12), there exists a unique algebra homomorphism $f:\mathbf{A}[\mathscr{E}; L]\rightarrow A$ such that $f\circ u(x) =x$ and $f\circ v(\gamma ) =\gamma$ for $x\in L$ and $\gamma \in \Gamma$ . Since the algebra $\mathbf{A}[\mathscr{E}; L]$ is simple, the homomorphism $f$ is injective. Now, the algebra $\mathbf{A}[\mathscr{E}; L]$ has degree $n^2$ over K and so does A because L is a maximal commutative semisimple subalgebra of A and we have $n= [L : K]$ (VIII, p. 262, Proposition 3, (ii)). Hence $f$ is bijective.

Definition 2 00JL

Let $\mathscr{S}$ be the set of maximal ideals of L. We define

Br(L $/K$ ) $=\bigcap_{\mathfrak{m}\in\mathscr{S}}$ Ker( $r_{(L/\mathfrak{m})/K}$ ), where $r_{(L/\mathfrak{m})/K}:$ Br(K) $\rightarrow$ Br(L $/\mathfrak{m}$ ) is the extension of scalars homomorphism (VIII, p. 281).

Theorem 5 00JM

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

w(\gamma_1, \gamma_2)(x_1\otimes x_2) =v_1(\gamma_1)x_1\otimes v_2(\gamma_2)x_2

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.

Remark 1 00JN

If L is an étale algebra over K and G is the automorphism group of L, then it is not always true that the algebra $\mathbf{A}[\mathscr{E}; L]$ is central simple (for example, we can take $L = K^n$ and $G =\mathfrak{S}_n$ ).

Remark 2 00JO

We can calculate a 2-cocycle $c$ associated with an algebra A split by a finite Galois extension L with group G as follows. First, there exists a K-homomorphism $\varphi : A\rightarrow \mathbf{M}_m$ (L), where $[A : K] =m^2$ . For $g\in G$ , let $\varphi^g$ be the homomorphism from A to $\mathbf{M}_m(L)$ given by $a\mapsto \varphi (g^{-1}ag)$ . By the Skolem–Noether theorem (VIII, p. 256, Theorem 3), for every $g\in G$ , there exists an element $u_g$ of $\mathbf{G}\mathbf{L}_m(L)$ such that

\varphi^g(a) =u_g\varphi (a)u^{-1}_g

for $a\in A$ . We then set

c(g, g') =u_gu_{g'}u^{-1}_{gg'}

We can also define an extension of G by $L^*$ using $\varphi :$ we consider the group $\Gamma \subset \mathbf{G}\mathbf{L}_m(L)$ consisting of the $\gamma$ for which there exists a $g\in G$ with

\varphi^g(a) =\gamma \varphi (a)\gamma^{-1}

for every $a\in A$ . The class of this extension is the inverse image by Ψ of the class of A in Br(L $/K$ ).

Corollary 00JP

The mapping $\Phi_{L/K}= \Theta \circ \Psi^{-1}$ defines a group isomorphism from Br(L $/K$ ) to $H^2(G,L^*)$ .

Let $K'$ be an extension of K and $\varphi : K'\rightarrow L$ a morphism of K-algebras. The set H of elements $h$ of G such that $\lambda (h)\circ \varphi =\varphi$ is a subgroup of G, and the $K'$ -algebra L endowed with the restriction of $\lambda$ to H is a Galois algebra over $K'$ .

Proposition 14 00JQ

The following diagram commutes:

Br(L $/K$ ) $^{\Phi_{L/K}}/$ / $H_2(G,L_*)$

$r_{K'/K}$ Res $^G_H$

Br(L $/K'$ ) $^{\Phi_{L/K'}}/$ / $H_2(H,L_*)$ .

This follows from Propositions 7 of VIII, p. 299 and 13 of VIII, p. 317.

11. Index and Exponent

Theorem 6 00JR

Let K be a commutative field, and let A be a central simple A-algebra of finite degree over K. Let L be a separable extension of K of finite degree that is a splitting field for the algebra A. Then [L : K][A] is zero in Br(K).

There exists an extension M of L that is a Galois extension of K of finite degree (V, §10, No. 1, p. 57, Proposition 2). The class [A] of A in the Brauer group of K belongs to the subgroup Br(M $/K$ ). Let G be the Galois group of M over K, and let $\alpha$ be the image of [A] in $H^2(G,M^*)$ (VIII, p. 321, Corollary). Let H be the Galois group of M over L. Then H is a subgroup of index [L : K] in G (V, §10, No. 7, p. 68, Corollary 5). Since Res $^G_H(\alpha ) = \Phi_{M/L}(A_{(L)})$ (Proposition 14), we have Res $^G_H(\alpha ) = 0$ . By Proposition 8 of VIII, p. 303, it follows that $[L : K]\alpha = 0$ , and consequently [L : K][A] = 0.

Let K be a commutative field, and let A be a central simple algebra of finite degree over K. Then A is isomorphic to an algebra of the form $\mathbf{M}_n$ (D), where D is a field with center K, and [A] = [D] in Br(K). The reduced degree of D depends only on A. We call this reduced degree the index of A. The index of A divides the reduced degree of A. The exponent of A is the order of the class of A in the Brauer group of K.

Corollary 1 00JS

The exponent of a central simple algebra of finite degree over a commutative field divides its index.

It suffices to prove this for a field D of finite degree over its center K. Let L be a maximal commutative subfield of D that is a separable extension of K; we have $[D : K] = [L : K]^2$ (VIII, p. 265, Corollary 2, b) and c)). Then the extension L of K is a splitting field for the algebra D (VIII, p. 281, Proposition 5), and [L : K] coincides with the reduced degree of D. We then apply Theorem 6.

Corollary 2 00JT

Let K be a commutative field, and let A be a central simple algebra of finite degree over K. Let $p$ be a prime number. If $p$ divides the index of A, then it divides the exponent of A.

Let us suppose that the prime number $p$ does not divide the exponent of A and prove that it does not divide its index. It suffices to prove this result in the case when A is a field. Let L be a Galois extension of K of finite degree that splits A. Let G be the Galois group of L over K, and let H be a Sylow $p$ -subgroup of G (I, §6, No. 6, p. 78). We denote by $L'= L^H$ the subfield of L of invariants of H. The exponent of $A_{(L')}$ divides that of A and is therefore prime to $p$ . By Theorem 6, we have $[L : L'][A_{(L')}] = 0$ in the Brauer group of $L'$ . It follows that $[A_{(L')}] = 0$ and that the field $L'$ splits A. We then apply Corollary 2 of VIII, p. 283; it follows that the index of A divides $[L': K]$ and is therefore not divisible by $p$ .

Corollary 3 00JU

Let $p$ be a prime number and let K be a perfect field of characteristic $p$ . Let A be a central simple algebra of finite degree over K. Then $p$ does not divide the index of A.

Let us prove that the Brauer group of K does not contain any element of order $p$ . Every Galois extension M of K of finite degree is a perfect field (V, §7, No. 1, p. 36, Proposition 2). Hence taking the $p$ -th power is an automorphism of the group $M^*$ . It follows that multiplication by $p$ is an automorphism of the group $H^2$ (Gal(M $/K$ ) $,M^*$ ) that is isomorphic to Br(M $/K$ ).

Consequently, the order of [A] is prime to $p$ and, by Corollary 2, its index is not divisible by $p$ .

Remark 00JV

By considering tensor products of quaternion algebras, it is possible to construct fields with center K, exponent 2, and arbitrarily large index (cf. VIII, p. 371, Exercises 7 and 8).

$*$ Conversely, if K is a finite Galois extension of a $p$ -adic field or a field of formal power series over a finite field, then the exponent of a central simple algebra of finite degree over K is equal to its index (VIII, p. 332, Exercise 17, e)). $*$

Exercises

See the exercises for § 16.