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§ 16。 布劳尔群中的其他描述
在这个截面中，如果F是F的阿贝尔的群和g g g g ( f ) g(f) g ( f ) g ⋅ f g\cdot f g ⋅ f
1. τ \tau τ
在这个小节中，我们将群G，阿贝尔的群F固定为乘法，并将群同态τ \tau τ e e e
回想一下 (I, §6, No. 1, p. 65)，F 的 G 的扩张 E \mathscr{E} E ( Γ , ι , π ) (\Gamma , \iota , \pi ) ( Γ , ι , π ) π : Γ → G \pi : \Gamma \rightarrow G π : Γ → G ι \iota ι ι \iota ι = = = π \pi π E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) γ ∈ Γ \gamma \in \Gamma γ ∈ Γ φ γ : F → F \varphi_{\gamma}: F\rightarrow F φ γ : F → F
ι ( φ γ ( f ) ) = γ ι ( f ) γ − 1
\iota (\varphi_{\gamma}(f)) =\gamma \iota (f)\gamma^{-1}
ι ( φ γ ( f )) = γ ι ( f ) γ − 1 因为 f ∈ F f\in F f ∈ F f ∈ F f\in F f ∈ F ι ( f ) \iota (f) ι ( f ) E _{\mathscr{E}} E
γ ι ( f ) γ − 1 = ι \gamma \iota (f)\gamma^{-1}=\iota γ ι ( f ) γ − 1 = ι E ( π ( γ ) ) ⋅ f _{\mathscr{E}}(\pi (\gamma ))\cdot f E ( π ( γ )) ⋅ f
对于 γ ∈ Γ \gamma \in \Gamma γ ∈ Γ f ∈ F f\in F f ∈ F
F 对 G 的A τ A\tau A τ E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) E _{\mathscr{E}} E τ \tau τ
γ ι ( f ) γ − 1 = ι ( τ ( π ( γ ) ) ⋅ f ) (1)
\gamma \iota (f)\gamma^{-1}=\iota (\tau (\pi (\gamma ))\cdot f) \tag{1}
γ ι ( f ) γ − 1 = ι ( τ ( π ( γ )) ⋅ f ) ( 1 ) 对于 γ ∈ Γ \gamma \in \Gamma γ ∈ Γ f ∈ F f\in F f ∈ F E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) E ′ = ( Γ ′ , ι ′ , π ′ ) \mathscr{E}'= (\Gamma ', \iota ', \pi ') E ′ = ( Γ ′ , ι ′ , π ′ ) τ \tau τ E \mathscr{E} E E ′ \mathscr{E}' E ′ τ \tau τ E \mathscr{E} E E ′ \mathscr{E}' E ′ u : Γ → Γ ′ u: \Gamma \rightarrow \Gamma ' u : Γ → Γ ′ π ′ ∘ u = π \pi '\circ u=\pi π ′ ∘ u = π ι ′ = u ∘ ι \iota '=u\circ \iota ι ′ = u ∘ ι τ \tau τ τ ( E , F ) _{\tau}(\mathscr{E},\mathscr{F}) τ ( E , F )
“E \mathscr{E} E F \mathscr{F} F τ \tau τ
这是等价关系； 扩张 E \mathscr{E} E τ ( _{\tau}( τ ( E \mathscr{E} E
“α \alpha α τ _{\tau} τ τ \tau τ
α \alpha α
集合E 0 = F × G E_0= F\times G E 0 = F × G ι 0 : F → E 0 \iota_0: F\rightarrow E_0 ι 0 : F → E 0 π 0 : E 0 → G \pi_0: E_0\rightarrow G π 0 : E 0 → G ι 0 ( f ) = ( f , e ) \iota_0(f) = (f, e) ι 0 ( f ) = ( f , e ) f ∈ F f\in F f ∈ F π 0 ( f , g ) = g \pi_0(f, g) =g π 0 ( f , g ) = g ( f , g ) ∈ F × G (f, g)\in F\times G ( f , g ) ∈ F × G E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ π \pi π σ : G → Γ \sigma : G\rightarrow \Gamma σ : G → Γ σ ( e ) = e \sigma (e) =e σ ( e ) = e F × G F\times G F × G u : ( f , g ) ↦ ι ( f ) σ ( g ) u: (f, g)\mapsto \iota (f)\sigma (g) u : ( f , g ) ↦ ι ( f ) σ ( g ) F × G F\times G F × G ( E 0 , ι 0 , π 0 ) (E_0, \iota_0, \pi_0) ( E 0 , ι 0 , π 0 ) E \mathscr{E} E τ \tau τ
我们用τ \tau τ τ ( G , F ) _{\tau}(G,F) τ ( G , F ) τ _{\tau} τ
外部半直积 ( F × τ G , i , p ) (F\times_{\tau}G, i, p) ( F × τ G , i , p ) τ \tau τ I τ \mathscr{I}_{\tau} I τ τ \tau τ τ \tau τ I τ \mathscr{I}_{\tau} I τ
引用了 1 引用了 命题 3, I § 6
对于i ∈ { 1 , 2 } i\in \{1,2\} i ∈ { 1 , 2 } G i G_i G i F i F_i F i G i G_i G i F i F_i F i τ i \tau_i τ i τ 1 × τ 2 : G 1 × G 2 → \tau_1\times \tau_2: G_1\times G_2\rightarrow τ 1 × τ 2 : G 1 × G 2 → 1 × F 2 _1\times F_2 1 × F 2
( τ 1 × τ 2 ) ( g 1 , g 2 ) ⋅ ( f 1 , f 2 ) = ( τ 1 ( g 1 ) ⋅ f 1 , τ 2 ( g 2 ) ⋅ f 2 )
(\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)
( τ 1 × τ 2 ) ( g 1 , g 2 ) ⋅ ( f 1 , f 2 ) = ( τ 1 ( g 1 ) ⋅ f 1 , τ 2 ( g 2 ) ⋅ f 2 ) 对于 g 1 ∈ G 1 , g 2 ∈ G 2 , f 1 ∈ F 1 g_1\in G_1,g_2\in G_2,f_1\in F_1 g 1 ∈ G 1 , g 2 ∈ G 2 , f 1 ∈ F 1 f 2 ∈ F 2 f_2\in F_2 f 2 ∈ F 2 E 1 = ( Γ 1 , ι 1 , π 1 ) \mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1) E 1 = ( Γ 1 , ι 1 , π 1 ) G 1 G_1 G 1 F 1 F_1 F 1 τ 1 \tau_1 τ 1 E 2 = ( Γ 2 , ι 2 , π 2 ) \mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2) E 2 = ( Γ 2 , ι 2 , π 2 ) G 2 G_2 G 2 F 2 F_2 F 2 τ 2 \tau_2 τ 2 ( Γ 1 × Γ 2 , ι 1 × ι 2 , π 1 × π 2 ) (\Gamma_1\times \Gamma_2, \iota_1\times \iota_2, \pi_1\times \pi_2) ( Γ 1 × Γ 2 , ι 1 × ι 2 , π 1 × π 2 ) G 1 × G 2 G_1\times G_2 G 1 × G 2 τ 1 × τ 2 \tau_1\times \tau_2 τ 1 × τ 2 F 1 × F 2 F_1\times F_2 F 1 × F 2 E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 E 1 × E 2 \mathscr{E}_1\times \mathscr{E}_2 E 1 × E 2
2. τ \tau τ
令 G 和G ′ G' G ′ τ : G → \tau : G\rightarrow τ : G → u : G ′ → G u: G'\rightarrow G u : G ′ → G τ ′ = τ ∘ u : G ′ → \tau '=\tau \circ u: G'\rightarrow τ ′ = τ ∘ u : G ′ → Γ ′ \Gamma ' Γ ′ π \pi π u u u Γ × G G ′ \Gamma \times_GG' Γ × G G ′ ( γ , g ′ ) (\gamma , g') ( γ , g ′ ) Γ × G ′ \Gamma \times G' Γ × G ′ π ( γ ) = u ( g ′ ) \pi (\gamma ) =u(g') π ( γ ) = u ( g ′ ) ι ′ : F → Γ ′ \iota ': F\rightarrow \Gamma ' ι ′ : F → Γ ′ ι ′ ( α ) = ( ι ( α ) , e ) \iota '(\alpha ) = (\iota (\alpha ), e) ι ′ ( α ) = ( ι ( α ) , e ) α ∈ F \alpha \in F α ∈ F π ′ : Γ ′ → G ′ \pi ': \Gamma '\rightarrow G' π ′ : Γ ′ → G ′ ι ′ \iota ' ι ′ π ′ \pi ' π ′ π \pi π ι ′ \iota ' ι ′ π ′ \pi ' π ′ α ∈ F \alpha \in F α ∈ F γ ′ = ( γ , g ) ∈ Γ ′ \gamma '= (\gamma , g)\in \Gamma ' γ ′ = ( γ , g ) ∈ Γ ′
γ ′ ι ′ ( α ) γ ′ − 1 = ( γ , g ) ( ι ( α ) , e ) ( γ , g ) − 1 = ( ι ( τ ( π ( γ ) ) . α ) , e ) = ι ′ ( τ ′ ( π ′ ( γ ′ ) ) . α )
\gamma '\iota '(\alpha )\gamma^{'-1}= (\gamma , g)(\iota (\alpha ), e)(\gamma , g)^{-1}= (\iota (\tau (\pi (\gamma )).\alpha ), e) =\iota '(\tau '(\pi '(\gamma ')).\alpha )
γ ′ ι ′ ( α ) γ ′ − 1 = ( γ , g ) ( ι ( α ) , e ) ( γ , g ) − 1 = ( ι ( τ ( π ( γ )) . α ) , e ) = ι ′ ( τ ′ ( π ′ ( γ ′ )) . α ) 因而，( Γ ′ , ι ′ , π ′ ) (\Gamma ', \iota ', \pi ') ( Γ ′ , ι ′ , π ′ ) G ′ G' G ′ τ ′ \tau ' τ ′ u u u E \mathscr{E} E u ∗ ( E ) u^*(\mathscr{E}) u ∗ ( E ) φ : Γ ′ → Γ \varphi : \Gamma '\rightarrow \Gamma φ : Γ ′ → Γ
以下图表通勤：
引用了 1 · 被以下引用 1 引用了 引理 2
被以下引用 推论 1
F ι ′ ^{\iota'} ι ′ Γ ′ π ′ {\Gamma'}^{\pi'} Γ ′ π ′ G ′ G' G ′
(2)
φ u \varphi u φ u F ι ^{\iota} ι π ^{\pi} π
此外，如果E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) \mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1) E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) τ ′ \tau ' τ ′ φ 1 : Γ 1 ′ → Γ \varphi_1: \Gamma '_1\rightarrow \Gamma φ 1 : Γ 1 ′ → Γ
F ι 1 ′ ^{\iota'_1} ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi'_1} Γ 1 ′ π 1 ′ G ′ G' G ′
φ 1 u \varphi_1u φ 1 u F ι ^{\iota} ι π ^{\pi} π
通勤，那么存在τ ′ \tau ' τ ′ ψ \psi ψ E 1 ′ \mathscr{E}_1' E 1 ′ u ∗ ( E ) u^*(\mathscr{E}) u ∗ ( E ) φ 1 = φ ∘ ψ \varphi_1=\varphi \circ \psi φ 1 = φ ∘ ψ
第一个图表的交换律源自φ \varphi φ ψ \psi ψ
令 F 1 ′ F'_1 F 1 ′ w : F 1 ′ → w: F'_1\rightarrow w : F 1 ′ → τ 1 : G ′ → \tau_1: G'\rightarrow τ 1 : G ′ → 1 ′ '_1 1 ′
w ( τ 1 ( g ) ( f ) ) = τ ( u ( g ) ) ( w ( f ) )
w(\tau_1(g)(f)) =\tau (u(g))(w(f))
w ( τ 1 ( g ) ( f )) = τ ( u ( g )) ( w ( f )) 被以下引用 2 被以下引用 命题 1 , 命题 3
对于每个f ∈ F 1 ′ f\in F'_1 f ∈ F 1 ′ g ∈ G ′ g\in G' g ∈ G ′ E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) \mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1) E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) G ′ G' G ′ F 1 ′ F'_1 F 1 ′ φ 1 : Γ 1 ′ → Γ \varphi_1: \Gamma '_1\rightarrow \Gamma φ 1 : Γ 1 ′ → Γ τ 1 \tau_1 τ 1
F 1 ′ ι 1 ′ {F'_1}^{\iota'_1} F 1 ′ ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi'_1} Γ 1 ′ π 1 ′ G ′ G' G ′
w φ 1 u w\varphi_1u w φ 1 u F ι ^{\iota} ι π ^{\pi} π
那么存在唯一群同态ψ : Γ 1 ′ → Γ ′ \psi : \Gamma '_1\rightarrow \Gamma ' ψ : Γ 1 ′ → Γ ′
F 1 ′ ι 1 ′ {F'_1}^{\iota'_1} F 1 ′ ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi_1'} Γ 1 ′ π 1 ′ G ′ G' G ′
w ψ w\psi w ψ F ι ′ ^{\iota'} ι ′ Γ ′ π ′ {\Gamma'}^{\pi'} Γ ′ π ′ G ′ G' G ′
通勤和φ 1 = φ ∘ ψ \varphi_1=\varphi \circ \psi φ 1 = φ ∘ ψ
如果群同态 ψ : Γ 1 ′ → Γ ′ \psi : \Gamma '_1\rightarrow \Gamma ' ψ : Γ 1 ′ → Γ ′
ψ ( γ ) = ( φ ∘ ψ ( γ ) , π ′ ∘ ψ ( γ ) ) = ( φ 1 ( γ ) , π 1 ′ ( γ ) )
\psi (\gamma ) = (\varphi \circ \psi (\gamma ), \pi '\circ \psi (\gamma )) = (\varphi_1(\gamma ), \pi '_1(\gamma ))
ψ ( γ ) = ( φ ∘ ψ ( γ ) , π ′ ∘ ψ ( γ )) = ( φ 1 ( γ ) , π 1 ′ ( γ )) 每γ ∈ Γ 1 ′ \gamma \in \Gamma '_1 γ ∈ Γ 1 ′ γ ↦ ( φ 1 ( γ ) , π 1 ′ ( γ ) ) \gamma \mapsto (\varphi_1(\gamma ), \pi_1'(\gamma )) γ ↦ ( φ 1 ( γ ) , π 1 ′ ( γ )) Γ 1 ′ \Gamma '_1 Γ 1 ′ Γ × G ′ \Gamma \times G' Γ × G ′ Γ × G G ′ \Gamma \times_GG' Γ × G G ′ γ ∈ Γ 1 ′ \gamma \in \Gamma '_1 γ ∈ Γ 1 ′ π ∘ φ 1 ( γ ) = u ∘ π 1 ′ ( γ ) \pi \circ \varphi_1(\gamma ) =u\circ \pi_1'(\gamma ) π ∘ φ 1 ( γ ) = u ∘ π 1 ′ ( γ )
令E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 τ \tau τ ψ \psi ψ τ \tau τ E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 φ 1 ( \varphi_1( φ 1 ( φ 2 ) \varphi_2) φ 2 ) u ∗ ( E 1 ) ( u^*(\mathscr{E}_1) ( u ∗ ( E 1 ) ( u ∗ ( E 2 ) ) u^*(\mathscr{E}_2)) u ∗ ( E 2 )) τ ′ \tau ' τ ′ u ∗ ( E 1 ) u^*(\mathscr{E}_1) u ∗ ( E 1 ) u ∗ ( E 2 ) u^*(\mathscr{E}_2) u ∗ ( E 2 ) u ∗ ( ψ ) u^*(\psi ) u ∗ ( ψ ) φ 2 ∘ u ∗ ( ψ ) = ψ ∘ φ 1 \varphi_2\circ u^*(\psi ) =\psi \circ \varphi_1 φ 2 ∘ u ∗ ( ψ ) = ψ ∘ φ 1
引用了 1 · 被以下引用 1 引用了 命题 1
被以下引用 § 16. 。 布劳尔群中的其他描述
应用命题1至ψ ∘ φ 1 \psi \circ \varphi_1 ψ ∘ φ 1
因此，τ ′ \tau ' τ ′ u ∗ ( E ) u^*(\mathscr{E}) u ∗ ( E ) E \mathscr{E} E u ∗ : u^*: u ∗ : τ ( G , F ) → _{\tau}(G,F)\rightarrow τ ( G , F ) → τ ′ ( G ′ , F ) _{\tau'}(G',F) τ ′ ( G ′ , F ) τ \tau τ E \mathscr{E} E τ ′ \tau ' τ ′ u ∗ ( E ) u^*(\mathscr{E}) u ∗ ( E )
令u ′ : G ′ ′ → G ′ u': G''\rightarrow G' u ′ : G ′′ → G ′ E \mathscr{E} E τ \tau τ τ ′ ′ = τ ′ ∘ u ′ \tau ''=\tau '\circ u' τ ′′ = τ ′ ∘ u ′ φ ( \varphi ( φ ( φ ′ , φ ′ ′ ) \varphi ',\varphi '') φ ′ , φ ′′ ) τ ′ \tau ' τ ′ u ∗ ( E ) ( u^*(\mathscr{E}) ( u ∗ ( E ) ( τ ′ ′ \tau '' τ ′′ u ′ ∗ ( u ∗ ( E ) ) u^{'*}(u^*(\mathscr{E})) u ′ ∗ ( u ∗ ( E )) τ ′ ′ \tau '' τ ′′ ( u ∘ u ′ ) ∗ ( E ) ) (u\circ u')^*(\mathscr{E})) ( u ∘ u ′ ) ∗ ( E )) τ ′ ′ \tau '' τ ′′ u ′ ∗ ( u ∗ ( E ) ) u^{'*}(u^*(\mathscr{E})) u ′ ∗ ( u ∗ ( E )) τ ′ ′ \tau '' τ ′′ ( u ∘ u ′ ) ∗ ( E ) (u\circ u')^*(\mathscr{E}) ( u ∘ u ′ ) ∗ ( E ) ψ \psi ψ φ ′ ′ ∘ ψ = φ ∘ φ ′ \varphi ''\circ \psi =\varphi \circ \varphi ' φ ′′ ∘ ψ = φ ∘ φ ′
被以下引用 1 被以下引用 命题 4
令 H 为 G 的子群和 j : H → G j: H\rightarrow G j : H → G τ \tau τ E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ ∘ j \tau \circ j τ ∘ j j ∗ ( E ) j^*(\mathscr{E}) j ∗ ( E ) ( π − 1 ( H ) , ι ′ , π ′ ) (\overset{-1}{\pi}(H), \iota ', \pi ') ( π − 1 ( H ) , ι ′ , π ′ ) ι ′ : F → π − 1 ( H ) \iota ': F\rightarrow \overset{-1}{\pi}(H) ι ′ : F → π − 1 ( H ) π ′ : π − 1 ( H ) → H \pi ':\overset{-1}{\pi}(H)\rightarrow H π ′ : π − 1 ( H ) → H f ↦ ι ( f ) f\mapsto \iota (f) f ↦ ι ( f ) γ ↦ π ( γ ) \gamma \mapsto \pi (\gamma ) γ ↦ π ( γ ) u : G ′ → G u: G'\rightarrow G u : G ′ → G φ \varphi φ π − 1 ( u ( G ′ ) ) \overset{-1}{\pi}(u(G')) π − 1 ( u ( G ′ ))
3. τ \tau τ
令G为群，令F和F ′ F' F ′ τ \tau τ τ ′ \tau ' τ ′ F ′ F' F ′ v : F → F ′ v: F\rightarrow F' v : F → F ′
v ( τ ( g ) ⋅ f ) = τ ′ ( g ) ⋅ v ( f ) (3)
v(\tau (g)\cdot f) =\tau '(g)\cdot v(f) \tag{3}
v ( τ ( g ) ⋅ f ) = τ ′ ( g ) ⋅ v ( f ) ( 3 ) 对于每个g ∈ G g\in G g ∈ G f ∈ F f\in F f ∈ F E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ Γ ~ \widetilde{\Gamma} Γ F ′ × τ ′ ∘ π Γ F'\times_{\tau'\circ\pi}\Gamma F ′ × τ ′ ∘ π Γ ı ~ : F ′ → Γ ~ \widetilde{\imath}: F'\rightarrow \widetilde{\Gamma} : F ′ → Γ f ↦ ( f , e ) f\mapsto (f, e) f ↦ ( f , e ) p ~ : Γ ~ → Γ \widetilde{p}:\widetilde{\Gamma}\rightarrow \Gamma p : Γ → Γ j : F → Γ ~ j: F\rightarrow \widetilde{\Gamma} j : F → Γ f ↦ ( v ( f ) , ι ( f ) − 1 ) f\mapsto (v(f), \iota (f)^{-1}) f ↦ ( v ( f ) , ι ( f ) − 1 ) ι \iota ι τ ′ ∘ π \tau '\circ \pi τ ′ ∘ π j j j ι \iota ι
( f ′ , γ ) j ( f ) ( f ′ , γ ) − 1 = ( f ′ , γ ) ( v ( f ) , ι ( f ) − 1 ) ( f ′ , γ ) − 1
(f', \gamma )j(f)(f', \gamma )^{-1}= (f', \gamma )(v(f), \iota (f)^{-1})(f', \gamma )^{-1}
( f ′ , γ ) j ( f ) ( f ′ , γ ) − 1 = ( f ′ , γ ) ( v ( f ) , ι ( f ) − 1 ) ( f ′ , γ ) − 1 = ( τ ′ ( π ( γ ) ) ⋅ v ( f ) , ι ( τ ( π ( γ ) ) ⋅ f ) − 1 ) (4)
= (\tau '(\pi (\gamma ))\cdot v(f), \iota (\tau (\pi (\gamma ))\cdot f)^{-1}) \tag{4}
= ( τ ′ ( π ( γ )) ⋅ v ( f ) , ι ( τ ( π ( γ )) ⋅ f ) − 1 ) ( 4 ) = j ( τ ( π ( γ ) ) ⋅ f )
=j(\tau (\pi (\gamma ))\cdot f)
= j ( τ ( π ( γ )) ⋅ f ) 对于f ∈ F , γ ∈ Γ f\in F,\gamma \in \Gamma f ∈ F , γ ∈ Γ f ′ ∈ F ′ f'\in F' f ′ ∈ F ′ j j j Γ ~ \widetilde{\Gamma} Γ Γ ′ \Gamma ' Γ ′ Γ ~ \widetilde{\Gamma} Γ j j j Γ ~ \widetilde{\Gamma} Γ Γ ′ \Gamma ' Γ ′ ı ~ \widetilde{\imath} ι ′ \iota ' ι ′ π ∘ p ~ \pi \circ \widetilde{p} π ∘ p F ′ × ι F'\times \iota F ′ × ι j j j π ′ : Γ ′ → G \pi ': \Gamma '\rightarrow G π ′ : Γ ′ → G π ∘ p ~ \pi \circ \widetilde{p} π ∘ p ι \iota ι j ( F ) j(F) j ( F ) ı ~ ( F ′ ) \widetilde{\imath}(F') ( F ′ ) Γ ~ \widetilde{\Gamma} Γ ι ′ \iota ' ι ′ F ′ × F F'\times F F ′ × F F ′ × ι ( F ) = F'\times \iota (F) = F ′ × ι ( F ) = π ∘ p ~ \pi \circ \widetilde{p} π ∘ p
( f ′ , f ) ⟼ ( f ′ v ( f ) , ι ( f ) − 1 )
(f', f)\longmapsto (f'v(f), \iota (f)^{-1})
( f ′ , f ) ⟼ ( f ′ v ( f ) , ι ( f ) − 1 ) 是群同构。 像为 ι ′ \iota ' ι ′ π ′ \pi ' π ′ π \pi π p ~ \widetilde{p} p π ′ \pi ' π ′ E ′ = ( Γ ′ , ι , π ′ ) \mathscr{E}'= (\Gamma ', \iota , \pi ') E ′ = ( Γ ′ , ι , π ′ ) τ ′ \tau ' τ ′ F ′ F' F ′ v v v E \mathscr{E} E v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) γ ↦ ( 1 , γ ) \gamma \mapsto (1, \gamma ) γ ↦ ( 1 , γ ) Γ ~ \widetilde{\Gamma} Γ Γ ′ \Gamma ' Γ ′ Γ ~ \widetilde{\Gamma} Γ φ : Γ → Γ ′ \varphi : \Gamma \rightarrow \Gamma ' φ : Γ → Γ ′
F ι ^{\iota} ι π ^{\pi} π
v φ (5)
v\varphi \tag{5}
v φ ( 5 ) F ′ ι ′ {F'}^{\iota'} F ′ ι ′ Γ ′ π ′ {\Gamma'}^{\pi'} Γ ′ π ′
令E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) \mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi_1') E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) τ ′ \tau ' τ ′ F ′ F' F ′ φ 1 : Γ → Γ 1 ′ \varphi_1: \Gamma \rightarrow \Gamma '_1 φ 1 : Γ → Γ 1 ′
F ι ^{\iota} ι π ^{\pi} π
v φ 1 v\varphi_1 v φ 1 F ′ ι 1 ′ {F'}^{\iota'_1} F ′ ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi'_1} Γ 1 ′ π 1 ′
那么存在τ ′ \tau ' τ ′ ψ \psi ψ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) E 1 ′ \mathscr{E}_1' E 1 ′ φ 1 = ψ ∘ φ \varphi_1=\psi \circ \varphi φ 1 = ψ ∘ φ
第一个图表的交换律之后是构造。 ψ \psi ψ
令G 1 ′ G'_1 G 1 ′ w : G → G 1 ′ w: G\rightarrow G'_1 w : G → G 1 ′ τ 1 : G 1 ′ → \tau_1: G'_1\rightarrow τ 1 : G 1 ′ → ′ ' ′ τ ′ = τ 1 ∘ w \tau '=\tau_1\circ w τ ′ = τ 1 ∘ w E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) \mathscr{E}_1'= (\Gamma '_1, \iota '_1, \pi '_1) E 1 ′ = ( Γ 1 ′ , ι 1 ′ , π 1 ′ ) G 1 ′ G'_1 G 1 ′ F ′ F' F ′ τ 1 \tau_1 τ 1 φ 1 : Γ → Γ 1 ′ \varphi_1: \Gamma \rightarrow \Gamma '_1 φ 1 : Γ → Γ 1 ′
被以下引用 1 被以下引用 命题 2
F ι ^{\iota} ι π ^{\pi} π
v φ 1 w v\varphi_1w v φ 1 w F ′ ι 1 ′ {F'}^{\iota'_1} F ′ ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi'_1} Γ 1 ′ π 1 ′ G 1 ′ G'_1 G 1 ′
那么存在唯一群同态ψ : Γ ′ → Γ 1 ′ \psi : \Gamma '\rightarrow \Gamma '_1 ψ : Γ ′ → Γ 1 ′
F ′ ι ′ {F'}^{\iota'} F ′ ι ′ Γ ′ π ′ {\Gamma'}^{\pi'} Γ ′ π ′
ψ w \psi w ψ w F ′ ι 1 ′ {F'}^{\iota'_1} F ′ ι 1 ′ Γ 1 ′ π 1 ′ {\Gamma'_1}^{\pi'_1} Γ 1 ′ π 1 ′ G 1 ′ G'_1 G 1 ′ φ 1 = ψ ∘ φ \varphi_1=\psi \circ \varphi φ 1 = ψ ∘ φ
对于任何( f , γ ) ∈ F ′ × Γ (f, \gamma )\in F'\times \Gamma ( f , γ ) ∈ F ′ × Γ ( f , γ ) (f, \gamma ) ( f , γ ) Γ ′ \Gamma ' Γ ′ ( f , γ ) (f, \gamma ) ( f , γ ) ψ : Γ ′ → Γ 1 ′ \psi : \Gamma '\rightarrow \Gamma '_1 ψ : Γ ′ → Γ 1 ′
ψ ( f ′ , γ ) = ψ ( ι ′ ( f ′ ) φ ( γ ) ) = ι 1 ′ ( f ′ ) φ 1 ( γ )
\psi (f', \gamma )=\psi (\iota '(f')\varphi (\gamma )) =\iota '_1(f')\varphi_1(\gamma )
ψ ( f ′ , γ ) = ψ ( ι ′ ( f ′ ) φ ( γ )) = ι 1 ′ ( f ′ ) φ 1 ( γ ) 对于每个f ′ ∈ F ′ f'\in F' f ′ ∈ F ′ γ ∈ Γ \gamma \in \Gamma γ ∈ Γ ( f , γ ) ↦ ι 1 ′ ( f ) φ 1 ( γ ) (f, \gamma )\mapsto \iota '_1(f)\varphi_1(\gamma ) ( f , γ ) ↦ ι 1 ′ ( f ) φ 1 ( γ ) F ′ × τ ′ ∘ π Γ F'\times_{\tau'\circ\pi}\Gamma F ′ × τ ′ ∘ π Γ Γ 1 ′ \Gamma '_1 Γ 1 ′ ψ ~ \widetilde{\psi} ψ
ι 1 ′ ( f ) φ 1 ( γ ) ι 1 ′ ( f ′ ) φ 1 ( γ ′ ) = ι 1 ′ ( f τ 1 ( π 1 ′ ( φ 1 ( γ ) ) ) ⋅ f ′ ) φ 1 ( γ γ ′ )
\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 ')
ι 1 ′ ( f ) φ 1 ( γ ) ι 1 ′ ( f ′ ) φ 1 ( γ ′ ) = ι 1 ′ ( f τ 1 ( π 1 ′ ( φ 1 ( γ ))) ⋅ f ′ ) φ 1 ( γ γ ′ ) = ι 1 ′ ( f τ ′ ( π ( γ ) ) ⋅ f ′ ) φ 1 ( γ γ ′ )
=\iota '_1(f \tau '(\pi (\gamma ))\cdot f')\varphi_1(\gamma \gamma ')
= ι 1 ′ ( f τ ′ ( π ( γ )) ⋅ f ′ ) φ 1 ( γ γ ′ ) 对于 f , f ′ ∈ F ′ f, f'\in F' f , f ′ ∈ F ′ γ , γ ′ ∈ Γ \gamma , \gamma '\in \Gamma γ , γ ′ ∈ Γ ψ ~ \widetilde{\psi} ψ j j j ι 1 ′ ( v ( f ) ) = φ 1 ( ι ( f ) ) \iota '_1(v(f)) =\varphi_1(\iota (f)) ι 1 ′ ( v ( f )) = φ 1 ( ι ( f )) f ∈ F f\in F f ∈ F ψ \psi ψ ψ ~ \widetilde{\psi} ψ
令E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 τ \tau τ ψ \psi ψ τ \tau τ E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 φ 1 ( \varphi_1( φ 1 ( φ 2 ) \varphi_2) φ 2 ) v ∗ ( E 1 ) ( v_*(\mathscr{E}_1) ( v ∗ ( E 1 ) ( v ∗ ( E 2 ) ) v_*(\mathscr{E}_2)) v ∗ ( E 2 )) τ ′ \tau ' τ ′ v ∗ ( E 1 ) v_*(\mathscr{E}_1) v ∗ ( E 1 ) v ∗ ( E 2 ) v_*(\mathscr{E}_2) v ∗ ( E 2 ) v ∗ ( ψ ) v_*(\psi ) v ∗ ( ψ ) φ 2 ∘ ψ = v ∗ ( ψ ) ∘ φ 1 \varphi_2\circ \psi =v_*(\psi )\circ \varphi_1 φ 2 ∘ ψ = v ∗ ( ψ ) ∘ φ 1
引用了 1 · 被以下引用 1 引用了 命题 2
被以下引用 § 16. 。 布劳尔群中的其他描述
应用命题2至φ 2 ∘ ψ \varphi_2\circ \psi φ 2 ∘ ψ
因此，τ ′ \tau ' τ ′ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) E \mathscr{E} E v ∗ : v_*: v ∗ : τ ( G , F ) → _{\tau}(G,F)\rightarrow τ ( G , F ) → τ ′ ( G , F ′ ) _{\tau'}(G,F') τ ′ ( G , F ′ ) τ \tau τ E \mathscr{E} E τ ′ \tau ' τ ′ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E )
我们保留命题中的记号。令 F ′ ′ F'' F ′′ τ ′ ′ : G → \tau '': G\rightarrow τ ′′ : G → ′ ′ '' ′′ v ′ : F ′ → F ′ ′ v': F'\rightarrow F'' v ′ : F ′ → F ′′
τ ′ ′ ( g ) ⋅ v ′ ( f ) = v ′ ( τ ′ ( g ) ⋅ f )
\tau ''(g)\cdot v'(f) =v'(\tau '(g)\cdot f)
τ ′′ ( g ) ⋅ v ′ ( f ) = v ′ ( τ ′ ( g ) ⋅ f ) 被以下引用 1 被以下引用 命题 4
对于 g ∈ G g\in G g ∈ G f ∈ F ′ f\in F' f ∈ F ′ E \mathscr{E} E τ \tau τ φ ( \varphi ( φ ( φ ′ , φ ′ ′ ) \varphi ',\varphi '') φ ′ , φ ′′ ) v ∗ ( E ) ( v_*(\mathscr{E}) ( v ∗ ( E ) ( v ∗ ′ ( v ∗ ( E ) ) v'_*(v_*(\mathscr{E})) v ∗ ′ ( v ∗ ( E )) ( v ′ ∘ v ) ∗ ( E ) ) (v'\circ v)_*(\mathscr{E})) ( v ′ ∘ v ) ∗ ( E )) τ ′ ′ \tau '' τ ′′ v ∗ ′ ( v ∗ ( E ) ) v'_*(v_*(\mathscr{E})) v ∗ ′ ( v ∗ ( E )) F ′ ′ F'' F ′′ τ ′ ′ \tau '' τ ′′ ( v ′ ∘ v ) ∗ ( E ) (v'\circ v)_*(\mathscr{E}) ( v ′ ∘ v ) ∗ ( E ) ψ \psi ψ φ ′ ′ = ψ ∘ φ ′ ∘ φ \varphi ''=\psi \circ \varphi '\circ \varphi φ ′′ = ψ ∘ φ ′ ∘ φ
令 j : { 1 } → F j:\{1\} \rightarrow F j : { 1 } → F I τ \mathscr{I}_{\tau} I τ j ∗ ( ( G , i j_*((G, i j ∗ (( G , i G ) ) _G)) G )) i : { e } → G i:\{e\} \rightarrow G i : { e } → G c : F → F c: F\rightarrow F c : F → F f ↦ 1 f\mapsto 1 f ↦ 1 E \mathscr{E} E τ \tau τ τ \tau τ c ∗ ( E ) c_*(\mathscr{E}) c ∗ ( E ) I τ \mathscr{I}_{\tau} I τ
被以下引用 1 被以下引用 命题 4
设 E 为 F 的子群在 G 的作用下稳定。用 F ′ F' F ′ v : F → F ′ v: F\rightarrow F' v : F → F ′ τ ′ : G → \tau ': G\rightarrow τ ′ : G → ′ ' ′ F ′ F' F ′
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f )
\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f ) 对于 g ∈ G g\in G g ∈ G f ∈ F f\in F f ∈ F E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ ι ( E ) \iota (E) ι ( E ) F ′ F' F ′ τ ′ \tau ' τ ′ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) ( Γ / ι ( E ) , ι , π ) (\Gamma /\iota (E), \iota , \pi ) ( Γ/ ι ( E ) , ι , π ) ι ‾ \overline{\iota} ι π ‾ \overline{\pi} π ι \iota ι π \pi π φ \varphi φ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) Γ / ι ( E ) \Gamma /\iota (E) Γ/ ι ( E )
设G和G ′ G' G ′ F ′ F' F ′ τ : G → \tau : G\rightarrow τ : G → , τ ′ : G → ,\tau ': G\rightarrow , τ ′ : G → ′ ' ′ , u : G ′ → G ,u: G'\rightarrow G , u : G ′ → G v : F → F ′ v: F\rightarrow F' v : F → F ′
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f )
\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f ) 引用了 2 · 被以下引用 1 引用了 引理 2 , 命题 2
被以下引用 命题 4
对于 g ∈ G g\in G g ∈ G f ∈ F f\in F f ∈ F τ ′ ′ = τ ′ ∘ u \tau ''=\tau '\circ u τ ′′ = τ ′ ∘ u E \mathscr{E} E τ \tau τ φ u ( \varphi_u( φ u ( φ v , φ u ′ , φ v ′ ) \varphi_v,\varphi '_u,\varphi '_v) φ v , φ u ′ , φ v ′ ) τ ∘ u \tau \circ u τ ∘ u u ∗ ( E ) ( u^*(\mathscr{E}) ( u ∗ ( E ) ( τ ′ \tau ' τ ′ v ∗ ( E ) v_*(\mathscr{E}) v ∗ ( E ) τ ′ ′ \tau '' τ ′′ u ∗ ( v ∗ ( E ) ) u^*(v_*(\mathscr{E})) u ∗ ( v ∗ ( E )) v ∗ ( u ∗ ( E ) ) ) v_*(u^*(\mathscr{E}))) v ∗ ( u ∗ ( E ))) τ ′ ′ \tau '' τ ′′ ψ \psi ψ v ∗ ( u ∗ ( E ) ) v_*(u^*(\mathscr{E})) v ∗ ( u ∗ ( E )) u ∗ ( v ∗ ( E ) ) u^*(v_*(\mathscr{E})) u ∗ ( v ∗ ( E )) φ v ∘ φ u = \varphi_v\circ \varphi_u= φ v ∘ φ u = φ u ′ ∘ ψ ∘ φ v ′ \varphi '_u\circ \psi \circ \varphi '_v φ u ′ ∘ ψ ∘ φ v ′
我们用( Γ u , ι u , π u ) (\Gamma_u, \iota_u, \pi_u) ( Γ u , ι u , π u ) ( Γ u ′ , ι u ′ , π u ′ ) (\Gamma '_u, \iota '_u, \pi '_u) ( Γ u ′ , ι u ′ , π u ′ ) τ ∘ u \tau \circ u τ ∘ u u ∗ ( E ) u^*(\mathscr{E}) u ∗ ( E ) τ ′ ′ \tau '' τ ′′ u ∗ ( v ∗ ( E ) ) u^*(v_*(\mathscr{E})) u ∗ ( v ∗ ( E )) φ v ∘ φ u \varphi_v\circ \varphi_u φ v ∘ φ u ψ 1 : Γ u → Γ u ′ \psi_1: \Gamma_u\rightarrow \Gamma '_u ψ 1 : Γ u → Γ u ′
F ι u ^{\iota_u} ι u Γ u π u \Gamma_u^{\pi_u} Γ u π u G ′ G' G ′
v ψ 1 v\psi_1 v ψ 1 F ′ ι u ′ {F'}^{\iota'_u} F ′ ι u ′ Γ u ′ π u ′ {\Gamma'_u}^{\pi'_u} Γ u ′ π u ′ G ′ G' G ′
通勤和φ u ′ ∘ ψ 1 = φ v ∘ φ u \varphi '_u\circ \psi_1=\varphi_v\circ \varphi_u φ u ′ ∘ ψ 1 = φ v ∘ φ u ψ \psi ψ ψ 1 \psi_1 ψ 1 ψ ′ \psi ' ψ ′ ψ ′ ∘ φ v ′ = ψ 1 \psi '\circ \varphi '_v=\psi_1 ψ ′ ∘ φ v ′ = ψ 1 ψ ′ = ψ \psi '=\psi ψ ′ = ψ
4. 群运算 on 类 of τ \tau τ
设G为群，F为阿贝尔的群，τ : G → \tau : G\rightarrow τ : G → δ : G → G × G \delta : G\rightarrow G\times G δ : G → G × G g ↦ g\mapsto g ↦ ( g , g ) (g, g) ( g , g ) m : F × F → F m: F\times F\rightarrow F m : F × F → F ( f 1 , f 2 ) ↦ (f_1, f_2)\mapsto ( f 1 , f 2 ) ↦ f 1 f 2 f_1f_2 f 1 f 2 s : F → F s: F\rightarrow F s : F → F f ↦ f − 1 f\mapsto f^{-1} f ↦ f − 1 E 1 = ( Γ 1 , ι 1 , π 1 ) \mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1) E 1 = ( Γ 1 , ι 1 , π 1 ) E 2 = ( Γ 2 , ι 2 , π 2 ) \mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2) E 2 = ( Γ 2 , ι 2 , π 2 ) τ \tau τ
m ( ( ( τ × τ ) ∘ δ ) ( g ) ⋅ ( f 1 , f 2 ) ) = τ ( g ) ⋅ m ( f 1 , f 2 )
m(((\tau \times \tau )\circ \delta )(g)\cdot (f_1, f_2)) =\tau (g)\cdot m(f_1, f_2)
m ((( τ × τ ) ∘ δ ) ( g ) ⋅ ( f 1 , f 2 )) = τ ( g ) ⋅ m ( f 1 , f 2 ) 对于每个g ∈ G g\in G g ∈ G f 1 , f 2 ∈ F f_1, f_2\in F f 1 , f 2 ∈ F m ∗ ( δ ∗ ( E 1 × E 2 ) ) m_*(\delta^*(\mathscr{E}_1\times \mathscr{E}_2)) m ∗ ( δ ∗ ( E 1 × E 2 )) τ \tau τ τ \tau τ E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 E 1 E 2 \mathscr{E}_1\mathscr{E}_2 E 1 E 2 E 1 \mathscr{E}_1 E 1 E 2 \mathscr{E}_2 E 2 τ ( G , F ) _{\tau}(G,F) τ ( G , F )
τ \tau τ τ ( G , F ) _{\tau}(G,F) τ ( G , F ) I τ \mathscr{I}_{\tau} I τ τ \tau τ E \mathscr{E} E s ∗ ( E ) s_*(\mathscr{E}) s ∗ ( E )
引用了 5 引用了 推论 2 , 推论 2 , 命题 3 , 命题 2 , 例 1
律的结合性源自图表的交换律
G → δ G × G ↓ δ ↓ Id × δ G × G → δ × Id G × G × G
\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}
G ↓ δ G × G δ δ × Id G × G ↓ Id × δ G × G × G 和
F × F × F → m × Id F × F ↓ Id × m ↓ m F × F → m F
\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}
F × F × F ↓ Id × m F × F m × Id m F × F ↓ m F 以及VIII、p. 288的推论2； VIII、p. 291的推论2；以及 VIII、p. 292 中的命题 3 个。
令 Δ : F → F × F \Delta : F \to F \times F Δ : F → F × F f ↦ ( f , f ) f \mapsto (f, f) f ↦ ( f , f ) E = ( Γ , ι , π ) \mathscr{E} = (\Gamma, \iota, \pi) E = ( Γ , ι , π ) τ \tau τ Δ ~ : Γ → Γ × G Γ \tilde{\Delta} : \Gamma \to \Gamma \times_G \Gamma Δ ~ : Γ → Γ × G Γ γ ↦ ( γ , γ ) \gamma \mapsto (\gamma, \gamma) γ ↦ ( γ , γ )
F → ι Γ → π G ↓ Δ ↓ Δ ~ ∥ F × F ⟶ Γ × G Γ ⟶ G .
\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}
F ↓ Δ F × F ι ⟶ Γ ↓ Δ ~ Γ × G Γ π ⟶ G ∥ G . 由VIII、p. 290的命题 2，可知( τ × τ ) ∘ δ (\tau \times \tau) \circ \delta ( τ × τ ) ∘ δ δ ∗ ( E × E ) \delta^*(\mathscr{E} \times \mathscr{E}) δ ∗ ( E × E ) Δ ∗ ( E ) \Delta_*(\mathscr{E}) Δ ∗ ( E )
用 c : F → F c : F \to F c : F → F f ↦ 1 f \mapsto 1 f ↦ 1 I τ \mathscr{I}_\tau I τ δ ∗ ( E × E ) \delta^*(\mathscr{E} \times \mathscr{E}) δ ∗ ( E × E ) Δ ∗ ( E ) \Delta_*(\mathscr{E}) Δ ∗ ( E )
F → ( Id F × c ) ∘ Δ F × F ( c × Id F ) ∘ Δ ↓ Id F ↓ m F × F → m F .
\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}
F ( c × Id F ) ∘ Δ F × F ( Id F × c ) ∘ Δ m F × F ↓ Id F ↓ m F . 最后一个断言由交换律图表推出
F → ( Id F × s ) ∘ Δ F × F ( s × Id F ) ∘ Δ ↓ c ↓ m F × F → m F .
\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}
F ( s × Id F ) ∘ Δ F × F ( Id F × s ) ∘ Δ m F × F ↓ c ↓ m F . 令E 1 = ( Γ 1 , ι 1 , π 1 ) \mathscr{E}_1 = (\Gamma_1, \iota_1, \pi_1) E 1 = ( Γ 1 , ι 1 , π 1 ) E 2 = ( Γ 2 , ι 2 , π 2 ) \mathscr{E}_2 = (\Gamma_2, \iota_2, \pi_2) E 2 = ( Γ 2 , ι 2 , π 2 ) τ \tau τ Γ 1 × Γ 2 → Γ 2 × Γ 1 \Gamma_1 \times \Gamma_2 \to \Gamma_2 \times \Gamma_1 Γ 1 × Γ 2 → Γ 2 × Γ 1 ( γ 1 , γ 2 ) ↦ ( γ 2 , γ 1 ) (\gamma_1, \gamma_2) \mapsto (\gamma_2, \gamma_1) ( γ 1 , γ 2 ) ↦ ( γ 2 , γ 1 ) σ : Γ 1 × G Γ 2 → Γ 2 × G Γ 1 \sigma : \Gamma_1 \times_G \Gamma_2 \to \Gamma_2 \times_G \Gamma_1 σ : Γ 1 × G Γ 2 → Γ 2 × G Γ 1
σ ( ι 1 ( f ) , ι 2 ( f ) − 1 ) = ( ι 2 ( f − 1 ) , ι 1 ( f − 1 ) − 1 )
\sigma(\iota_1(f), \iota_2(f)^{-1}) = (\iota_2(f^{-1}), \iota_1(f^{-1})^{-1})
σ ( ι 1 ( f ) , ι 2 ( f ) − 1 ) = ( ι 2 ( f − 1 ) , ι 1 ( f − 1 ) − 1 ) 对于 f ∈ F f\in F f ∈ F σ \sigma σ τ \tau τ E 1 E 2 \mathscr{E}_1\mathscr{E}_2 E 1 E 2 E 2 E 1 \mathscr{E}_2\mathscr{E}_1 E 2 E 1
a) 令G和G ′ G' G ′ τ : G → \tau : G\rightarrow τ : G → u : G ′ → G u: G'\rightarrow G u : G ′ → G u ∗ : u^*: u ∗ : τ ( G , F ) → _{\tau}(G,F)\rightarrow τ ( G , F ) → τ ∘ u ( G ′ , F ) _{\tau\circ u}(G',F) τ ∘ u ( G ′ , F )
b) 令G为群，令F和F ′ F' F ′ τ : G → \tau : G\rightarrow τ : G → , τ ′ : G → ,\tau ': G\rightarrow , τ ′ : G → ′ ' ′ v : F → F ′ v: F\rightarrow F' v : F → F ′
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f )
\tau '(g)\cdot v(f) =v(\tau (g)\cdot f)
τ ′ ( g ) ⋅ v ( f ) = v ( τ ( g ) ⋅ f ) 对于 g ∈ G g\in G g ∈ G f ∈ F f\in F f ∈ F v ∗ : v_*: v ∗ : τ ( G , F ) → _{\tau}(G,F)\rightarrow τ ( G , F ) → τ ′ ( G , F ′ ) _{\tau'}(G,F') τ ′ ( G , F ′ )
这是从图表的交换律得出的
G ′ δ {G'}^{\delta} G ′ δ G ′ × G ′ F × F m G'\times G'F\times F^m G ′ × G ′ F × F m
u u × u _uu_{\times u} u u × u v × v v v_{\times v}v v × v v
G δ ^{\delta} δ G × G F ′ × F ′ m G\times GF'\times {F'}^m G × G F ′ × F ′ m F ′ F' F ′
5. 上同调描述
令G为群，令F为阿贝尔的群，令τ : G → \tau : G\rightarrow τ : G → g ∈ G g\in G g ∈ G f ∈ F f\in F f ∈ F τ ( g ) ⋅ f \tau (g)\cdot f τ ( g ) ⋅ f g f ^gf g f c : G × G → F c: G\times G\rightarrow F c : G × G → F ( g 1 , g 2 , g 3 ) ∈ G × G × G (g_1, g_2, g_3)\in G\times G\times G ( g 1 , g 2 , g 3 ) ∈ G × G × G
g 1 c ( g 2 , g 3 ) c ( g 1 , g 2 g 3 ) = c ( g 1 , g 2 ) c ( g 1 g 2 , g 3 ) (7)
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}
g 1 c ( g 2 , g 3 ) c ( g 1 , g 2 g 3 ) = c ( g 1 , g 2 ) c ( g 1 g 2 , g 3 ) ( 7 ) 由于F是可交换的，2-余循环的集合从G × G G\times G G × G Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F ) C 1 ( G , F ) C^1(G,F) C 1 ( G , F ) h ∈ C 1 ( G , F ) h\in C^1(G,F) h ∈ C 1 ( G , F ) ( g 1 , g 2 ) ∈ G × G (g_1, g_2)\in G\times G ( g 1 , g 2 ) ∈ G × G
( ∂ h ) ( g 1 , g 2 ) = g 1 h ( g 2 ) h ( g 1 g 2 ) − 1 h ( g 1 ) (8)
(\partial h)(g_1, g_2) =^{g_1}h(g_2)h(g_1g_2)^{-1}h(g_1) \tag{8}
( ∂ h ) ( g 1 , g 2 ) = g 1 h ( g 2 ) h ( g 1 g 2 ) − 1 h ( g 1 ) ( 8 ) 简单计算可知映射∂ h : G × G → F \partial h: G\times G\rightarrow F ∂ h : G × G → F ∂ : C 1 ( G , F ) → Z 2 ( G , F ) \partial : C^1(G,F)\rightarrow Z^2(G,F) ∂ : C 1 ( G , F ) → Z 2 ( G , F ) h ∈ C 1 ( G , F ) h\in C^1(G,F) h ∈ C 1 ( G , F ) ∂ h \partial h ∂ h Z 2 ( G , F ) / ∂ ( C 1 ( G , F ) ) Z^2(G,F)/\partial (C^1(G,F)) Z 2 ( G , F ) / ∂ ( C 1 ( G , F )) H 2 ( G , F ) H^2(G,F) H 2 ( G , F )
∗ * ∗ o 8 ^o8 o 8 ∗ * ∗
令 E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ σ \sigma σ π ( \pi ( π ( π ( σ ( g ) ) = g \pi (\sigma (g)) =g π ( σ ( g )) = g g g g g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 \sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1} σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 π \pi π c σ : G × G → F c_{\sigma}: G\times G\rightarrow F c σ : G × G → F
ι ( c σ ( g 1 , g 2 ) ) = σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 (9)
\iota (c_{\sigma}(g_1, g_2)) =\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1} \tag{9}
ι ( c σ ( g 1 , g 2 )) = σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 ( 9 ) 对于所有g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G σ \sigma σ c σ c_{\sigma} c σ
映射c σ c_{\sigma} c σ Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F ) H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) τ \tau τ E \mathscr{E} E
引用了 1 引用了 § 16. 。 布劳尔群中的其他描述
我们说映射c σ c_{\sigma} c σ σ \sigma σ H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) τ \tau τ E \mathscr{E} E
我们首先验证c σ c_{\sigma} c σ g 1 , g 2 g_1,g_2 g 1 , g 2 g 3 g_3 g 3
ι ( g 1 c σ ( g 2 , g 3 ) c σ ( g 1 , g 2 g 3 ) )
\iota (^{g_1}c_{\sigma}(g_2, g_3)c_{\sigma}(g_1, g_2g_3))
ι ( g 1 c σ ( g 2 , g 3 ) c σ ( g 1 , g 2 g 3 )) = σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 2 g 3 ) − 1 σ ( g 1 ) − 1 σ ( g 1 ) σ ( g 2 g 3 ) σ ( g 1 g 2 g 3 ) − 1
=\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}
= σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 2 g 3 ) − 1 σ ( g 1 ) − 1 σ ( g 1 ) σ ( g 2 g 3 ) σ ( g 1 g 2 g 3 ) − 1 = σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1
=\sigma (g_1)\sigma (g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}
= σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1 和
ι ( c σ ( g 1 , g 2 ) c σ ( g 1 g 2 , g 3 ) )
\iota (c_{\sigma}(g_1, g_2)c_{\sigma}(g_1g_2, g_3))
ι ( c σ ( g 1 , g 2 ) c σ ( g 1 g 2 , g 3 )) = σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 σ ( g 1 g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1
=\sigma (g_1)\sigma (g_2)\sigma (g_1g_2)^{-1}\sigma (g_1g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}
= σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 σ ( g 1 g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1 = σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1
=\sigma (g_1)\sigma (g_2)\sigma (g_3)\sigma (g_1g_2g_3)^{-1}
= σ ( g 1 ) σ ( g 2 ) σ ( g 3 ) σ ( g 1 g 2 g 3 ) − 1 因此，映射 c σ c_{\sigma} c σ Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F )
现在让我们证明H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c σ c_{\sigma} c σ σ \sigma σ σ \sigma σ σ ′ \sigma ' σ ′ g g g a g a_g a g σ ′ ( g ) = ι ( a g ) σ ( g ) \sigma '(g) =\iota (a_g)\sigma (g) σ ′ ( g ) = ι ( a g ) σ ( g ) g 1 g_1 g 1 g 2 g_2 g 2
ι ( c σ ′ ( g 1 , g 2 ) ) = σ ′ ( g 1 ) σ ′ ( g 2 ) σ ′ ( g 1 g 2 ) − 1
\iota (c_{\sigma'}(g_1, g_2)) =\sigma '(g_1)\sigma '(g_2)\sigma '(g_1g_2)^{-1}
ι ( c σ ′ ( g 1 , g 2 )) = σ ′ ( g 1 ) σ ′ ( g 2 ) σ ′ ( g 1 g 2 ) − 1 = ι ( a g 1 ) σ ( g 1 ) ι ( a g 2 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 ι ( a g 1 g 2 ) − 1 (10)
=\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}
= ι ( a g 1 ) σ ( g 1 ) ι ( a g 2 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 ι ( a g 1 g 2 ) − 1 ( 10 ) = ι ( a g 1 ) ι ( g 1 a g 2 ) ι ( c σ ( g 1 , g 2 ) ) ι ( a g 1 g 2 ) − 1
=\iota (a_{g_1})\iota (^{g_1}a_{g_2})\iota (c_{\sigma}(g_1, g_2))\iota (a_{g_1g_2})^{-1}
= ι ( a g 1 ) ι ( g 1 a g 2 ) ι ( c σ ( g 1 , g 2 )) ι ( a g 1 g 2 ) − 1 由于群 F 是可交换的，所以我们有关系
c σ ′ ( g 1 , g 2 ) c σ ( g 1 , g 2 ) − 1 = ( g 1 a g 2 ) a g 1 g 2 − 1 a g 1 = ( ∂ a ) ( g 1 , g 2 ) (11)
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}
c σ ′ ( g 1 , g 2 ) c σ ( g 1 , g 2 ) − 1 = ( g 1 a g 2 ) a g 1 g 2 − 1 a g 1 = ( ∂ a ) ( g 1 , g 2 ) ( 11 ) 由(8)得出如下。这就证明了H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c σ ′ c_{\sigma'} c σ ′ c σ c_{\sigma} c σ
最后，令E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) E ′ = ( Γ ′ , ι ′ , π ′ ) \mathscr{E}'= (\Gamma ', \iota ', \pi ') E ′ = ( Γ ′ , ι ′ , π ′ ) τ \tau τ u : E → E ′ u:\mathscr{E}\rightarrow \mathscr{E}' u : E → E ′ τ \tau τ σ \sigma σ π \pi π u ∘ σ u\circ \sigma u ∘ σ π ′ \pi ' π ′ c σ = c u ∘ σ c_{\sigma}=c_{u\circ\sigma} c σ = c u ∘ σ H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c σ c_{\sigma} c σ τ ( G , F ) _{\tau}(G,F) τ ( G , F ) E \mathscr{E} E
我们用 Θ τ \Theta_{\tau} Θ τ τ ( G , F ) → H 2 ( G , F ) _{\tau}(G,F)\rightarrow H^2(G,F) τ ( G , F ) → H 2 ( G , F ) τ \tau τ ( Γ , ι , π ) (\Gamma , \iota , \pi ) ( Γ , ι , π ) c σ c_{\sigma} c σ σ \sigma σ π \pi π
我们首先证明 θ 是群同态。 令E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) E ′ = ( Γ ′ , ι ′ , π ′ ) \mathscr{E}'= (\Gamma ', \iota ', \pi ') E ′ = ( Γ ′ , ι ′ , π ′ ) τ \tau τ σ \sigma σ σ ′ \sigma ' σ ′ π \pi π π ′ \pi ' π ′ τ \tau τ E \mathscr{E} E E ′ \mathscr{E}' E ′ E E ′ = ( Γ ′ ′ , ι ′ ′ , π ′ ′ ) \mathscr{E} \mathscr{E}'= (\Gamma '', \iota '', \pi '') E E ′ = ( Γ ′′ , ι ′′ , π ′′ ) [ γ , γ ′ ] [\gamma , \gamma '] [ γ , γ ′ ] Γ ′ ′ \Gamma '' Γ ′′ ( γ , γ ′ ) (\gamma , \gamma ') ( γ , γ ′ ) Γ × G Γ ′ \Gamma \times_G\Gamma ' Γ × G Γ ′ Γ ′ ′ \Gamma '' Γ ′′ g g g [ σ ( g ) , σ ′ ( g ) ] [\sigma (g), \sigma '(g)] [ σ ( g ) , σ ′ ( g )] π ′ ′ \pi '' π ′′ σ ′ ′ \sigma '' σ ′′ g 1 g_1 g 1 g 2 g_2 g 2
ι ′ ′ ( c σ ′ ′ ( g 1 , g 2 ) ) = σ ′ ′ ( g 1 ) σ ′ ′ ( g 2 ) σ ′ ′ ( g 1 g 2 ) − 1
\iota ''(c_{\sigma''}(g_1, g_2)) =\sigma ''(g_1)\sigma ''(g_2)\sigma ''(g_1g_2)^{-1}
ι ′′ ( c σ ′′ ( g 1 , g 2 )) = σ ′′ ( g 1 ) σ ′′ ( g 2 ) σ ′′ ( g 1 g 2 ) − 1 = [ ι ( c σ ( g 1 , g 2 ) ) , ι ′ ( c σ ′ ( g 1 , g 2 ) ]
= [\iota (c_{\sigma}(g_1, g_2)), \iota '(c_{\sigma'}(g_1, g_2)]
= [ ι ( c σ ( g 1 , g 2 )) , ι ′ ( c σ ′ ( g 1 , g 2 )] = ι ′ ′ ( c σ ( g 1 , g 2 ) c σ ′ ( g 1 , g 2 ) )
=\iota ''(c_{\sigma}(g_1, g_2)c_{\sigma'}(g_1, g_2))
= ι ′′ ( c σ ( g 1 , g 2 ) c σ ′ ( g 1 , g 2 )) 因此我们有c σ ′ ′ ( g 1 , g 2 ) = c σ ( g 1 , g 2 ) c σ ′ ( g 1 , g 2 ) c_{\sigma''}(g_1, g_2) =c_{\sigma}(g_1, g_2)c_{\sigma'}(g_1, g_2) c σ ′′ ( g 1 , g 2 ) = c σ ( g 1 , g 2 ) c σ ′ ( g 1 , g 2 )
让我们证明映射 θ 是单射的。令E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ σ \sigma σ π \pi π H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c σ c_{\sigma} c σ a : G → F a: G\rightarrow F a : G → F
c σ ( g 1 , g 2 ) = ( g 1 a ( g 2 ) ) a ( g 1 g 2 ) − 1 a ( g 1 )
c_{\sigma}(g_1, g_2) = (^{g_1}a(g_2))a(g_1g_2)^{-1}a(g_1)
c σ ( g 1 , g 2 ) = ( g 1 a ( g 2 )) a ( g 1 g 2 ) − 1 a ( g 1 ) 对于所有g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G σ ′ : G → Γ \sigma ': G\rightarrow \Gamma σ ′ : G → Γ
σ ′ ( g ) = ι ( a ( g ) − 1 ) σ ( g )
\sigma '(g) =\iota (a(g)^{-1})\sigma (g)
σ ′ ( g ) = ι ( a ( g ) − 1 ) σ ( g ) 对于g ∈ G g\in G g ∈ G c σ ′ c_{\sigma'} c σ ′ σ ′ \sigma ' σ ′ τ \tau τ E \mathscr{E} E
让我们证明映射 θ 是满射。设c c c Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F ) Γ = F × G \Gamma = F\times G Γ = F × G ( f 1 , g 1 ) ( f 2 , g 2 ) = ( f 1 ( g 1 f 2 ) c ( g 1 , g 2 ) , g 1 g 2 ) (f_1, g_1)(f_2, g_2) =(f_1(^{g_1}f_2)c(g_1, g_2), g_1g_2) ( f 1 , g 1 ) ( f 2 , g 2 ) = ( f 1 ( g 1 f 2 ) c ( g 1 , g 2 ) , g 1 g 2 )
对于所有 f 1 , f 2 ∈ F f_1, f_2\in F f 1 , f 2 ∈ F g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G
( 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 2 g 3 )
(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 2 ) ( f 3 , g 3 )) = ( f 1 , g 1 ) ( f 2 ( g 2 f 3 ) c ( g 2 , g 3 ) , g 2 g 3 ) = ( f 1 ( g 1 f 2 ) ( g 1 g 2 f 3 ) ( g 1 c ( g 2 , g 3 ) ) c ( g 1 , g 2 g 3 ) , g 1 g 2 g 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)
= ( f 1 ( g 1 f 2 ) ( g 1 g 2 f 3 ) ( g 1 c ( g 2 , g 3 )) c ( g 1 , g 2 g 3 ) , g 1 g 2 g 3 ) 和
( ( f 1 , g 1 ) ( f 2 , g 2 ) ) ( f 3 , g 3 ) = ( f 1 ( g 1 f 2 ) c ( g 1 , g 2 ) , g 1 g 2 ) ( f 3 , g 3 )
((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 2 )) ( f 3 , g 3 ) = ( f 1 ( g 1 f 2 ) c ( g 1 , g 2 ) , g 1 g 2 ) ( f 3 , g 3 ) = ( f 1 ( g 1 f 2 ) ( g 1 g 2 f 3 ) c ( g 1 , g 2 ) c ( g 1 g 2 , g 3 ) , g 1 g 2 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)
= ( f 1 ( g 1 f 2 ) ( g 1 g 2 f 3 ) c ( g 1 , g 2 ) c ( g 1 g 2 , g 3 ) , g 1 g 2 g 3 ) 对于所有 f 1 , f 2 , f 3 ∈ F f_1, f_2, f_3\in F f 1 , f 2 , f 3 ∈ F g 1 , g 2 , g 3 ∈ G g_1,g_2, g_3\in G g 1 , g 2 , g 3 ∈ G c c c ( f , g ) ∈ Γ (f, g)\in \Gamma ( f , g ) ∈ Γ
( f , g ) ( c ( e , e ) − 1 , e ) = ( f ( g c ( e , e ) − 1 ) c ( g , e ) , g )
(f, g)(c(e, e)^{-1}, e) = (f(^gc(e, e)^{-1})c(g, e), g)
( f , g ) ( c ( e , e ) − 1 , e ) = ( f ( g c ( e , e ) − 1 ) c ( g , e ) , g ) 现在，从 2-cocycle 的定义可以得出 c ( g , e ) = g c ( e , e ) c(g, e) =^gc(e, e) c ( g , e ) = g c ( e , e ) ( f , g ) ( c ( e , e ) − 1 , e ) = ( f , g ) (f, g)(c(e, e)^{-1}, e) = (f, g) ( f , g ) ( c ( e , e ) − 1 , e ) = ( f , g )
( c ( e , e ) − 1 , e ) ( f , g ) = ( f c ( e , e ) − 1 c ( e , g ) , g ) = ( f , g )
(c(e, e)^{-1}, e)(f, g) = (f c(e, e)^{-1}c(e, g), g) = (f, g)
( c ( e , e ) − 1 , e ) ( f , g ) = ( f c ( e , e ) − 1 c ( e , g ) , g ) = ( f , g ) 因而，由 (12) 定义的合成的律有单位元素 ( c ( e , e ) − 1 , e ) (c(e, e)^{-1}, e) ( c ( e , e ) − 1 , e ) ( f , g ) (f, g) ( f , g )
( g − 1 ( f − 1 c ( e , e ) − 1 c ( g , g − 1 ) − 1 ) , g − 1 )
(^{g^{-1}}(f^{-1}c(e, e)^{-1}c(g, g^{-1})^{-1}), g^{-1})
( g − 1 ( f − 1 c ( e , e ) − 1 c ( g , g − 1 ) − 1 ) , g − 1 ) 所以我们赋予 Γ 群结构。让我们用ι : F → G \iota : F\rightarrow G ι : F → G f f f ( c ( e , e ) − 1 f , e ) (c(e, e)^{-1}f, e) ( c ( e , e ) − 1 f , e ) π : Γ → G \pi : \Gamma \rightarrow G π : Γ → G σ \sigma σ g ↦ ( 1 , g ) g\mapsto (1, g) g ↦ ( 1 , g ) ι \iota ι π \pi π E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ σ \sigma σ π \pi π c σ c_{\sigma} c σ c c c
σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 = ( 1 , g 1 ) ( 1 , g 2 ) ( 1 , g 1 g 2 ) − 1 = ( c ( g 1 , g 2 ) , g 1 g 2 ) ( 1 , g 1 g 2 ) − 1
\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}
σ ( g 1 ) σ ( g 2 ) σ ( g 1 g 2 ) − 1 = ( 1 , g 1 ) ( 1 , g 2 ) ( 1 , g 1 g 2 ) − 1 = ( c ( g 1 , g 2 ) , g 1 g 2 ) ( 1 , g 1 g 2 ) − 1 = ( c ( g 1 , g 2 ) c ( e , g 1 g 2 ) − 1 , e ) = ( c ( e , e ) − 1 c ( g 1 , g 2 ) , e )
= (c(g_1, g_2)c(e, g_1g_2)^{-1}, e) = (c(e, e)^{-1}c(g_1, g_2), e)
= ( c ( g 1 , g 2 ) c ( e , g 1 g 2 ) − 1 , e ) = ( c ( e , e ) − 1 c ( g 1 , g 2 ) , e ) g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G
6. 限制与余限制
令G和G ′ G' G ′ τ \tau τ u : G ′ → G u: G'\rightarrow G u : G ′ → G τ ′ = τ ∘ u \tau '=\tau \circ u τ ′ = τ ∘ u
如果ψ : G × G → F \psi : G\times G\rightarrow F ψ : G × G → F ( g 1 , g 2 ) ↦ ψ ( u ( g 1 ) , u ( g 2 ) ) (g_1, g_2)\mapsto \psi (u(g_1), u(g_2)) ( g 1 , g 2 ) ↦ ψ ( u ( g 1 ) , u ( g 2 )) G ′ × G ′ G'\times G' G ′ × G ′ ψ ∘ ( u × u ) \psi \circ (u\times u) ψ ∘ ( u × u ) G ′ G' G ′ ψ ↦ ψ ∘ ( u × u ) \psi \mapsto \psi \circ (u\times u) ψ ↦ ψ ∘ ( u × u ) Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F ) Z 2 ( G ′ , F ) Z^2(G',F) Z 2 ( G ′ , F ) u ∗ : H 2 ( G , F ) → H 2 ( G ′ , F ) u^*: H^2(G,F)\rightarrow H^2(G',F) u ∗ : H 2 ( G , F ) → H 2 ( G ′ , F ) λ ∈ H 2 ( G , F ) \lambda \in H^2(G,F) λ ∈ H 2 ( G , F ) u ∗ ( λ ) u^*(\lambda ) u ∗ ( λ ) u u u λ \lambda λ G ′ G' G ′ u : G ′ → G u: G'\rightarrow G u : G ′ → G G ′ G' G ′ u ∗ u^* u ∗ G ′ G ^G_{G'} G ′ G G ′ G' G ′ u : G ′ → G u: G'\rightarrow G u : G ′ → G G ′ G' G ′ u ∗ u^* u ∗
例τ ( G , F ) u ∗ _{\tau}(G,F)^{u^*} τ ( G , F ) u ∗ τ ( G ′ , F ) _{\tau}(G',F) τ ( G ′ , F )
Θ τ Θ τ ′ \Theta_{\tau}\Theta_{\tau'} Θ τ Θ τ ′ H 2 ( G , F ) u ∗ H_2(G,F)^{u^*} H 2 ( G , F ) u ∗ H 2 ( G ′ , F ) H_2(G',F) H 2 ( G ′ , F )
令 H 为有限指标的 G 的子群。令s s s H \ G H\backslash G H \ G H \ G H\backslash G H \ G ( g , x ) ↦ x ⋅ g (g, x)\mapsto x\cdot g ( g , x ) ↦ x ⋅ g x ∈ H \ G x\in H\backslash G x ∈ H \ G g ∈ G g\in G g ∈ G s ( x ) g s ( x ⋅ g ) − 1 s(x)gs(x\cdot g)^{-1} s ( x ) g s ( x ⋅ g ) − 1 c : H × H → F c: H\times H\rightarrow F c : H × H → F c ~ s : G × G → F \widetilde{c}_s: G\times G\rightarrow F c s : G × G → F
c ~ s ( g 1 , g 2 ) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 )
\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})
c s ( g 1 , g 2 ) = x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) g 1 , g 2 ∈ G g_1,g_2\in G g 1 , g 2 ∈ G F H × H F^{H\times H} F H × H F G × G F^{G\times G} F G × G c ↦ c ~ s c\mapsto \widetilde{c}_s c ↦ c s
如果c : H × H → F c: H\times H\rightarrow F c : H × H → F c ~ s \widetilde{c}_s c s
令g 1 , g 2 g_1,g_2 g 1 , g 2 g 3 g_3 g 3 i ∈ { 1 , 2 , 3 } i\in \{1,2,3\} i ∈ { 1 , 2 , 3 } h i : H \ G → H h_i: H\backslash G\rightarrow H h i : H \ G → H
h i ( x ) = s ( x ⋅ g 1 ⋯ g i − 1 ) g i s ( x ⋅ g 1 ⋯ g i ) − 1
h_i(x) =s(x\cdot g_1\cdots g_{i-1})g_is(x\cdot g_1\cdots g_i)^{-1}
h i ( x ) = s ( x ⋅ g 1 ⋯ g i − 1 ) g i s ( x ⋅ g 1 ⋯ g i ) − 1 x ∈ H \ G x\in H\backslash G x ∈ H \ G
h 1 ( x ) h 2 ( x ) = s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1
h_1(x)h_2(x) =s(x)g_1g_2s(x\cdot g_1g_2)^{-1}
h 1 ( x ) h 2 ( x ) = s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 h 2 ( x ) h 3 ( x ) = s ( x ⋅ g 1 ) g 2 g 3 s ( x ⋅ g 1 g 2 g 3 ) − 1
h_2(x)h_3(x) =s(x\cdot g_1)g_2g_3s(x\cdot g_1g_2g_3)^{-1}
h 2 ( x ) h 3 ( x ) = s ( x ⋅ g 1 ) g 2 g 3 s ( x ⋅ g 1 g 2 g 3 ) − 1 x ∈ H \ G x\in H\backslash G x ∈ H \ G
g 1 c ~ s ( g 2 , g 3 ) c ~ s ( g 1 g 2 , g 3 ) − 1 c ~ s ( g 1 , g 2 g 3 ) c ~ s ( g 1 , g 2 ) − 1
^{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}
g 1 c s ( g 2 , g 3 ) c s ( g 1 g 2 , g 3 ) − 1 c s ( g 1 , g 2 g 3 ) c s ( g 1 , g 2 ) − 1 = ∏ x ∈ H \ G g 1 s ( x ) − 1 c ( s ( x ) g 2 s ( x ⋅ g 2 ) − 1 , s ( x ⋅ g 2 ) g 3 s ( x ⋅ g 2 g 3 ) − 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})
= x ∈ H \ G ∏ g 1 s ( x ) − 1 c ( s ( x ) g 2 s ( x ⋅ g 2 ) − 1 , s ( x ⋅ g 2 ) g 3 s ( x ⋅ g 2 g 3 ) − 1 ) × ∏ x ∈ H \ G s ( x ) − 1 ( c ( h 1 ( x ) h 2 ( x ) , h 3 ( x ) ) − 1
\times \prod_{x\in H\backslash G}^{s(x)^{-1}}(c(h_1(x)h_2(x), h_3(x))^{-1}
× x ∈ H \ G ∏ s ( x ) − 1 ( c ( h 1 ( x ) h 2 ( x ) , h 3 ( x ) ) − 1 × c ( h 1 ( x ) , h 2 ( x ) h 3 ( x ) ) c ( h 1 ( x ) , h 2 ( x ) ) − 1 )
\times c(h_1(x), h_2(x)h_3(x))c(h_1(x), h_2(x))^{-1})
× c ( h 1 ( x ) , h 2 ( x ) h 3 ( x )) c ( h 1 ( x ) , h 2 ( x ) ) − 1 ) = ∏ x ∈ H \ G s ( x ⋅ g 1 − 1 ) − 1 h 1 ( x ⋅ g 1 − 1 ) c ( h 2 ( x ⋅ g 1 − 1 ) , h 3 ( x ⋅ g 1 − 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}))
= x ∈ H \ G ∏ s ( x ⋅ g 1 − 1 ) − 1 h 1 ( x ⋅ g 1 − 1 ) c ( h 2 ( x ⋅ g 1 − 1 ) , h 3 ( x ⋅ g 1 − 1 )) × ∏ x ∈ H \ G s ( x ) − 1 h 1 ( x ) c ( h 2 ( x ) , h 3 ( x ) ) − 1
\times \prod_{x\in H\backslash G}^{s(x)^{-1}h_1(x)}c(h_2(x), h_3(x))^{-1}
× x ∈ H \ G ∏ s ( x ) − 1 h 1 ( x ) c ( h 2 ( x ) , h 3 ( x ) ) − 1 其中第二个等式由 c c c
如果c c c c ~ s \widetilde{c}_s c s
令t : H → F t: H\rightarrow F t : H → F c = ∂ t c=\partial t c = ∂ t t ~ s : G → F \widetilde{t}_s: G\rightarrow F t s : G → F
t ~ s ( g ) = ∏ x ∈ H \ G s ( x ) − 1 t ( s ( x ) g s ( x ⋅ g ) − 1 )
\widetilde{t}_s(g) =\prod_{x\in H\backslash G}^{s(x)^{-1}}t(s(x)gs(x\cdot g)^{-1})
t s ( g ) = x ∈ H \ G ∏ s ( x ) − 1 t ( s ( x ) g s ( x ⋅ g ) − 1 ) g ∈ G g\in G g ∈ G c ~ s = ∂ t ~ s \widetilde{c}_s=\partial \widetilde{t}_s c s = ∂ t s c ~ s ( g 1 , g 2 ) = ∏ x ∈ H \ G s ( x ) − 1 s ( x ) g 1 s ( x ⋅ g 1 ) − 1 t ( s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) \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}) c s ( g 1 , g 2 ) = ∏ x ∈ H \ G s ( x ) − 1 s ( x ) g 1 s ( x ⋅ g 1 ) − 1 t ( s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 )
× ∏ x ∈ H \ G s ( x ) − 1 ( t ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 t ( s ( x ) g 1 s ( x ⋅ g 1 ) − 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}))
× x ∈ H \ G ∏ s ( x ) − 1 ( t ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 t ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 )) = ∂ t ~ s ( g 1 , g 2 )
=\partial \widetilde{t}_s(g_1, g_2)
= ∂ t s ( g 1 , g 2 ) g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G
令 c c c H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c ~ s \widetilde{c}_s c s s s s
引用了 1 引用了 § 16. 。 布劳尔群中的其他描述
令s ′ s' s ′ H \ G H\backslash G H \ G h : H \ G → H h: H\backslash G\rightarrow H h : H \ G → H
s ′ ( x ) = h ( x ) s ( x )
s'(x) =h(x)s(x)
s ′ ( x ) = h ( x ) s ( x ) x ∈ H \ G x\in H\backslash G x ∈ H \ G c ~ s ′ \widetilde{c}_{s'} c s ′
c ~ s ′ ( g 1 , g 2 )
\widetilde{c}_{s'}(g_1, g_2)
c s ′ ( g 1 , g 2 ) = ∏ x ∈ H \ G s ( x ) − 1 h ( x ) − 1 c ( h ( x ) s ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 , s ′ ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 )
=\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})
= x ∈ H \ G ∏ s ( x ) − 1 h ( x ) − 1 c ( h ( x ) s ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 , s ′ ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 h ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 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})
= x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 h ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) s ( x ) − 1 ( − 1 ′ ′ − 1 − 1 s(x)^{-1}(-1''-1-1 s ( x ) − 1 ( − 1 ′′ − 1 − 1 × ∏ x ∈ H \ G c ( h ( x ) , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) )
\times \prod_{x\in H\backslash G}c(h(x), s(x)g_1g_2s(x\cdot g_1g_2))
× x ∈ H \ G ∏ c ( h ( x ) , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 )) × c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 ) )
\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))
× c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 )) = ∏ x ∈ H \ G g 1 s ( x ⋅ g 1 ) − 1 c ( h ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 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}) = ∏ x ∈ H \ G g 1 s ( x ⋅ g 1 ) − 1 c ( h ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) × ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 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})
× x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) s ( x ) − 1 − 1 − 1 − 1 s(x)^{-1}-1-1-1 s ( x ) − 1 − 1 − 1 − 1 × ∏ x ∈ H \ G c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 ) ) \times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1)) × ∏ x ∈ H \ G c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 )) × ∏ x ∈ H \ G s ( x ) − 1 ( c ( h ( x ) − 1 , s ′ ( x ) g 1 g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) − 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}
× x ∈ H \ G ∏ s ( x ) − 1 ( c ( h ( x ) − 1 , s ′ ( x ) g 1 g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) − 1 × c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 ) )
\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))
× c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 )) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 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}) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 ) s ( x ) − 1 − 1 − 1 − 1 s(x)^{-1}-1-1-1 s ( x ) − 1 − 1 − 1 − 1 × ∏ x ∈ H \ G c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 ) ) \times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1)) × ∏ x ∈ H \ G c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 )) × ∏ x ∈ H \ G g 1 s ( x ) − 1 c ( h ( x ) − 1 , h ( x ) s ( x ) g 2 s ′ ( x ⋅ g 2 ) − 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}) × ∏ x ∈ H \ G g 1 s ( x ) − 1 c ( h ( x ) − 1 , h ( x ) s ( x ) g 2 s ′ ( x ⋅ g 2 ) − 1 ) s ( x ) − 1 ( − 1 ′ ′ − 1 − 1 s(x)^{-1}(-1''-1-1 s ( x ) − 1 ( − 1 ′′ − 1 − 1 × ∏ x ∈ H \ G c ( h ( x ) , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) )
\times \prod_{x\in H\backslash G}c(h(x), s(x)g_1g_2s(x\cdot g_1g_2))
× x ∈ H \ G ∏ c ( h ( x ) , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 )) × c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 ) )
\times c(h(x)^{-1}, s'(x)g_1s'(x\cdot g_1)^{-1}))
× c ( h ( x ) − 1 , s ′ ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 )) 对于所有g 1 , g 2 ∈ G g_1, g_2\in G g 1 , g 2 ∈ G
h ( x ) − 1 , h ( x ) s ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 h(x)^{-1},h(x)s(x)g_1s'(x\cdot g_1)^{-1} h ( x ) − 1 , h ( x ) s ( x ) g 1 s ′ ( x ⋅ g 1 ) − 1 s ′ ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 s'(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1} s ′ ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1
第二个是通过应用余循环关系到元素获得
s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) − 1 s(x)g_1s(x\cdot g_1)^{-1},h(x\cdot g_1)^{-1} s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) − 1 h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 h(x\cdot g_1)s(x\cdot g_1)g_2s'(x\cdot g_1g_2)^{-1} h ( x ⋅ g 1 ) s ( x ⋅ g 1 ) g 2 s ′ ( x ⋅ g 1 g 2 ) − 1 x ↦ x ⋅ g 1 x\mapsto x\cdot g_1 x ↦ x ⋅ g 1 H \ G H\backslash G H \ G
获得的表达式对应的最后两行为 2-coboundary。我们发现c ~ s ′ \widetilde{c}_{s'} c s ′ H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) ( g 1 , g 2 ) ∈ G 2 (g_1, g_2)\in G^2 ( g 1 , g 2 ) ∈ G 2
∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 h ( x ⋅ g 1 g 2 ) − 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}h(x\cdot g_1g_2)^{-1})
x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 h ( x ⋅ g 1 g 2 ) − 1 ) s ( x ) − 1 − 1 − 1 − 1 s(x)^{-1}-1-1-1 s ( x ) − 1 − 1 − 1 − 1 × ∏ x ∈ H \ G c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 ) )
\times \prod_{x\in H\backslash G}c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1))
× x ∈ H \ G ∏ c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 )) = ∏ x ∈ H \ G g 1 s ( x ⋅ g 1 ) − 1 c ( s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 2 ) − 1 ) − 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} = ∏ x ∈ H \ G g 1 s ( x ⋅ g 1 ) − 1 c ( s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 2 ) − 1 ) − 1 × ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 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})
× x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) × ∏ x ∈ H \ G s ( x ) − 1 ( c ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 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})
× x ∈ H \ G ∏ s ( x ) − 1 ( c ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 2 ) − 1 ) − 1 − 1 − 1 ) -1-1-1) − 1 − 1 − 1 ) × c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 ) ) \times c(s(x)g_1s(x\cdot g_1), h(x\cdot g_1)) × c ( s ( x ) g 1 s ( x ⋅ g 1 ) , h ( x ⋅ g 1 )) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 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}) = ∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) g s ( x ) − 1 − 1 − 1 − 1 gs(x)^{-1}-1-1-1 g s ( x ) − 1 − 1 − 1 − 1 × ∏ x ∈ H \ G 1 c ( s ( x ) g 2 s ( x ⋅ g 2 ) , h ( x ⋅ g 2 ) )
\times \prod_{x\in H\backslash G^1}c(s(x)g_2s(x\cdot g_2), h(x\cdot g_2))
× x ∈ H \ G 1 ∏ c ( s ( x ) g 2 s ( x ⋅ g 2 ) , h ( x ⋅ g 2 )) × ∏ x ∈ H \ G s ( x ) − 1 ( c ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 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})
× x ∈ H \ G ∏ s ( x ) − 1 ( c ( s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 , h ( x ⋅ g 1 g 2 ) − 1 ) × c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) − 1 ) − 1 )
\times c(s(x)g_1s(x\cdot g_1)^{-1}, h(x\cdot g_1)^{-1})^{-1})
× c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , h ( x ⋅ g 1 ) − 1 ) − 1 ) 其中第一个等式源自应用于元素的余循环关系
s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 s(x)g_1s(x\cdot g_1)^{-1},s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1} s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 h ( x ⋅ g 1 g 2 ) − 1 h(x\cdot g_1g_2)^{-1} h ( x ⋅ g 1 g 2 ) − 1
获得的表达式对应的最后两行为 2-coboundary。我们发现c ~ s ′ \widetilde{c}_{s'} c s ′ c ~ s \widetilde{c}_s c s
我们构造了从群H 2 ( H , F ) H^2(H,F) H 2 ( H , F ) H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) H G ^G_H H G
令 H 为有限指标的 G 的子群。 自同态 CorH G ∘ ^G_H\circ H G ∘ H G ^G_H H G H 2 ( G , F ) H^2(G,F) H 2 ( G , F )
引用了 1 · 被以下引用 1 引用了 § 16. 。 布劳尔群中的其他描述
被以下引用 定理 6
令α \alpha α H 2 ( G , F ) H^2(G,F) H 2 ( G , F ) c c c α \alpha α Z 2 ( G , F ) Z^2(G,F) Z 2 ( G , F ) H G ∘ ^G_H\circ H G ∘ H G ( α ) ^G_H(\alpha ) H G ( α ) ( g 1 , g 2 ) ∈ G 2 (g_1, g_2)\in G^2 ( g 1 , g 2 ) ∈ G 2
∏ x ∈ H \ G s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 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})
x ∈ H \ G ∏ s ( x ) − 1 c ( s ( x ) g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) = ∏ x ∈ H \ G c ( g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 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})
= x ∈ H \ G ∏ c ( g 1 s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) × ∏ x ∈ H \ G ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 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}))
× x ∈ H \ G ∏ ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 1 ) − 1 )) = ∏ x ∈ H \ G g 1 c ( s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 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})
= x ∈ H \ G ∏ g 1 c ( s ( x ⋅ g 1 ) − 1 , s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 ) × ∏ x ∈ H \ G c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 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}
× x ∈ H \ G ∏ c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1 × ∏ x ∈ H \ G ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 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}))
× x ∈ H \ G ∏ ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 1 ) − 1 )) = ∏ x ∈ H \ G c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 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} = ∏ x ∈ H \ G c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1 × ∏ x ∈ H \ G g 1 c ( s ( x ) − 1 , s ( x ) g 2 s ( x g 2 ) − 1 )
\times \prod_{x\in H\backslash G}^{g_1}c(s(x)^{-1}, s(x)g_2s(xg_2)^{-1})
× x ∈ H \ G ∏ g 1 c ( s ( x ) − 1 , s ( x ) g 2 s ( x g 2 ) − 1 ) × ∏ x ∈ H \ G ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 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}))
× x ∈ H \ G ∏ ( c ( s ( x ) − 1 , s ( x ) g 1 g 2 s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 1 ) − 1 )) 第一个等式来自余循环关系（VIII，p. 295，公式（7））应用于元素
s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 1 ) − 1 s(x)^{-1},s(x)g_1s(x\cdot g_1)^{-1} s ( x ) − 1 , s ( x ) g 1 s ( x ⋅ g 1 ) − 1 s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1} s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1
第二个由应用余循环关系到元素获得
g 1 , s ( x ⋅ g 1 ) − 1 g_1,s(x\cdot g_1)^{-1} g 1 , s ( x ⋅ g 1 ) − 1 s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1 s(x\cdot g_1)g_2s(x\cdot g_1g_2)^{-1} s ( x ⋅ g 1 ) g 2 s ( x ⋅ g 1 g 2 ) − 1
通过删除余界，我们得到 CorH G ∘ ^G_H\circ H G ∘ H G ( α ) ^G_H(\alpha ) H G ( α ) ( g 1 , g 2 ) ∈ G 2 (g_1, g_2)\in G^2 ( g 1 , g 2 ) ∈ G 2
∏ x ∈ H \ G c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 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}
x ∈ H \ G ∏ c ( g 1 , g 2 s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1 = ∏ x ∈ H \ G g 1 c ( g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( g 1 g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , g 2 )
=\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)
= x ∈ H \ G ∏ g 1 c ( g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) − 1 c ( g 1 g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , g 2 ) × ∏ x ∈ H \ G c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1
\times \prod_{x\in H\backslash G}c(g_1, s(x\cdot g_1)^{-1})^{-1}
× x ∈ H \ G ∏ c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1 = ∏ x ∈ H \ G g 1 c ( g 2 , s ( x ⋅ g 2 ) − 1 ) − 1 c ( g 1 g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ 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}
= x ∈ H \ G ∏ g 1 c ( g 2 , s ( x ⋅ g 2 ) − 1 ) − 1 c ( g 1 g 2 , s ( x ⋅ g 1 g 2 ) − 1 ) c ( g 1 , s ( x ⋅ g 1 ) − 1 ) − 1 × c ( g 1 , g 2 ) ( G : H )
\times c(g_1, g_2)^{(G:H)}
× c ( g 1 , g 2 ) ( G : H ) 这证明了结果。
7. 伽罗瓦代数
令 K 为交换域。如果E是K-代数，那么我们用AutK ( E ) _K(E) K ( E ) K ( E ) _K(E) K ( E ) / K /K / K
令 G 为群。 A ( K , G ) A (K,G) A ( K , G ) λ : G → \lambda : G\rightarrow λ : G → K ( E ) _K(E) K ( E ) λ \lambda λ
( ( ∑ g ∈ G μ g g ) x = ∑ g ∈ G μ g λ ( g ) . x (14)
((\sum_{g\in G}\mu_gg)x=\sum_{g\in G}\mu_g\lambda (g).x \tag{14}
(( g ∈ G ∑ μ g g ) x = g ∈ G ∑ μ g λ ( g ) . x ( 14 ) 对于 K[G] 的每 x ∈ L x\in L x ∈ L ( μ g ) g ∈ G (\mu_g)_{g\in G} ( μ g ) g ∈ G ( K , G ) (K,G) ( K , G )
对于( K (K ( K ( E i ) i ∈ I (E_i)_{i\in I} ( E i ) i ∈ I ∏ i ∈ I E i \prod_{i\in I}E_i ∏ i ∈ I E i ( K (K ( K
如果E是( K (K ( K E G E^G E G
令 E 为 ( K (K ( K λ : G → \lambda : G\rightarrow λ : G → K ( E ) _K(E) K ( E ) K ′ K' K ′ g ∈ G g\in G g ∈ G λ ′ ( g ) \lambda '(g) λ ′ ( g ) K ′ K' K ′ L ( K ′ ) L_{(K')} L ( K ′ ) K ′ ⊗ λ ( g ) _{K'}\otimes \lambda (g) K ′ ⊗ λ ( g ) λ ′ : g ↦ λ ′ ( g ) \lambda ':g\mapsto \lambda '(g) λ ′ : g ↦ λ ′ ( g ) L ( K ′ ) L_{(K')} L ( K ′ ) ( K ′ (K' ( K ′
给定群 H和H-集合X和Y，我们用从X到Y的H-集合态射的F H ( X , Y ) \mathscr{F}_H(X,Y) F H ( X , Y ) f : X → Y f: X\rightarrow Y f : X → Y h ∈ H h\in H h ∈ H x ∈ X x\in X x ∈ X f ( h x ) = h f ( x ) f(hx) =hf(x) f ( h x ) = h f ( x )
令G为群且单位元素e e e ( K (K ( K ( K , G ) (K,G) ( K , G ) H G ^G_H H G F H ( G , E ) \mathscr{F}_H(G,E) F H ( G , E ) λ \lambda λ K _K K H G ( E ) ^G_H(E) H G ( E )
( λ ( g ) ⋅ f ) ( g ′ ) = f ( g ′ g ) (15)
(\lambda (g)\cdot f)(g') =f(g'g) \tag{15}
( λ ( g ) ⋅ f ) ( g ′ ) = f ( g ′ g ) ( 15 ) 对于f ∈ f\in f ∈ H G ( E ) ^G_H(E) H G ( E ) g , g ′ ∈ G g, g'\in G g , g ′ ∈ G
a) 令 S 为 G 的子集，使得 G 的每个元素都可以唯一地写为 h s hs h s h ∈ H h\in H h ∈ H s ∈ S s\in S s ∈ S f f f ( f ( s ) ) s ∈ S (f(s))_{s\in S} ( f ( s ) ) s ∈ S H G ( E ) ^G_H(E) H G ( E ) F ( S , E ) \mathscr{F}(S,E) F ( S , E )
被以下引用 3 被以下引用 引理 9 , 命题 9 , 定理 2
b) 代数 CoindH G ( E ) ^G_H(E) H G ( E )
[硬币H G ( E ) : K ] = ( G : H ) [ E : K ] ^G_H(E) : K] = (G : H)[E : K] H G ( E ) : K ] = ( G : H ) [ E : K ]
c) 设E H E^H E H H G ( E ) G ^G_H(E)^G H G ( E ) G H G ( E ) ^G_H(E) H G ( E ) f ↦ f ( e ) f\mapsto f(e) f ↦ f ( e ) H G ( E ) ^G_H(E) H G ( E ) H G ( E ) G ^G_H(E)^G H G ( E ) G E H E^H E H
断言 a) 由定义推出并隐含 b)。由(15)可知，从G到E的映射是G下CoindH G ( E ) ^G_H(E) H G ( E )
[ L σ : K ] = [L_{\sigma}: K] = [ L σ : K ] = σ \sigma σ [ L / m : K ] [L/\mathfrak{m}: K] [ L / m : K ]
b) 映射 π : x ↦ ( π σ ( x ) ) σ ∈ O \pi :x\mapsto (\pi_{\sigma}(x))_{\sigma\in\mathscr{O}} π : x ↦ ( π σ ( x ) ) σ ∈ O ∏ σ ∈ O L σ \prod_{\sigma\in\mathscr{O}}L_{\sigma} ∏ σ ∈ O L σ ( K , G ) (K,G) ( K , G )
c) 分别用L G ( L^G( L G ( L σ L_{\sigma} L σ L σ G ) L^G_{\sigma}) L σ G ) L G L^G L G ∏ σ ∈ O L σ G \prod_{\sigma\in\mathscr{O}}L^G_{\sigma} ∏ σ ∈ O L σ G π \pi π
由于代数 L 减少，并且有限次数的减少，L 最大理想的交减少到 0（VIII，p. 173，推论 2）。 此外，如果m \mathfrak{m} m m ′ \mathfrak{m}' m ′ m + m ′ = L \mathfrak{m}+\mathfrak{m}'= L m + m ′ = L ∏ m ∈ S L / m \prod_{\mathfrak{m}\in\mathscr{S}}L/\mathfrak{m} ∏ m ∈ S L / m σ ∈ O \sigma \in \mathscr{O} σ ∈ O L / a σ L/\mathfrak{a}_{\sigma} L / a σ ∏ m ∈ σ L / m \prod_{\mathfrak{m}\in\sigma}L/\mathfrak{m} ∏ m ∈ σ L / m O \mathscr{O} O S \mathscr{S} S
现在让我们修复 σ \sigma σ σ ∈ O \sigma \in \mathscr{O} σ ∈ O m \mathfrak{m} m F m F_{\mathfrak{m}} F m G m G ( L / m ) ^G_{G_{\mathfrak{m}}}(L/\mathfrak{m}) G m G ( L / m ) F m F_{\mathfrak{m}} F m λ F m \lambda_{F_{\mathfrak{m}}} λ F m x ∈ L x\in L x ∈ L L / m L/\mathfrak{m} L / m x ‾ \overline{x} x g g g π m ( g x ) \pi_{\mathfrak{m}}(gx) π m ( g x ) L / m L/\mathfrak{m} L / m G m G_{\mathfrak{m}} G m x ‾ \overline{x} x F m F_{\mathfrak{m}} F m F m F_{\mathfrak{m}} F m u : x ↦ x ‾ u:x\mapsto \overline{x} u : x ↦ x g ∈ G g\in G g ∈ G λ F m ( g ) ∘ u = u ∘ λ ( g ) \lambda_{F_{\mathfrak{m}}}(g)\circ u=u\circ \lambda (g) λ F m ( g ) ∘ u = u ∘ λ ( g ) u u u ( K (K ( K
态射 u u u a σ \mathfrak{a}_{\sigma} a σ u u u ( K , G ) (K,G) ( K , G ) L σ L_{\sigma} L σ F m F_{\mathfrak{m}} F m
引用了 2 · 被以下引用 1 引用了 引理 8 , 引理 7
被以下引用 定理 2
Since the kernel of π m \pi_{\mathfrak{m}} π m m \mathfrak{m} m u u u ⋂ g ∈ G g − 1 . m = a σ \bigcap_{g\in G}g^{-1}.\mathfrak{m}=\mathfrak{a}_{\sigma} ⋂ g ∈ G g − 1 . m = a σ u u u L σ = L_{\sigma}= L σ = L / a σ L/\mathfrak{a}_{\sigma} L / a σ F m F_{\mathfrak{m}} F m g ⋅ m g\cdot \mathfrak{m} g ⋅ m
[ L σ : K ] = [L_{\sigma}: K] = [ L σ : K ] = σ \sigma σ . [ L / m : K ] .[L/\mathfrak{m}: K] . [ L / m : K ]
此外，到引理 7, b)，我们有
[ F m : K ] = ( G : G m ) [ L / m : K ]
[F_{\mathfrak{m}}: K] = (G : G_{\mathfrak{m}})[L/\mathfrak{m}: K]
[ F m : K ] = ( G : G m ) [ L / m : K ] 由于我们有 Card(σ \sigma σ = ( G : G m ) = (G : G_{\mathfrak{m}}) = ( G : G m ) [ L σ : K ] = [L_{\sigma}: K] = [ L σ : K ] = [ F m : K ] [F_{\mathfrak{m}}: K] [ F m : K ]
此外，我们现在假设从 G 到 AutK ( L ) _K(L) K ( L ) λ \lambda λ ξ ↦ ξ ⋅ 1 \xi \mapsto \xi \cdot 1 ξ ↦ ξ ⋅ 1 F ( G , Ω ) \mathscr{F}(G,\Omega ) F ( G , Ω ) ( Ω (\Omega ( Ω { e } G ( Ω ) ^G_{\{e\}}(\Omega ) { e } G ( Ω ) H \mathscr{H} H H \mathscr{H} H ( g , χ ) ↦ χ ∘ λ ( g ) (g, \chi )\mapsto \chi \circ \lambda (g) ( g , χ ) ↦ χ ∘ λ ( g ) F ( H , Ω ) \mathscr{F}(\mathscr{H},\Omega ) F ( H , Ω ) H \mathscr{H} H ( Ω (\Omega ( Ω
令 L 为 ( K , G ) (K,G) ( K , G ) ψ \psi ψ L ( Ω ) L_{(\Omega )} L ( Ω ) F ( H , Ω ) \mathscr{F}(\mathscr{H},\Omega ) F ( H , Ω )
ψ ( ξ ⊗ x ) = ( ξ χ ( x ) ) χ ∈ H
\psi (\xi \otimes x) = (\xi \chi (x))_{\chi\in\mathscr{H}}
ψ ( ξ ⊗ x ) = ( ξ χ ( x ) ) χ ∈ H 引用了 2 · 被以下引用 1 引用了 命题 1, V § 6 , 命题 2, V § 6
被以下引用 定理 2
是( K , G ) (K,G) ( K , G )
由于L是étale，映射ψ \psi ψ
ψ \psi ψ ⊗ λ ( g ) \otimes \lambda (g) ⊗ λ ( g ) ( ξ ⊗ x ) (\xi \otimes x) ( ξ ⊗ x ) = ( ξ ( χ ∘ λ ( g ) ) ( x ) ) χ ∈ H = (\xi (\chi \circ \lambda (g))(x))_{\chi\in\mathscr{H}} = ( ξ ( χ ∘ λ ( g )) ( x ) ) χ ∈ H
对于ξ ∈ Ω , x ∈ L \xi \in \Omega ,x\in L ξ ∈ Ω , x ∈ L g ∈ G g\in G g ∈ G ψ \psi ψ ( Ω (\Omega ( Ω
设G为有限群，并令L为有限次数的交换K-代数，赋予G的作用，由单同态λ \lambda λ K ( L ) _K(L) K ( L )
引用了 8 · 被以下引用 9 引用了 定义 7, I § 5 , 引理 7 , 定理 3, V § 10 , 命题 4, V § 6 , 引理 10 , 定理 3, VIII § 2 , 引理 8 , 引理 9
被以下引用 定义 1 , 引理 12 , 例 1 , 例 2 , 注 2 , 注 3 , 注 5 , 命题 10 , 命题 9
(i) 从 H 到 Gal(E/ K /K / K H G ( E ) ^G_H(E) H G ( E ) ( K , G ) (K,G) ( K , G )
(ii) 代数 L 是étale，H \mathscr{H} H
(iii) 存在( Ω , G ) (\Omega ,G) ( Ω , G ) ψ : L ( Ω ) → F ( G , Ω ) \psi : L_{(\Omega )}\rightarrow \mathscr{F}(G,\Omega ) ψ : L ( Ω ) → F ( G , Ω ) g ∈ G g\in G g ∈ G Ω G \Omega^G Ω G ψ ∘ λ ( g ) ( Ω ) ∘ ψ − 1 \psi \circ \lambda (g)_{(\Omega )}\circ \psi^{-1} ψ ∘ λ ( g ) ( Ω ) ∘ ψ − 1
( x h ) h ∈ G ⟼ ( x h g ) h ∈ G
(x_h)_{h\in G}\longmapsto (x_{hg})_{h\in G}
( x h ) h ∈ G ⟼ ( x h g ) h ∈ G (iv) 代数L减少，L是秩1的自由K[G]模。
(v) 代数 L 约简，我们有 Card(G) = [L : K]，并且 K 是 G 的作用下元素 L 不变的子环。
(vi) 减少代数 L，群 G 传递作用于 L 最大理想集合，并且对于 G 中L 的每个最大理想 m \mathfrak{m} m m \mathfrak{m} m G m G_{\mathfrak{m}} G m L / m L/\mathfrak{m} L / m
(i)⇒ \Rightarrow ⇒ K ( E ) _K(E) K ( E ) τ \tau τ H G ( E ) ^G_H(E) H G ( E ) F ( S , E ) \mathscr{F}(S,E) F ( S , E ) λ F \lambda_F λ F ψ \psi ψ χ 0 \chi_0 χ 0 f ↦ ψ ( f ( e ) ) f\mapsto \psi (f(e)) f ↦ ψ ( f ( e )) g ∈ G g\in G g ∈ G χ 0 ∘ λ F ( g ) = χ 0 \chi_0\circ \lambda_F(g) =\chi_0 χ 0 ∘ λ F ( g ) = χ 0 ψ \psi ψ f ∈ F f\in F f ∈ F f ( g ) = f ( e ) f(g) =f(e) f ( g ) = f ( e ) g ∈ H g\in H g ∈ H
f ( h ) = τ ( h ) ⋅ f ( e ) (18)
f(h) =\tau (h)\cdot f(e) \tag{18}
f ( h ) = τ ( h ) ⋅ f ( e ) ( 18 ) 对于每个 h ∈ H h\in H h ∈ H g = e g=e g = e
[F : K] = (G : H)[E : K] = (G : H) 卡(H) = 卡(G)。
从 F 到 Ω 的 K-同态的集合 K \mathscr{K} K K \mathscr{K} K = = = χ 0 \chi_0 χ 0 { e } \{e\} { e } K \mathscr{K} K
(ii)⇒ \Rightarrow ⇒ H \mathscr{H} H ( Ω (\Omega ( Ω L ( Ω ) L_{(\Omega )} L ( Ω ) F ( H , Ω ) \mathscr{F}(\mathscr{H},\Omega ) F ( H , Ω ) H \mathscr{H} H ( Ω (\Omega ( Ω F ( H , Ω ) \mathscr{F}(\mathscr{H},\Omega ) F ( H , Ω ) F ( G , Ω ) \mathscr{F}(G,\Omega ) F ( G , Ω )
(iii)⇒ \Rightarrow ⇒ L ( Ω ) L_{(\Omega )} L ( Ω ) K [ G ] ( Ω ) K[G]_{(\Omega )} K [ G ] ( Ω )
蕴涵(iv)⇒ \Rightarrow ⇒
让我们证明蕴涵 (v)⇒ \Rightarrow ⇒ S \mathscr{S} S m \mathfrak{m} m S \mathscr{S} S ⋂ n ∈ S n = { 0 } \bigcap_{\mathfrak{n}\in\mathscr{S}}\mathfrak{n}=\{0\} ⋂ n ∈ S n = { 0 } G m G ( L / m ) ^G_{G_{\mathfrak{m}}}(L/\mathfrak{m}) G m G ( L / m ) L / m L/\mathfrak{m} L / m G m G_{\mathfrak{m}} G m G m G_{\mathfrak{m}} G m / m /\mathfrak{m} / m / K /K / K λ m \lambda_{\mathfrak{m}} λ m
卡(G)= [ L : K ] = ( G : G m ) [ L / m : K ] = [L : K] = (G : G_{\mathfrak{m}})[L/\mathfrak{m}: K] = [ L : K ] = ( G : G m ) [ L / m : K ] m _{\mathfrak{m}} m = [ L / m : K ] = [L/\mathfrak{m}: K] = [ L / m : K ] λ m \lambda_{\mathfrak{m}} λ m
仍有待证明蕴涵(vi)⇒ \Rightarrow ⇒ m \mathfrak{m} m ( K (K ( K G m G ( L / m ) ^G_{G_{\mathfrak{m}}}(L/\mathfrak{m}) G m G ( L / m ) G m G_{\mathfrak{m}} G m L / m L/\mathfrak{m} L / m λ m \lambda_{\mathfrak{m}} λ m G m G_{\mathfrak{m}} G m / m /\mathfrak{m} / m / K /K / K
设 G 为有限群，并令 L 为有限次数的非零的交换代数，赋予 K 赋予 ( K , G ) (K,G) ( K , G ) K ( L ) _K(L) K ( L )
引用了 1 引用了 定理 2
设 n n n n n n μ n \mu_n μ n n n n n n n d d d d d d μ d \mu_d μ d d d d a a a K [ X ] / ( X n − a ) K[X]/(X^n-a) K [ X ] / ( X n − a ) x x x ( 1 , x , . . . , x n − 1 ) (1, x, . . . , x^{n-1}) ( 1 , x , ... , x n − 1 ) x n = a x^n=a x n = a X n − a X^n-a X n − a n X n − 1 nX^{n-1} n X n − 1 μ n \mu_n μ n ζ \zeta ζ P ( X ) ↦ P ( ζ X ) P(X)\mapsto P(\zeta X) P ( X ) ↦ P ( ζ X ) λ ( ζ ) \lambda (\zeta ) λ ( ζ ) ( ζ X ) n − a = X n − a (\zeta X)^n-a= X^n-a ( ζ X ) n − a = X n − a
λ ( ζ ) x i = ζ i x i (19)
\lambda (\zeta )x^i=\zeta^ix^i \tag{19}
λ ( ζ ) x i = ζ i x i ( 19 ) 对于0 ⩽ i < n 0\leqslant i < n 0 ⩽ i < n μ n \mu_n μ n K _K K λ : ζ ↦ λ ( ζ ) \lambda :\zeta \mapsto \lambda (\zeta ) λ : ζ ↦ λ ( ζ ) λ ( μ n ) \lambda (\mu_n) λ ( μ n ) K ⋅ 1 K\cdot 1 K ⋅ 1 μ n \mu_n μ n n = [ L : K ] n= [L : K] n = [ L : K ] λ \lambda λ
令r r r a r a^r a r K ∗ n K^{*n} K ∗ n n n n K ∗ K^* K ∗ b b b a = b n / r a=b^{n/r} a = b n / r X r − b X^r-b X r − b X n − a = ∏ ζ ∈ μ n / r ( X r − ζ b ) X^n-a=\prod_{\zeta\in\mu_{n/r}}(X^r-\zeta b) X n − a = ∏ ζ ∈ μ n / r ( X r − ζ b ) K [ Y ] / ( Y r − b ) K[Y]/(Y^r-b) K [ Y ] / ( Y r − b ) y y y μ n / r \mu_{n/r} μ n / r / K /K / K θ \theta θ θ ( ξ ) ( y ) = ξ y \theta (\xi )(y) =\xi y θ ( ξ ) ( y ) = ξ y ( K , μ n (K, \mu_n ( K , μ n μ n / r μ n ( E ) ^{\mu_n}_{\mu_{n/r}}(E) μ n / r μ n ( E )
引用了 2 引用了 命题 3, V § 7 , 定理 2
现在假设域 K 具有特征 p ≠ 0 p\not= 0 p = 0 c c c f = X p − X − c f= X^p-X-c f = X p − X − c f ′ = − 1 f'=-1 f ′ = − 1 L = K [ X ] / ( f ) L = K[X]/(f) L = K [ X ] / ( f ) x x x x p = x + c x^p=x+c x p = x + c ( 1 , x , . . . , x p − 1 ) (1, x, . . . , x^{p-1}) ( 1 , x , ... , x p − 1 )
令 P 为 K 素数子域的加法群；它是阶p p p j j j j p = j j^p=j j p = j f ( X + j ) = f ( X ) f(X +j) =f(X) f ( X + j ) = f ( X ) γ ( j ) \gamma (j) γ ( j ) γ ( j ) ( x ) = x + j \gamma (j)(x) =x+j γ ( j ) ( x ) = x + j K ( L ) _K(L) K ( L ) γ \gamma γ
设 Ω 为 K 的代数闭扩张，并令 ξ \xi ξ f f f ξ p = ξ + c \xi^p=\xi +c ξ p = ξ + c
X p − X − c = ( X p − ξ p ) − ( X − ξ ) = ( X − ξ ) p − ( X − ξ ) = ∏ j ∈ P ( X − ξ − j )
X^p-X-c= (X^p-\xi^p)-(X-\xi ) = (X-\xi )^p-(X-\xi ) =\prod_{j\in P}(X-\xi -j)
X p − X − c = ( X p − ξ p ) − ( X − ξ ) = ( X − ξ ) p − ( X − ξ ) = j ∈ P ∏ ( X − ξ − j ) 由 V, §12, No. 1、p. 94、公式 (1)。对于P中的每个j j j χ j : L → Ω \chi_j: L\rightarrow \Omega χ j : L → Ω x x x ξ + j \xi +j ξ + j χ j \chi_j χ j χ j = χ 0 ∘ γ ( j ) \chi_j=\chi_0\circ \gamma (j) χ j = χ 0 ∘ γ ( j ) γ \gamma γ
为了描述L的结构，我们必须区分两种情况：
a) 我们有 ξ ∉ K \xi \notin K ξ ∈ / K f ( X ) f(X) f ( X ) p p p γ \gamma γ / K /K / K
b) 我们有 ξ ∈ K \xi \in K ξ ∈ K ψ : y ↦ ( χ j ( y ) ) j ∈ P \psi :y\mapsto (\chi_j(y))_{j\in P} ψ : y ↦ ( χ j ( y ) ) j ∈ P K P K^P K P k ∈ P k\in P k ∈ P ψ ∘ γ ( k ) ∘ ψ − 1 \psi \circ \gamma (k)\circ \psi^{-1} ψ ∘ γ ( k ) ∘ ψ − 1 K P K^P K P ( x j ) j ∈ P ↦ ( x j + k ) j ∈ P (x_j)_{j\in P}\mapsto (x_{j+k})_{j\in P} ( x j ) j ∈ P ↦ ( x j + k ) j ∈ P
引用了 2 引用了 命题 3, V § 7 , 定理 2
8. 作用于伽罗瓦代数
设G为有限群，H为G的子群，E为域K上的伽罗瓦代数和群H。则通过从H到G的共归纳，从E推导出( K , G ) (K,G) ( K , G ) H G ( E ) ^G_H(E) H G ( E )
引用了 2 引用了 引理 7 , 定理 2
由于 E 在 K 上伽罗瓦代数，通过 VIII、p. 308 的定理 2 的性质 (v) 进行约减，我们有 Card(H) = [E : K]，并且 K 是 E 中 H 的不变量的环。但是通过引理 VIII、p. 305、代数 F = CoindH G ( E ) ^G_H(E) H G ( E )
[F : K] = (G : H)[E : K] = (G : H) 卡(H) = 卡(G)，
K 是 F 中 G 的不变量的环。因此，根据定理 2, (v) 中给出的标准，F 是伽罗瓦代数。
设G为有限群。令 L 为域 K 上的群 G 上的伽罗瓦代数，并令 K ′ K' K ′ ( K ′ , G ) (K',G) ( K ′ , G ) L ( K ′ ) L_{(K')} L ( K ′ ) K ′ K' K ′
引用了 2 · 被以下引用 1 引用了 推论 2, V § 6 , 定理 2
被以下引用 注
我们通过注意到在定理2中使用性质(v)，如果K-代数L是étale，那么K ′ K' K ′ L ( K ′ ) L_{(K')} L ( K ′ ) [ L ( K ′ ) : K ′ ] = [ L : K ] [L_{(K')}: K'] = [L : K] [ L ( K ′ ) : K ′ ] = [ L : K ] L ( K ′ ) L_{(K')} L ( K ′ ) ( L G ) ( K ′ ) (L^G)_{(K')} ( L G ) ( K ′ ) L G L^G L G
令G 1 G_1 G 1 G 2 G_2 G 2 L 1 L_1 L 1 L 2 L_2 L 2 λ 1 : G 1 → \lambda_1: G_1\rightarrow λ 1 : G 1 → K ( L 1 ) _K(L_1) K ( L 1 ) λ 2 : G 2 → \lambda_2: G_2\rightarrow λ 2 : G 2 → K ( L 2 ) _K(L_2) K ( L 2 ) L = L 1 ⊗ K L 2 , G = G 1 × G 2 L = L_1\otimes_KL_2,G = G_1\times G_2 L = L 1 ⊗ K L 2 , G = G 1 × G 2 λ ( g 1 , g 2 ) = λ 1 ( g 1 ) ⊗ λ 2 ( g 2 ) \lambda (g_1, g_2) =\lambda_1(g_1)\otimes \lambda_2(g_2) λ ( g 1 , g 2 ) = λ 1 ( g 1 ) ⊗ λ 2 ( g 2 ) ( g 1 , g 2 ) ∈ G (g_1, g_2)\in G ( g 1 , g 2 ) ∈ G λ \lambda λ
引用了 2 · 被以下引用 1 引用了 推论 1, V § 6 , 引理 11
被以下引用 注
我们将之前进行推理，并考虑以下因素：如果 L 1 L_1 L 1 L 2 L_2 L 2 L = L 1 ⊗ K L 2 L = L_1\otimes_KL_2 L = L 1 ⊗ K L 2
[ L : K ] = [ L 1 : K ] [ L 2 : K ] [L : K] = [L_1: K][L_2: K] [ L : K ] = [ L 1 : K ] [ L 2 : K ] 1 _1 1 2 _2 2
此外，如果L i G i L^{G_i}_i L i G i L i L_i L i G i G_i G i L 1 G 1 ⊗ K L 2 G 2 L^{G_1}_1\otimes_KL^{G_2}_2 L 1 G 1 ⊗ K L 2 G 2 L 1 ⊗ K L 2 L_1\otimes_KL_2 L 1 ⊗ K L 2 G 1 × G 2 G_1\times G_2 G 1 × G 2
令G 1 G_1 G 1 G 2 G_2 G 2 W 1 W_1 W 1 W 2 W_2 W 2 W 1 ( W_1( W 1 ( W 2 ) W_2) W 2 ) G 1 ( G_1( G 1 ( G 2 ) G_2) G 2 ) ρ 1 : G 1 → \rho_1: G_1\rightarrow ρ 1 : G 1 → K ( W 1 ) ( _K(W_1) ( K ( W 1 ) ( ρ 2 : G 2 → \rho_2: G_2\rightarrow ρ 2 : G 2 → K ( W 2 ) ) _K(W_2)) K ( W 2 ))
引用了 1 · 被以下引用 1 引用了 引理 1, VIII § 12
被以下引用 命题 11
ρ 1 ⊗ ρ 2 : G 1 × G 2 ⟶ \rho_1\otimes \rho_2: G_1\times G_2\longrightarrow ρ 1 ⊗ ρ 2 : G 1 × G 2 ⟶ K ( W 1 ⊗ K W 2 ) _K(W_1\otimes_KW_2) K ( W 1 ⊗ K W 2 )
对于g 1 ∈ G 1 , g 2 ∈ G 2 , w 1 ∈ W 1 g_1\in G_1,g_2\in G_2,w_1\in W_1 g 1 ∈ G 1 , g 2 ∈ G 2 , w 1 ∈ W 1 w 2 ∈ W 2 w_2\in W_2 w 2 ∈ W 2 ( ρ 1 ⊗ ρ 2 ) ( g 1 , g 2 ) ( w 1 ⊗ w 2 ) = ρ 1 ( g 1 ) ( w 1 ) ⊗ ρ 2 ( g 2 ) ( w 2 ) (\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) ( ρ 1 ⊗ ρ 2 ) ( g 1 , g 2 ) ( w 1 ⊗ w 2 ) = ρ 1 ( g 1 ) ( w 1 ) ⊗ ρ 2 ( g 2 ) ( w 2 ) W 1 G 1 ⊗ K W 2 G 2 W^{G_1}_1\otimes_KW^{G_2}_2 W 1 G 1 ⊗ K W 2 G 2 W 1 ⊗ K W 2 W_1\otimes_KW_2 W 1 ⊗ K W 2 W 1 G 1 ⊗ K W 2 G 2 W_1^{G_1}\otimes_KW^{G_2}_2 W 1 G 1 ⊗ K W 2 G 2 ( W 1 ⊗ K W 2 ) G 1 × G 2 (W_1\otimes_KW_2)^{G_1\times G_2} ( W 1 ⊗ K W 2 ) G 1 × G 2
这是从应用于 K[G]-模块 M 1 = M 2 = K M_1= M_2= K M 1 = M 2 = K G , N 1 = W 1 G, N_1= W_1 G , N 1 = W 1 N 2 = W 2 N_2= W_2 N 2 = W 2
9. 叉积
令K为域，令G为群，其单位元素我们用e e e λ \lambda λ g g g τ ( g ) \tau (g) τ ( g ) λ ( g ) \lambda (g) λ ( g ) L ∗ L^* L ∗
设E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ L ∗ L^* L ∗ L × Γ L\times \Gamma L × Γ L ∗ L^* L ∗
( β , γ ) . α = ( β α , ι ( α ) − 1 γ ) (20)
(\beta , \gamma ).\alpha = (\beta \alpha , \iota (\alpha )^{-1}\gamma ) \tag{20}
( β , γ ) . α = ( β α , ι ( α ) − 1 γ ) ( 20 ) 对于 α ∈ L ∗ , β ∈ L \alpha \in L^*,\beta \in L α ∈ L ∗ , β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ L × Γ L\times \Gamma L × Γ L ∗ L^* L ∗ ( β , γ ) (\beta , \gamma ) ( β , γ ) [ β ; γ ] [\beta ;\gamma ] [ β ; γ ]
[ β α ; γ ] = [ β ; ι ( α ) γ ] (21)
[\beta \alpha ;\gamma ] = [\beta ;\iota (\alpha )\gamma ] \tag{21}
[ β α ; γ ] = [ β ; ι ( α ) γ ] ( 21 ) 对于 α ∈ L ∗ , β ∈ L \alpha \in L^*,\beta \in L α ∈ L ∗ , β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ
给定 L 中的 β \beta β γ \gamma γ λ ∘ π ( γ ) \lambda \circ \pi (\gamma ) λ ∘ π ( γ ) β \beta β γ β ^{\gamma}\beta γ β
(22)γ ( γ ′ β ) = γ γ ′ β , γ ( β + β ′ ) = γ β + γ β ′ ^{\gamma}(^{\gamma'}\beta ) =^{\gamma \gamma'}\beta ,^{\gamma}(\beta +\beta ') =^{\gamma}\beta +^{\gamma}\beta ' γ ( γ ′ β ) = γ γ ′ β , γ ( β + β ′ ) = γ β + γ β ′ γ ( β β ′ ) = γ β γ β ′ ^{\gamma}(\beta \beta ') =^{\gamma}\beta^{\gamma}\beta ' γ ( β β ′ ) = γ β γ β ′
对于 Γ 中的 γ , γ ′ \gamma , \gamma ' γ , γ ′ β , β ′ \beta , \beta ' β , β ′
[ β ; γ ] [ β ′ ; γ ′ ] = [ β γ β ′ ; γ γ ′ ] (23)
[\beta ;\gamma ][\beta ';\gamma '] = [\beta^{\gamma}\beta ';\gamma \gamma '] \tag{23}
[ β ; γ ] [ β ′ ; γ ′ ] = [ β γ β ′ ; γ γ ′ ] ( 23 ) 事实上，如果我们分别将替换、β , γ , β ′ \beta ,\gamma ,\beta ' β , γ , β ′ γ ′ \gamma ' γ ′ β α , ι ( α ) − 1 γ , β ′ α ′ \beta \alpha ,\iota (\alpha )^{-1}\gamma ,\beta '\alpha ' β α , ι ( α ) − 1 γ , β ′ α ′ ι ( α ′ ) − 1 γ ′ \iota (\alpha ')^{-1}\gamma ' ι ( α ′ ) − 1 γ ′ L ∗ L^* L ∗ α , α ′ \alpha ,\alpha ' α , α ′ E \mathscr{E} E γ ∈ Γ \gamma \in \Gamma γ ∈ Γ α ∈ L ∗ \alpha \in L^* α ∈ L ∗ γ ι ( α ) = ι ( γ α ) γ \gamma \iota (\alpha ) =\iota (^{\gamma}\alpha )\gamma γ ι ( α ) = ι ( γ α ) γ [ 1 ; e ] [1;e] [ 1 ; e ]
由于π ∘ ι \pi \circ \iota π ∘ ι e e e π ~ \widetilde{\pi} π
π ~ ( [ β ; γ ] ) = π ( γ ) (24)
\widetilde{\pi}([\beta ;\gamma ]) =\pi (\gamma ) \tag{24}
π ([ β ; γ ]) = π ( γ ) ( 24 ) 对于 β ∈ L \beta \in L β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ
令g g g E g = π ~ − 1 ( g ) E_g=\widetilde{\pi}^{-1}(g) E g = π − 1 ( g ) γ 0 \gamma_0 γ 0 π − 1 ( g ) \pi^{-1}(g) π − 1 ( g ) β ↦ [ β ; γ 0 ] \beta \mapsto [\beta ;\gamma_0] β ↦ [ β ; γ 0 ] E g E_g E g E g E_g E g π − 1 ( g ) \pi^{-1}(g) π − 1 ( g ) ι ( α ) γ 0 \iota (\alpha )\gamma_0 ι ( α ) γ 0 α \alpha α L ∗ L^* L ∗
[ β ; ι ( α ) γ 0 ] = [ β α ; γ 0 ]
[\beta ;\iota (\alpha )\gamma_0] = [\beta \alpha ;\gamma_0]
[ β ; ι ( α ) γ 0 ] = [ β α ; γ 0 ] E g E_g E g γ 0 \gamma_0 γ 0
设g g g g ′ g' g ′ E g × E g ′ E_g\times E_{g'} E g × E g ′ E g g ′ E_{gg'} E g g ′ P = ⨁ g ∈ G E g P =\bigoplus_{g\in G}E_g P = ⨁ g ∈ G E g E g × E g ′ E_g\times E_{g'} E g × E g ′ E g g ′ E_{gg'} E g g ′ g g g g ′ g' g ′ E \mathscr{E} E A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] E e E_e E e [ 1 ; e ] [1;e] [ 1 ; e ]
集合
u ( β ) = [ β ; e ] (25)
u(\beta ) = [\beta ;e] \tag{25}
u ( β ) = [ β ; e ] ( 25 ) L 中的 β \beta β u : L → A [ E ; L ] u: L\rightarrow \mathbf{A}[\mathscr{E}; L] u : L → A [ E ; L ] γ ∈ Γ \gamma \in \Gamma γ ∈ Γ [ 1 ; γ ] [1;\gamma ] [ 1 ; γ ] A [ E , L ] \mathbf{A}[\mathscr{E},L] A [ E , L ] γ \gamma γ [ 1 ; γ ] [1;\gamma ] [ 1 ; γ ] v : Γ → A [ E , L ] ∗ v: \Gamma \rightarrow \mathbf{A}[\mathscr{E},L]^* v : Γ → A [ E , L ] ∗ u u u v v v
u ( α ) = v ( ι ( α ) ) (26)
u(\alpha ) =v(\iota (\alpha )) \tag{26}
u ( α ) = v ( ι ( α )) ( 26 ) u ( γ β ) = v ( γ ) u ( β ) v ( γ ) − 1 (27)
u(^{\gamma}\beta ) =v(\gamma )u(\beta )v(\gamma )^{-1} \tag{27}
u ( γ β ) = v ( γ ) u ( β ) v ( γ ) − 1 ( 27 ) [ β ; γ ] = u ( β ) v ( γ ) (28)
[\beta ;\gamma ] =u(\beta )v(\gamma ) \tag{28}
[ β ; γ ] = u ( β ) v ( γ ) ( 28 ) 对于 α ∈ L ∗ , β ∈ L \alpha \in L^*,\beta \in L α ∈ L ∗ , β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ
相反，我们有以下泛性质或代数 A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ]
设 B 为 K-代数、u ′ : L → B u': L\rightarrow B u ′ : L → B v ′ : Γ → B ∗ v': \Gamma \rightarrow B^* v ′ : Γ → B ∗
u ′ ( α ) = v ′ ( ι ( α ) ) (29)
u'(\alpha ) =v'(\iota (\alpha )) \tag{29}
u ′ ( α ) = v ′ ( ι ( α )) ( 29 ) u ′ ( γ β ) = v ′ ( γ ) u ′ ( β ) v ′ ( γ ) − 1 (30)
u'(^{\gamma}\beta ) =v'(\gamma )u'(\beta )v'(\gamma )^{-1} \tag{30}
u ′ ( γ β ) = v ′ ( γ ) u ′ ( β ) v ′ ( γ ) − 1 ( 30 ) 被以下引用 2 被以下引用 定理 4 , 定理 5
α ∈ L ∗ , β ∈ L \alpha \in L^*,\beta \in L α ∈ L ∗ , β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f u ′ = f ∘ u u'=f\circ u u ′ = f ∘ u v ′ = f ∘ v v'=f\circ v v ′ = f ∘ v
为了证明f f f A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] [ β ; γ ] = u ( β ) v ( γ ) [\beta ;\gamma ] =u(\beta )v(\gamma ) [ β ; γ ] = u ( β ) v ( γ ) f ′ : A [ E ; L ] → B f':\mathbf{A}[\mathscr{E}; L]\rightarrow B f ′ : A [ E ; L ] → B f ′ ∘ u = u ′ f'\circ u=u' f ′ ∘ u = u ′ f ′ ∘ v = v ′ f'\circ v=v' f ′ ∘ v = v ′ [ β ; γ ] [\beta ;\gamma ] [ β ; γ ] u ′ ( β ) v ′ ( γ ) u'(\beta )v'(\gamma ) u ′ ( β ) v ′ ( γ ) f f f
到公式 (29)，我们有
u ′ ( β α ) v ′ ( ι ( α ) − 1 γ ) = u ′ ( β ) v ′ ( γ ) (31)
u'(\beta \alpha )v'(\iota (\alpha )^{-1}\gamma ) =u'(\beta )v'(\gamma ) \tag{31}
u ′ ( β α ) v ′ ( ι ( α ) − 1 γ ) = u ′ ( β ) v ′ ( γ ) ( 31 ) 对于 L 中的 L ∗ , β L^*,\beta L ∗ , β α \alpha α γ \gamma γ f 0 : E → B f_0: E\rightarrow B f 0 : E → B f 0 ( [ β ; γ ] ) = u ′ ( β ) v ′ ( γ ) f_0([\beta ;\gamma ]) =u'(\beta )v'(\gamma ) f 0 ([ β ; γ ]) = u ′ ( β ) v ′ ( γ ) x x x x ′ x' x ′ f 0 ( x x ′ ) = f 0 ( x ) f 0 ( x ′ ) f_0(xx') =f_0(x)f_0(x') f 0 ( x x ′ ) = f 0 ( x ) f 0 ( x ′ ) g g g f 0 f_0 f 0 E g E_g E g A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f f 0 f_0 f 0 f f f u ′ = f ∘ u u'=f\circ u u ′ = f ∘ u v ′ = f ∘ v v'=f\circ v v ′ = f ∘ v
K ′ K' K ′ φ \varphi φ A [ E ; L ] ( K ′ ) \mathbf{A}[\mathscr{E}; L]_{(K')} A [ E ; L ] ( K ′ ) A [ E ′ ; L ′ ] \mathbf{A}[\mathscr{E}'; L'] A [ E ′ ; L ′ ]
u ′ ( h ( β ) ) = φ ( 1 ⊗ u ( β ) ) (36)
u'(h(\beta )) =\varphi (1\otimes u(\beta )) \tag{36}
u ′ ( h ( β )) = φ ( 1 ⊗ u ( β )) ( 36 )
对于 β ∈ L \beta \in L β ∈ L
v ′ ( h ( γ ) ) = φ ( 1 ⊗ v ( γ ) ) (37)
v'(h(\gamma )) =\varphi (1\otimes v(\gamma )) \tag{37}
v ′ ( h ( γ )) = φ ( 1 ⊗ v ( γ )) ( 37 ) γ ∈ Γ \gamma \in \Gamma γ ∈ Γ
证明紧接在构造之后。
同构 φ \varphi φ
10. 申请布劳尔群
在这个小节中，K是域，L是K上的伽罗瓦代数和作用λ : G → \lambda : G\rightarrow λ : G → K ( L ) _K(L) K ( L ) n n n n = n= n =
设E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ L ∗ L^* L ∗ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] n 2 n^2 n 2 u u u A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ]
被以下引用 1 被以下引用 引理 12
b) 令g g g a g \mathfrak{a}_g a g x − λ ( g ) x x-\lambda (g)x x − λ ( g ) x x x x a g = L \mathfrak{a}_g= L a g = L
令S \mathscr{S} S S \mathscr{S} S a ( S ) = ⋂ m ∈ S m \mathfrak{a}(S) =\bigcap_{\mathfrak{m}\in S}\mathfrak{m} a ( S ) = ⋂ m ∈ S m W ↦ a ( W ) W\mapsto \mathfrak{a}(W) W ↦ a ( W ) S \mathscr{S} S a ( S ) \mathfrak{a}(S) a ( S ) S \mathscr{S} S S \mathscr{S} S ∅ \emptyset ∅ S \mathscr{S} S a ( ∅ ) = L \mathfrak{a}(\emptyset ) = L a ( ∅ ) = L a ( S ) = { 0 } \mathfrak{a}(\mathscr{S}) =\{0\} a ( S ) = { 0 }
让我们用反证法来证明b)；假设我们有a g ≠ L \mathfrak{a}_g\not= L a g = L m \mathfrak{m} m a g \mathfrak{a}_g a g x x x λ ( g ) x ≡ x \lambda (g)x\equiv x λ ( g ) x ≡ x m \mathfrak{m} m m \mathfrak{m} m g g g λ ( g ) \lambda (g) λ ( g ) L / m L/\mathfrak{m} L / m G m G_{\mathfrak{m}} G m g g g
现在让我们证明定理 3. 向量空间 A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] : K ] = n 2 : K] =n^2 : K ] = n 2 = [ L : K ] = n = [L : K] =n = [ L : K ] = n
令a \mathfrak{a} a A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] a a a a = ∑ g ∈ G a g ε g a=\sum_{g\in G}a_g\varepsilon_g a = ∑ g ∈ G a g ε g g ∈ G g\in G g ∈ G a g ∈ L a_g\in L a g ∈ L g g g Φ ( a ) \Phi (a) Φ ( a ) a g ≠ 0 a_g\not= 0 a g = 0
ε g ε g ′ = c σ ( g , g ′ ) ε g g ′ (38)
\varepsilon_g\varepsilon_{g'}=c_{\sigma}(g, g')\varepsilon_{gg'} \tag{38}
ε g ε g ′ = c σ ( g , g ′ ) ε g g ′ ( 38 ) 对于所有 g , g ′ ∈ G g, g'\in G g , g ′ ∈ G
Φ ( a ε g ) = Φ ( a ) . g (39)
\Phi (a\varepsilon_g) = \Phi (a).g \tag{39}
Φ ( a ε g ) = Φ ( a ) . g ( 39 ) 对于每个 g ∈ G g\in G g ∈ G a ∈ A [ E ; L ] a\in \mathbf{A}[\mathscr{E}; L] a ∈ A [ E ; L ]
令a a a a \mathfrak{a} a Φ ( a ) \Phi (a) Φ ( a ) a a a a ε g − 1 a\varepsilon_{g^{-1}} a ε g − 1 g ∈ Φ ( a ) g\in \Phi (a) g ∈ Φ ( a ) e ∈ Φ ( a ) e\in \Phi (a) e ∈ Φ ( a ) s s s e e e Φ ( a ) \Phi (a) Φ ( a ) x x x a s ( x − λ ( s ) ⋅ x ) ≠ 0 a_s(x-\lambda (s)\cdot x)\not= 0 a s ( x − λ ( s ) ⋅ x ) = 0
x a − a x = ∑ g a g ( x − λ ( g ) ⋅ x ) ε g (40)
xa-ax=\sum_ga_g(x-\lambda (g)\cdot x)\varepsilon_g \tag{40}
x a − a x = g ∑ a g ( x − λ ( g ) ⋅ x ) ε g ( 40 ) 其中和取代了Φ ( a ) \Phi (a) Φ ( a ) g g g e e e Φ ( a ) \Phi (a) Φ ( a ) x a − a x = 0 xa-ax= 0 x a − a x = 0 x x x Φ ( a ) \Phi (a) Φ ( a ) e e e a ∈ L a\in L a ∈ L
因而 L ∩ a L\cap \mathfrak{a} L ∩ a { 0 } \{0\} { 0 } x x x ε g x ε g − 1 = λ ( g ) ⋅ x \varepsilon_gx\varepsilon^{-1}_g=\lambda (g)\cdot x ε g x ε g − 1 = λ ( g ) ⋅ x L ∩ a L\cap \mathfrak{a} L ∩ a L ∩ a = L L\cap \mathfrak{a}= L L ∩ a = L L ⊂ a L\subset \mathfrak{a} L ⊂ a A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] a = A [ E ; L ] \mathfrak{a}=\mathbf{A}[\mathscr{E}; L] a = A [ E ; L ]
由于代数 A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] { 0 } \{0\} { 0 } A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ]
让我们证明A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] a a a a a a ∑ g ∈ G a g ε g \sum_{g\in G}a_g\varepsilon_g ∑ g ∈ G a g ε g a g a_g a g x x x x a − a x = 0 xa-ax= 0 x a − a x = 0
a g ( x − λ ( g ) ⋅ x ) = 0 (41)
a_g(x-\lambda (g)\cdot x) = 0 \tag{41}
a g ( x − λ ( g ) ⋅ x ) = 0 ( 41 ) 对于每个g ∈ G g\in G g ∈ G x ∈ L x\in L x ∈ L a g = 0 a_g= 0 a g = 0 g ≠ e g\not=e g = e a ∈ L a\in L a ∈ L
最后，让我们来计算A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] z z z
0 = ε g z − z ε g = ( λ ( g ) ⋅ z − z ) ε g
0 =\varepsilon_gz-z\varepsilon_g= (\lambda (g)\cdot z-z)\varepsilon_g
0 = ε g z − z ε g = ( λ ( g ) ⋅ z − z ) ε g 对于G的每个g g g z z z λ ( G ) \lambda (G) λ ( G ) z ∈ K z\in K z ∈ K
不失一般性，我们可以假设 L 是 A 的最大值交换子代数。令 Γ 为 A 的可逆元 γ \gamma γ g g g
γ x γ − 1 = λ ( g ) . x (42)
\gamma x\gamma^{-1}=\lambda (g).x \tag{42}
γ x γ − 1 = λ ( g ) . x ( 42 ) 对于 L 中的每元素 x x x γ ∈ Γ \gamma \in \Gamma γ ∈ Γ g g g π ( γ ) \pi (\gamma ) π ( γ ) π \pi π L ∗ L^* L ∗ π \pi π
If we denote the canonical injection of L ∗ L^* L ∗ ι \iota ι E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) τ \tau τ L ∗ L^* L ∗
u : L → A [ E ; L ] u: L\rightarrow \mathbf{A}[\mathscr{E}; L] u : L → A [ E ; L ] v : Γ → A [ E ; L ] v: \Gamma \rightarrow \mathbf{A}[\mathscr{E}; L] v : Γ → A [ E ; L ]
是典范同态。通过A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f : A [ E ; L ] → A f:\mathbf{A}[\mathscr{E}; L]\rightarrow A f : A [ E ; L ] → A f ∘ u ( x ) = x f\circ u(x) =x f ∘ u ( x ) = x f ∘ v ( γ ) = γ f\circ v(\gamma ) =\gamma f ∘ v ( γ ) = γ x ∈ L x\in L x ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] n 2 n^2 n 2 n = [ L : K ] n= [L : K] n = [ L : K ] f f f
Br(L/ K /K / K = ⋂ m ∈ S =\bigcap_{\mathfrak{m}\in\mathscr{S}} = ⋂ m ∈ S r ( L / m ) / K r_{(L/\mathfrak{m})/K} r ( L / m ) / K r ( L / m ) / K : r_{(L/\mathfrak{m})/K}: r ( L / m ) / K : → \rightarrow → / m /\mathfrak{m} / m
Ψ : Exτ ( G , L ∗ ) ⟶ _{\tau}(G,L^*)\longrightarrow τ ( G , L ∗ ) ⟶ / K /K / K
将G的τ \tau τ E \mathscr{E} E L ∗ L^* L ∗ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] / K /K / K
为了定义Ψ并验证它是双射，我们必须建立以下点：
a) 如果E \mathscr{E} E E ′ \mathscr{E}' E ′ τ \tau τ L ∗ L^* L ∗ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] A [ E ′ ; L ] \mathbf{A}[\mathscr{E}'; L] A [ E ′ ; L ]
b) 相反，如果代数 A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] A [ E ′ ; L ] \mathbf{A}[\mathscr{E}'; L] A [ E ′ ; L ] L ∗ L^* L ∗ τ \tau τ E \mathscr{E} E E ′ \mathscr{E}' E ′
c) 在 Br(L/ K /K / K
d) 如果 E 是有限次数的中心单代数在 K 包含 L 上作为最大值交换子代数，则存在 G 的 τ \tau τ E \mathscr{E} E L ∗ L^* L ∗ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ]
断言 a) 由构造或叉积得出； b) 由VIII、p. 263推论得出。 断言 c) 是从 VIII、p. 281 的命题 5 得出的，而 d) 只是上面的定理 4。
还需要验证 Ψ 是否为群同态；为此，只要证明如果E 1 = ( Γ 1 , ι 1 , π 1 ) \mathscr{E}_1= (\Gamma_1, \iota_1, \pi_1) E 1 = ( Γ 1 , ι 1 , π 1 ) E 2 = ( Γ 2 , ι 2 , π 2 ) \mathscr{E}_2= (\Gamma_2, \iota_2, \pi_2) E 2 = ( Γ 2 , ι 2 , π 2 ) τ \tau τ A [ E 1 E 2 ; L ] \mathbf{A}[\mathscr{E}_1\mathscr{E}_2; L] A [ E 1 E 2 ; L ] A [ E 1 ; L ] ⊗ A [ E 2 ; L ] \mathbf{A}[\mathscr{E}_1; L]\otimes \mathbf{A}[\mathscr{E}_2; L] A [ E 1 ; L ] ⊗ A [ E 2 ; L ] E = ( Γ , ι , π ) \mathscr{E}= (\Gamma , \iota , \pi ) E = ( Γ , ι , π ) E 1 E 2 \mathscr{E}_1\mathscr{E}_2 E 1 E 2 L ∗ L^* L ∗ μ \mu μ ( ι 1 ( μ ) , ι 2 ( μ ) − 1 ) (\iota_1(\mu), \iota_2(\mu)^{-1}) ( ι 1 ( μ ) , ι 2 ( μ ) − 1 ) Γ 1 × G Γ 2 \Gamma_1\times_G\Gamma_2 Γ 1 × G Γ 2 ρ \rho ρ i ∈ { 1 , 2 } i\in \{1,2\} i ∈ { 1 , 2 } A i = A [ E i ; L ] A_i=\mathbf{A}[\mathscr{E}_i; L] A i = A [ E i ; L ] u i : L → A i u_i: L\rightarrow A_i u i : L → A i v i : Γ i → A i ∗ v_i: \Gamma_i\rightarrow A^*_i v i : Γ i → A i ∗ u i u_i u i A i A_i A i V i V_i V i A i A_i A i L ⊗ K A i o L\otimes_KA^o_i L ⊗ K A i o V i V_i V i n 2 n^2 n 2 L ⊗ K A i o L\otimes_KA^o_i L ⊗ K A i o L ( V i ) _L(V_i) L ( V i ) L ⊗ K A 1 o ⊗ K A 2 o L\otimes_KA^o_1\otimes_KA^o_2 L ⊗ K A 1 o ⊗ K A 2 o ( L ⊗ K A 1 o ) ⊗ L ( L ⊗ K A 2 o ) (L\otimes_KA^o_1)\otimes_L(L\otimes_KA^o_2) ( L ⊗ K A 1 o ) ⊗ L ( L ⊗ K A 2 o ) L ( V 1 ⊗ L V 2 ) _L(V_1\otimes_LV_2) L ( V 1 ⊗ L V 2 ) A 1 o ⊗ K A 2 o ( V 1 ⊗ L V 2 ) _{A^o_1\otimes_KA^o_2}(V_1\otimes_LV_2) A 1 o ⊗ K A 2 o ( V 1 ⊗ L V 2 ) A 1 ⊗ K A 2 A_1\otimes_KA_2 A 1 ⊗ K A 2 L ⊗ 1 ⊗ 1 L\otimes 1\otimes 1 L ⊗ 1 ⊗ 1 ( γ 1 , γ 2 ) ∈ Γ 1 × Γ 2 (\gamma_1, \gamma_2)\in \Gamma_1\times \Gamma_2 ( γ 1 , γ 2 ) ∈ Γ 1 × Γ 2 π 1 ( γ 1 ) = π 2 ( γ 2 ) \pi_1(\gamma_1) =\pi_2(\gamma_2) π 1 ( γ 1 ) = π 2 ( γ 2 ) w ( γ 1 , γ 2 ) w(\gamma_1, \gamma_2) w ( γ 1 , γ 2 ) λ ( π 1 ( γ 1 \lambda (\pi_1(\gamma_1 λ ( π 1 ( γ 1
w ( γ 1 , γ 2 ) ( x 1 ⊗ x 2 ) = v 1 ( γ 1 ) x 1 ⊗ v 2 ( γ 2 ) x 2
w(\gamma_1, \gamma_2)(x_1\otimes x_2) =v_1(\gamma_1)x_1\otimes v_2(\gamma_2)x_2
w ( γ 1 , γ 2 ) ( x 1 ⊗ x 2 ) = v 1 ( γ 1 ) x 1 ⊗ v 2 ( γ 2 ) x 2 对于 x 1 ∈ V 1 x_1\in V_1 x 1 ∈ V 1 x 2 ∈ V 2 x_2\in V_2 x 2 ∈ V 2 w ( γ 1 , γ 2 ) ∈ C ∗ w(\gamma_1, \gamma_2)\in C^* w ( γ 1 , γ 2 ) ∈ C ∗ w w w Γ 1 × G Γ 2 \Gamma_1\times_G\Gamma_2 Γ 1 × G Γ 2 C ∗ C^* C ∗ ρ \rho ρ v v v C ∗ C^* C ∗ u : L → C u: L\rightarrow C u : L → C u : ℓ ↦ ℓ ⊗ 1 ⊗ 1 u:\ell \mapsto \ell \otimes 1\otimes 1 u : ℓ ↦ ℓ ⊗ 1 ⊗ 1
u ( α ) = v ( ι ( α ) ) u(\alpha ) =v(\iota (\alpha )) u ( α ) = v ( ι ( α )) u ( γ β ) = v ( γ ) u ( β ) v ( γ ) − 1 u(^{\gamma}\beta ) =v(\gamma )u(\beta )v(\gamma )^{-1} u ( γ β ) = v ( γ ) u ( β ) v ( γ ) − 1
α ∈ L ∗ , β ∈ L \alpha \in L^*,\beta \in L α ∈ L ∗ , β ∈ L γ ∈ Γ \gamma \in \Gamma γ ∈ Γ A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f A [ E ; L ] \mathbf{A}[\mathscr{E}; L] A [ E ; L ] f f f
映射Φ L / K = Θ ∘ Ψ − 1 \Phi_{L/K}= \Theta \circ \Psi^{-1} Φ L / K = Θ ∘ Ψ − 1 / K /K / K H 2 ( G , L ∗ ) H^2(G,L^*) H 2 ( G , L ∗ )
设 K ′ K' K ′ φ : K ′ → L \varphi : K'\rightarrow L φ : K ′ → L h h h λ ( h ) ∘ φ = φ \lambda (h)\circ \varphi =\varphi λ ( h ) ∘ φ = φ λ \lambda λ K ′ K' K ′ K ′ K' K ′
Br(L/ K /K / K Φ L / K / ^{\Phi_{L/K}}/ Φ L / K / H 2 ( G , L ∗ ) H_2(G,L_*) H 2 ( G , L ∗ )
r K ′ / K r_{K'/K} r K ′ / K H G ^G_H H G
Br(L/ K ′ /K' / K ′ Φ L / K ′ / ^{\Phi_{L/K'}}/ Φ L / K ′ / H 2 ( H , L ∗ ) H_2(H,L_*) H 2 ( H , L ∗ )
这是从 VIII、p. 299 的命题 7 和VIII、p. 317 的 13 得出的。
11. 指标和指数
存在L的扩张M，即有限次数的K的伽罗瓦扩张(V, §10, No. 1，p. 57，命题2)。 K 属于到子群 Br(M/ K /K / K α \alpha α H 2 ( G , M ∗ ) H^2(G,M^*) H 2 ( G , M ∗ ) H G ( α ) = Φ M / L ( A ( L ) ) ^G_H(\alpha ) = \Phi_{M/L}(A_{(L)}) H G ( α ) = Φ M / L ( A ( L ) ) H G ( α ) = 0 ^G_H(\alpha ) = 0 H G ( α ) = 0 [ L : K ] α = 0 [L : K]\alpha = 0 [ L : K ] α = 0
设 K 为交换域，并令 A 为 K 上有限次数的中心单代数。则 A 同构于形式 M n \mathbf{M}_n M n
只要证明有限次数的域 D 超过其中心 K 就足够了。令 L 为 D 的极大交换子域，即 K 的可分扩张；我们有 [ D : K ] = [ L : K ] 2 [D : K] = [L : K]^2 [ D : K ] = [ L : K ] 2
让我们假设素数数p p p p p p L ′ = L H L'= L^H L ′ = L H A ( L ′ ) A_{(L')} A ( L ′ ) p p p L ′ L' L ′ [ L : L ′ ] [ A ( L ′ ) ] = 0 [L : L'][A_{(L')}] = 0 [ L : L ′ ] [ A ( L ′ ) ] = 0 [ A ( L ′ ) ] = 0 [A_{(L')}] = 0 [ A ( L ′ ) ] = 0 L ′ L' L ′ [ L ′ : K ] [L': K] [ L ′ : K ] p p p
让我们证明 K 的布劳尔群不包含阶 p p p p p p M ∗ M^* M ∗ p p p H 2 H^2 H 2 / K /K / K , M ∗ ,M^* , M ∗ / K /K / K
因而，[A] 的阶是 p p p p p p
习题
参见习题为 § 16 。