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§ 10。逆与直极限
在本段中，I将表示非空预订集合，α ≤ β \alpha \leq \beta α ≤ β Set Theory , III, § 7, no. 1 (resp. Set Theory , III, § 7, no. 5, under the hypothesis that I is right directed).
1. 逆系岩浆
An inverse system of magmas relative to the indexing set I is an inverse system of sets* ( E α , f α β ) (\mathbf{E}_\alpha, f_{\alpha\beta}) ( E α , f α β ) relative to I, each E α \mathbf{E}_\alpha E α having a magma structure and each f α β f_{\alpha\beta} f α β
引用了 1 引用了 III § 7. 。外观代数
令 ( E α , f α β ) (\mathbf{E}_\alpha, f_{\alpha\beta}) ( E α , f α β ) E = lim ← E α E = \lim_{\leftarrow} \mathbf{E}_\alpha E = lim ← E α ( x α ) α ∈ I (x_\alpha)_{\alpha \in I} ( x α ) α ∈ I ∏ α ∈ I E α \prod_{\alpha \in I} \mathbf{E}_\alpha ∏ α ∈ I E α x α = f α β ( x β ) x_\alpha = f_{\alpha\beta}(x_\beta) x α = f α β ( x β ) α ≤ β \alpha \leq \beta α ≤ β ( x α ) (x_\alpha) ( x α ) ( y α ) (y_\alpha) ( y α ) α ≤ β \alpha \leq \beta α ≤ β x α = f α β ( x β ) x_\alpha = f_{\alpha\beta}(x_\beta) x α = f α β ( x β ) y α = f α β ( y β ) y_\alpha = f_{\alpha\beta}(y_\beta) y α = f α β ( y β ) x α y α = f α β ( x β ) f α β ( y β ) = f α β ( x β y β ) x_\alpha y_\alpha = f_{\alpha\beta}(x_\beta) f_{\alpha\beta}(y_\beta) = f_{\alpha\beta}(x_\beta y_\beta) x α y α = f α β ( x β ) f α β ( y β ) = f α β ( x β y β ) ∏ α ∈ I E α \prod_{\alpha \in I} \mathbf{E}_\alpha ∏ α ∈ I E α ∏ α ∈ I E α \prod_{\alpha \in I} \mathbf{E}_\alpha ∏ α ∈ I E α
† 我们避免当前的术语，其中正数意味着 > 0 > 0 > 0
逆极限岩浆E α E_\alpha E α
(a) 对于所有α ∈ I \alpha \in I α ∈ I E E E E α E_\alpha E α f α f_\alpha f α E E E E α E_\alpha E α f α = f α β ∘ f β f_\alpha = f_{\alpha \beta} \circ f_\beta f α = f α β ∘ f β α ≤ β \alpha \leq \beta α ≤ β
(b) 假设给定岩浆F F F u α : F → E α u_\alpha : F \to E_\alpha u α : F → E α u α = f α β ∘ u β u_\alpha = f_{\alpha \beta} \circ u_\beta u α = f α β ∘ u β α ≤ β \alpha \leq \beta α ≤ β u : F → E u : F \to E u : F → E u α = f α ∘ u u_\alpha = f_\alpha \circ u u α = f α ∘ u α ∈ I \alpha \in I α ∈ I x ↦ u ( x ) = ( u α ( x ) ) α ∈ I x \mapsto u(x) = (u_\alpha(x))_{\alpha \in I} x ↦ u ( x ) = ( u α ( x ) ) α ∈ I
如果岩浆 E α E_\alpha E α E E E E α E_\alpha E α e α e_\alpha e α f α β f_{\alpha \beta} f α β e = ( e α ) α ∈ I e = (e_\alpha)_{\alpha \in I} e = ( e α ) α ∈ I E E E e α = f α β ( e β ) e_\alpha = f_{\alpha \beta}(e_\beta) e α = f α β ( e β ) α ≤ β \alpha \leq \beta α ≤ β E E E f α f_\alpha f α u α u_\alpha u α u u u E E E x = ( x α ) α ∈ I x = (x_\alpha)_{\alpha \in I} x = ( x α ) α ∈ I x α x_\alpha x α E α E_\alpha E α x − 1 = ( x α − 1 ) α ∈ I x^{-1} = (x_\alpha^{-1})_{\alpha \in I} x − 1 = ( x α − 1 ) α ∈ I f α β ( x β − 1 ) = f α β ( x β ) − 1 = x α − 1 f_{\alpha \beta}(x_\beta^{-1}) = f_{\alpha \beta}(x_\beta)^{-1} = x_\alpha^{-1} f α β ( x β − 1 ) = f α β ( x β ) − 1 = x α − 1 α ≤ β \alpha \leq \beta α ≤ β
从这些注可以推断，如果岩浆E α E_\alpha E α f α β f_{\alpha \beta} f α β E E E
留给读者去验证环( E α , f α β ) (E_\alpha, f_{\alpha \beta}) ( E α , f α β ) E = lim ← E α E = \lim \leftarrow E_\alpha E = lim ← E α ∏ α ∈ I E α \prod_{\alpha \in I} E_\alpha ∏ α ∈ I E α E α E_\alpha E α
令 C = ( E α , f α β ) \mathfrak{C} = (E_\alpha, f_{\alpha \beta}) C = ( E α , f α β ) C ′ = ( E α ′ , f α β ′ ) \mathfrak{C}' = (E'_\alpha, f'_{\alpha \beta}) C ′ = ( E α ′ , f α β ′ ) C \mathfrak{C} C C ′ \mathfrak{C}' C ′ u α : E α → E α ′ u_\alpha : E_\alpha \to E'_\alpha u α : E α → E α ′ ( u α ) α ∈ I (u_\alpha)_{\alpha \in I} ( u α ) α ∈ I u α u_\alpha u α lim ← E α \lim \leftarrow E_\alpha lim ← E α lim ← E α ′ \lim \leftarrow E'_\alpha lim ← E α ′ u = lim ← u α u = \lim \leftarrow u_\alpha u = lim ← u α
2. 逆极限作用
假设给定两个集合、( Ω α , ϕ α β ) (\Omega_\alpha, \phi_{\alpha \beta}) ( Ω α , ϕ α β ) ( E α , f α β ) (E_\alpha, f_{\alpha \beta}) ( E α , f α β ) I I I E α E_\alpha E α Ω α \Omega_\alpha Ω α α ∈ I \alpha \in I α ∈ I
(1)
f α β ( ω β x β ) = ϕ α β ( ω β ) ⋅ f α β ( x β )
f_{\alpha \beta}(\omega_\beta x_\beta) = \phi_{\alpha \beta}(\omega_\beta) \cdot f_{\alpha \beta}(x_\beta)
f α β ( ω β x β ) = ϕ α β ( ω β ) ⋅ f α β ( x β ) α ≤ β , x β ∈ E β , ω β ∈ Ω β \alpha \leq \beta, x_\beta \in E_\beta, \omega_\beta \in \Omega_\beta α ≤ β , x β ∈ E β , ω β ∈ Ω β Ω = lim ← Ω α \Omega = \lim \leftarrow \Omega_\alpha Ω = lim ← Ω α E = lim ← E α E = \lim \leftarrow E_\alpha E = lim ← E α x = ( x α ) α ∈ I x = (x_\alpha)_{\alpha \in I} x = ( x α ) α ∈ I E E E ω = ( ω α ) α ∈ I \omega = (\omega_\alpha)_{\alpha \in I} ω = ( ω α ) α ∈ I Ω \Omega Ω ω . x = ( ω α . x α ) α ∈ I \omega . x = (\omega_\alpha . x_\alpha)_{\alpha \in I} ω . x = ( ω α . x α ) α ∈ I Ω \Omega Ω E α E_\alpha E α Ω α \Omega_\alpha Ω α 上述情况尤其适用于Ω α \Omega_\alpha Ω α E α E_\alpha E α Ω α \Omega_\alpha Ω α
留给读者对群的逆系和定义逆极限进行运算符，并验证这个极限是否是群。
3. 直接系统岩浆
在这个没有。接下来我们假设 I 是*右向的*。
A direct system of magmas relative to the indexing set I is a direct system of sets* ( E α , f β α ) (E_\alpha, f_{\beta\alpha}) ( E α , f β α ) relative to I, each E α E_\alpha E α having a magma structure and each f β α f_{\beta\alpha} f β α
令( E α , f β α ) (E_\alpha, f_{\beta\alpha}) ( E α , f β α ) E α E_\alpha E α lim ⟶ E α \lim_{\longrightarrow} E_\alpha lim ⟶ E α f α f_\alpha f α
(2)
f β ∘ f β α = f α for α ≤ β ,
f_\beta \circ f_{\beta\alpha} = f_\alpha \quad \text{for } \alpha \leq \beta,
f β ∘ f β α = f α for α ≤ β , (3)
E = ⋃ α ∈ I f α ( E α ) .
E = \bigcup_{\alpha \in I} f_\alpha(E_\alpha).
E = α ∈ I ⋃ f α ( E α ) . 由(2) 也可得
(4)
f α ( E α ) ⊂ f β ( E β ) for α ≤ β .
f_\alpha(E_\alpha) \subset f_\beta(E_\beta) \quad \text{for } \alpha \leq \beta.
f α ( E α ) ⊂ f β ( E β ) for α ≤ β . 如果x α , y α ∈ E α x_\alpha, y_\alpha \in E_\alpha x α , y α ∈ E α f α ( x α ) = f α ( y α ) f_\alpha(x_\alpha) = f_\alpha(y_\alpha) f α ( x α ) = f α ( y α ) β ≥ α \beta \geq \alpha β ≥ α f β α ( x α ) = f β α ( y α ) f_{\beta\alpha}(x_\alpha) = f_{\beta\alpha}(y_\alpha) f β α ( x α ) = f β α ( y α )
E 上存在一种且仅有一种岩浆结构，其中映射* f α : E α → E f_\alpha : E_\alpha \to E f α : E α → E 为同态。如果岩浆 E α E_\alpha E α E α E_\alpha E α f β α f_{\beta\alpha} f β α f α f_\alpha f α
被以下引用 1 被以下引用 命题 2
岩浆E α E_\alpha E α
设x , y x, y x , y α \alpha α E α E_\alpha E α x α , y α x_\alpha, y_\alpha x α , y α x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) y = f α ( y α ) y = f_\alpha(y_\alpha) y = f α ( y α ) f α f_\alpha f α x . y = f α ( x α y α ) x.y = f_\alpha(x_\alpha y_\alpha) x . y = f α ( x α y α )
为了证明存在性，我们必须证明对于I中的α , β \alpha, \beta α , β E α E_\alpha E α x α , y α x_\alpha, y_\alpha x α , y α E β E_\beta E β x β ′ , y β ′ x'_\beta, y'_\beta x β ′ , y β ′
(5)
f α ( x α ) = f β ( x β ′ ) , f α ( y α ) = f β ( y β ′ )
f_\alpha(x_\alpha) = f_\beta(x'_\beta), \qquad f_\alpha(y_\alpha) = f_\beta(y'_\beta)
f α ( x α ) = f β ( x β ′ ) , f α ( y α ) = f β ( y β ′ ) 暗示
f α ( x α y α ) = f β ( x β ′ y β ′ ) f_\alpha(x_\alpha y_\alpha) = f_\beta(x'_\beta y'_\beta) f α ( x α y α ) = f β ( x β ′ y β ′ ) 。对于
γ ≥ α \gamma \geq \alpha γ ≥ α 和
γ ≥ β \gamma \geq \beta γ ≥ β ，我们集合
x γ = f γ α ( x α ) x_\gamma = f_{\gamma\alpha}(x_\alpha) x γ = f γ α ( x α ) ，
y γ = f γ α ( y α ) , x γ ′ = f γ β ( x β ′ ) , y γ ′ = f γ β ( y β ′ ) y_\gamma = f_{\gamma \alpha}(y_\alpha), x'_\gamma = f_{\gamma \beta}(x'_\beta), y'_\gamma = f_{\gamma \beta}(y'_\beta) y γ = f γ α ( y α ) , x γ ′ = f γ β ( x β ′ ) , y γ ′ = f γ β ( y β ′ ) 。到直极限的定义，
I I I 和
γ ≥ α , γ ≥ β , x γ = x γ ′ , y γ = y γ ′ \gamma \geq \alpha, \gamma \geq \beta, x_\gamma = x'_\gamma, y_\gamma = y'_\gamma γ ≥ α , γ ≥ β , x γ = x γ ′ , y γ = y γ ′ 中存在
γ \gamma γ 。然后
f α ( x α y α ) = f γ ( f γ α ( x α y α ) ) = f γ ( x γ y γ ) = f γ ( x γ ′ y γ ′ ) = f γ ( f γ β ( x β ′ y β ′ ) ) = f β ( x β ′ y β ′ ) .
f_\alpha(x_\alpha y_\alpha) = f_\gamma(f_{\gamma \alpha}(x_\alpha y_\alpha)) = f_\gamma(x_\gamma y_\gamma) = f_\gamma(x'_\gamma y'_\gamma) = f_\gamma(f_{\gamma \beta}(x'_\beta y'_\beta)) \\
= f_\beta(x'_\beta y'_\beta).
f α ( x α y α ) = f γ ( f γ α ( x α y α )) = f γ ( x γ y γ ) = f γ ( x γ ′ y γ ′ ) = f γ ( f γ β ( x β ′ y β ′ )) = f β ( x β ′ y β ′ ) . 假设岩浆
E α E_\alpha E α 是结合的。让
x , y , z x, y, z x , y , z 在
E E E 中。
E α E_\alpha E α 中存在
α ∈ I \alpha \in I α ∈ I 和元素
x α , y α , z α x_\alpha, y_\alpha, z_\alpha x α , y α , z α 使得
x = f α ( x α ) , y = f α ( y α ) , z = f α ( z α ) .
x = f_\alpha(x_\alpha), \quad y = f_\alpha(y_\alpha), \quad z = f_\alpha(z_\alpha).
x = f α ( x α ) , y = f α ( y α ) , z = f α ( z α ) . 然后
x y = f α ( x α y α ) xy = f_\alpha(x_\alpha y_\alpha) x y = f α ( x α y α ) 、从而
( x y ) z = f α ( ( x α y α ) z α ) (xy)z = f_\alpha((x_\alpha y_\alpha)z_\alpha) ( x y ) z = f α (( x α y α ) z α ) ；同样地
x ( y z ) = f α ( x α ( y α z α ) ) ,
x(yz) = f_\alpha(x_\alpha(y_\alpha z_\alpha)),
x ( y z ) = f α ( x α ( y α z α )) , 从而
( x y ) z = x ( y z ) (xy)z = x(yz) ( x y ) z = x ( y z ) ( x α y α ) z α = x α ( y α z α ) (x_\alpha y_\alpha)z_\alpha = x_\alpha(y_\alpha z_\alpha) ( x α y α ) z α = x α ( y α z α ) 。交换岩浆的情况也可以类似地处理。
最后假设每个岩浆E α E_\alpha E α e α e_\alpha e α f β α ( e α ) = e β f_{\beta \alpha}(e_\alpha) = e_\beta f β α ( e α ) = e β α ≤ β \alpha \leq \beta α ≤ β I I I α , β \alpha, \beta α , β I I I γ \gamma γ γ ≥ α \gamma \geq \alpha γ ≥ α γ ≥ β \gamma \geq \beta γ ≥ β
f α ( e α ) = f γ ( f γ α ( e α ) ) = f γ ( e γ ) = f γ ( f γ β ( e β ) ) = f β ( e β )
f_\alpha(e_\alpha) = f_\gamma(f_{\gamma \alpha}(e_\alpha)) = f_\gamma(e_\gamma) = f_\gamma(f_{\gamma \beta}(e_\beta)) = f_\beta(e_\beta)
f α ( e α ) = f γ ( f γ α ( e α )) = f γ ( e γ ) = f γ ( f γ β ( e β )) = f β ( e β ) 因此，
E E E 中存在元素
e e e ，使得
f α ( e α ) = e f_\alpha(e_\alpha) = e f α ( e α ) = e 对于所有
α ∈ I \alpha \in I α ∈ I 。设
x x x 在
E E E 中； 选择
α ∈ I \alpha \in I α ∈ I 和
x α ∈ E α x_\alpha \in E_\alpha x α ∈ E α 使得
x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) 。然后
e x = f α ( e α ) ⋅ f α ( x α ) = f α ( e α ⋅ x α ) = f α ( x α ) = x
ex = f_\alpha(e_\alpha) \cdot f_\alpha(x_\alpha) = f_\alpha(e_\alpha \cdot x_\alpha) = f_\alpha(x_\alpha) = x
e x = f α ( e α ) ⋅ f α ( x α ) = f α ( e α ⋅ x α ) = f α ( x α ) = x 类似地，
x ⋅ e = x x \cdot e = x x ⋅ e = x ，因此
e e e 是
E E E 的单位元素。
岩浆E E E E α E_\alpha E α
设* ( E α , f β α ) (E_\alpha, f_{\beta \alpha}) ( E α , f β α ) 为岩浆的直接系统，并让 E E E 为其直极限 f α : E α → E f_\alpha : E_\alpha \to E f α : E α → E 典范同态。假设给定岩浆 F F F u α : E α → F u_\alpha : E_\alpha \to F u α : E α → F u α = u β ∘ f β α u_\alpha = u_\beta \circ f_{\beta \alpha} u α = u β ∘ f β α α ≤ β \alpha \leq \beta α ≤ β 存在且仅有一个同态 u : E → F u : E \to F u : E → F 使得 u α = u ∘ f α u_\alpha = u \circ f_\alpha u α = u ∘ f α 对于所有 α ∈ I \alpha \in I α ∈ I 如果岩浆 E α E_\alpha E α 和 F F F 和同态 f β α f_{\beta \alpha} f β α 和 u α u_\alpha u α 是单一的，同态 u u u
引用了 3 · 被以下引用 1 引用了 命题 6, E III § 7 , 命题 1 , I § 2. 。 单位元素；可取消元素； 可逆元
被以下引用 § 10. 。逆与直极限
我们知道（*集合理论*，III, § 7, no. 6，命题6）存在且仅有一个映射u : E → F u : E \to F u : E → F u α = u ∘ f α u_\alpha = u \circ f_\alpha u α = u ∘ f α α ∈ I \alpha \in I α ∈ I u u u x , y x, y x , y E E E α \alpha α I I I x α , y α x_\alpha, y_\alpha x α , y α E α E_\alpha E α x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) y = f α ( y α ) y = f_\alpha(y_\alpha) y = f α ( y α ) x y = f α ( x α y α ) xy = f_\alpha(x_\alpha y_\alpha) x y = f α ( x α y α )
u ( x y ) = u ( f α ( x α y α ) ) = u α ( x α y α ) = u α ( x α ) u α ( y α ) = u ( f α ( u α ) ) u ( f α ( y α ) ) = u ( x ) u ( y ) .
u(xy) = u(f_\alpha(x_\alpha y_\alpha)) = u_\alpha(x_\alpha y_\alpha) = u_\alpha(x_\alpha)u_\alpha(y_\alpha) \\
= u(f_\alpha(u_\alpha))u(f_\alpha(y_\alpha)) = u(x)u(y).
u ( x y ) = u ( f α ( x α y α )) = u α ( x α y α ) = u α ( x α ) u α ( y α ) = u ( f α ( u α )) u ( f α ( y α )) = u ( x ) u ( y ) . 现在我们考虑单位情况，令
e α e_\alpha e α 表示
E α E_\alpha E α 的单位元素，
e e e 表示
E E E ，
e ′ e' e ′ 表示
F F F 。 选择
α ∈ I \alpha \in I α ∈ I ，然后
e = f α ( e α ) e = f_\alpha(e_\alpha) e = f α ( e α ) ，从而
u ( e ) = u ( f α ( e α ) ) = u α ( e α ) = e ′
u(e) = u(f_\alpha(e_\alpha)) = u_\alpha(e_\alpha) = e'
u ( e ) = u ( f α ( e α )) = u α ( e α ) = e ′ 因为
u α u_\alpha u α 是单位的。因此
u u u 是单位的。
类比岩浆的概念或直接系统，可以公式化幺半群的直接系统或群。 命题 1 表明幺半群( E α , f β α ) α , β ∈ I (E_\alpha, f_{\beta\alpha})_{\alpha, \beta \in I} ( E α , f β α ) α , β ∈ I E α E_\alpha E α x ∈ E , α ∈ I x \in E, \alpha \in I x ∈ E , α ∈ I x α ∈ E α x_\alpha \in E_\alpha x α ∈ E α x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) y = f α ( x α − 1 ) y = f_\alpha(x_\alpha^{-1}) y = f α ( x α − 1 )
留给读者的是环的定义直接系统。令( A α , f β α ) (A_\alpha, f_{\beta\alpha}) ( A α , f β α ) A = lim A α → A = \lim \overrightarrow{A_\alpha} A = lim A α f α : A α → A f_\alpha : A_\alpha \to A f α : A α → A A α A_\alpha A α x = f α ( x α ) , y = f α ( y α ) x = f_\alpha(x_\alpha), y = f_\alpha(y_\alpha) x = f α ( x α ) , y = f α ( y α ) α ∈ I , x α , y α \alpha \in I, x_\alpha, y_\alpha α ∈ I , x α , y α x + y = f α ( x α + y α ) , x y = f α ( x α y α ) x + y = f_\alpha(x_\alpha + y_\alpha), xy = f_\alpha(x_\alpha y_\alpha) x + y = f α ( x α + y α ) , x y = f α ( x α y α ) x , y , z x, y, z x , y , z α \alpha α x α , y α x_\alpha, y_\alpha x α , y α A α A_\alpha A α z α z_\alpha z α
x = f α ( x α ) , y = f α ( y α ) and z = f α ( z α ) .
x = f_\alpha(x_\alpha), \quad y = f_\alpha(y_\alpha) \quad \text{and} \quad z = f_\alpha(z_\alpha).
x = f α ( x α ) , y = f α ( y α ) and z = f α ( z α ) . 然后
( x + y ) ⋅ z = f α ( x α + y α ) f α ( z α ) = f α ( ( x α + y α ) z α ) = f α ( x α z α + y α z α ) = f α ( x α z α ) + f α ( y α z α ) = x z + y z
(x + y) \cdot z = f_\alpha(x_\alpha + y_\alpha)f_\alpha(z_\alpha) = f_\alpha((x_\alpha + y_\alpha)z_\alpha)
= f_\alpha(x_\alpha z_\alpha + y_\alpha z_\alpha) = f_\alpha(x_\alpha z_\alpha) + f_\alpha(y_\alpha z_\alpha) = xz + yz
( x + y ) ⋅ z = f α ( x α + y α ) f α ( z α ) = f α (( x α + y α ) z α ) = f α ( x α z α + y α z α ) = f α ( x α z α ) + f α ( y α z α ) = x z + y z 类似地证明关系
x ( y + z ) = x y + x z x(y + z) = xy + xz x ( y + z ) = x y + x z 。在其他字中，A具有环的结构，其特征在于，对于所有
α ∈ I \alpha \in I α ∈ I ，
f α f_\alpha f α 是环同态。
环 A 称为环 A α A_\alpha A α
(a) 如果 A α A_\alpha A α (b) 如果A α A_\alpha A α (c) *如果A α A_\alpha A α
引用了 1 引用了 E III § 7. 逆极限和直极限
设0 α , 1 α 0_\alpha, 1_\alpha 0 α , 1 α A α A_\alpha A α α ∈ I \alpha \in I α ∈ I f α ( 0 α ) = 0 , f α ( 1 α ) = 1 f_\alpha(0_\alpha) = 0, f_\alpha(1_\alpha) = 1 f α ( 0 α ) = 0 , f α ( 1 α ) = 1 0 = 1 0 = 1 0 = 1 β ≥ α \beta \geq \alpha β ≥ α f β α ( 0 α ) = f β α ( 1 α ) f_{\beta\alpha}(0_\alpha) = f_{\beta\alpha}(1_\alpha) f β α ( 0 α ) = f β α ( 1 α ) 0 β = 1 β 0_\beta = 1_\beta 0 β = 1 β
假设A α A_\alpha A α x , y x, y x , y x y = 0 xy = 0 x y = 0 α ∈ I \alpha \in I α ∈ I x α , y α ∈ A α x_\alpha, y_\alpha \in A_\alpha x α , y α ∈ A α x = f α ( x α ) , y = f α ( y α ) x = f_\alpha(x_\alpha), y = f_\alpha(y_\alpha) x = f α ( x α ) , y = f α ( y α ) f α ( x α y α ) = x y = 0 = f α ( 0 α ) f_\alpha(x_\alpha y_\alpha) = xy = 0 = f_\alpha(0_\alpha) f α ( x α y α ) = x y = 0 = f α ( 0 α ) β ≥ α \beta \geq \alpha β ≥ α f β α ( x α y α ) = f β α ( 0 α ) f_{\beta\alpha}(x_\alpha y_\alpha) = f_{\beta\alpha}(0_\alpha) f β α ( x α y α ) = f β α ( 0 α ) A β A_\beta A β f β α ( x α ) = 0 β f_{\beta\alpha}(x_\alpha) = 0_\beta f β α ( x α ) = 0 β f β α ( y α ) = 0 β f_{\beta\alpha}(y_\alpha) = 0_\beta f β α ( y α ) = 0 β x = 0 x = 0 x = 0 y = 0 y = 0 y = 0
假设A α A_\alpha A α A ≠ { 0 } A \neq \{0\} A = { 0 } x x x α ∈ I \alpha \in I α ∈ I x α ∈ A α x_\alpha \in A_\alpha x α ∈ A α x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) x α ≠ 0 x_\alpha \neq 0 x α = 0 f α ( x α − 1 ) f_\alpha(x_\alpha^{-1}) f α ( x α − 1 ) x x x
令E = ( E α , f β α ) \mathfrak{E} = (E_\alpha, f_{\beta\alpha}) E = ( E α , f β α ) E ′ = ( E α ′ , f β α ′ ) \mathfrak{E}' = (E'_\alpha, f'_{\beta\alpha}) E ′ = ( E α ′ , f β α ′ ) E \mathfrak{E} E E ′ \mathfrak{E}' E ′ u α : E α → E α ′ u_\alpha : E_\alpha \to E'_\alpha u α : E α → E α ′ ( u α ) α ∈ I (u_\alpha)_{\alpha \in I} ( u α ) α ∈ I u α u_\alpha u α E = lim E α E = \lim E_\alpha E = lim E α E ′ = lim E α ′ E' = \lim E'_\alpha E ′ = lim E α ′ u = lim u α u = \lim u_\alpha u = lim u α
4. 直极限作用
假设相对于相同索引集合I I I ( Ω α , ϕ β α ) (\Omega_\alpha, \phi_{\beta \alpha}) ( Ω α , ϕ β α ) ( E α , f β α ) (E_\alpha, f_{\beta \alpha}) ( E α , f β α ) E α E_\alpha E α Ω α \Omega_\alpha Ω α α ∈ I \alpha \in I α ∈ I
f β α ( ω α ⋅ x α ) = ϕ β α ( ω α ) ⋅ f β α ( x α )
f_{\beta \alpha}(\omega_\alpha \cdot x_\alpha) = \phi_{\beta \alpha}(\omega_\alpha) \cdot f_{\beta \alpha}(x_\alpha)
f β α ( ω α ⋅ x α ) = ϕ β α ( ω α ) ⋅ f β α ( x α ) 对于 α ≤ β , ω α ∈ Ω α \alpha \leq \beta, \omega_\alpha \in \Omega_\alpha α ≤ β , ω α ∈ Ω α x α ∈ E α x_\alpha \in E_\alpha x α ∈ E α E = lim E α E = \lim E_\alpha E = lim E α Ω = lim Ω α \Omega = \lim \Omega_\alpha Ω = lim Ω α h h h ω ∈ Ω \omega \in \Omega ω ∈ Ω x ∈ E x \in E x ∈ E α ∈ I \alpha \in I α ∈ I ω α ∈ Ω α , x α ∈ E α \omega_\alpha \in \Omega_\alpha, x_\alpha \in E_\alpha ω α ∈ Ω α , x α ∈ E α ω = ϕ α ( ω α ) \omega = \phi_\alpha(\omega_\alpha) ω = ϕ α ( ω α ) x = f α ( x α ) x = f_\alpha(x_\alpha) x = f α ( x α ) ϕ α : Ω α → Ω \phi_\alpha : \Omega_\alpha \to \Omega ϕ α : Ω α → Ω f α : E α → E f_\alpha : E_\alpha \to E f α : E α → E ω ⋅ x = f α ( ω α ⋅ x α ) \omega \cdot x = f_\alpha(\omega_\alpha \cdot x_\alpha) ω ⋅ x = f α ( ω α ⋅ x α ) E E E Ω \Omega Ω E α E_\alpha E α Ω α \Omega_\alpha Ω α
如果Ω α \Omega_\alpha Ω α E α E_\alpha E α Ω α \Omega_\alpha Ω α
留给读者对群的直接系统和定义直极限进行运算符，并验证这个极限是否是群。
习题
参见习题为 § 10 。