# 群上同調

## 起源

${\displaystyle M^{G}=\lbrace x\in M:\forall g\in G\ gx=x\rbrace .}$

${\displaystyle N\subset M}$ 是一個 ${\displaystyle G}$-子模（即：是 ${\displaystyle M}$ 的子群，且在 ${\displaystyle G}$ 的作用下不變），則 ${\displaystyle M/N}$ 上賦有自然的 ${\displaystyle G}$-模結構，${\displaystyle N^{G}\subset M^{G}}$，但是未必有 ${\displaystyle (M/N)^{G}=M^{G}/N^{G}}$。第一個群上同調群 ${\displaystyle H^{1}(G,N)}$ 可以設想為兩者間差異的某種量度。一般而言，可以定義一族函子 ${\displaystyle H^{n}(G,-)}$，其間關係可以由長正合序列表示。

## 形式建構

• ${\displaystyle H^{0}(G,M)=M^{G}}$
• 長正合序列：若 ${\displaystyle 0\to M'\to M\to M''\to 0}$${\displaystyle G}$-模的短正合序列，則導出相應的長正合序列
${\displaystyle \cdots \to H^{i-1}(G,M'')\to H^{i}(G,M')\to H^{i}(G,M)\to H^{i}(G,M'')\to H^{i+1}(M')\to H^{i+1}(M)\to \cdots }$

## 標準分解

${\displaystyle H^{i}(G,M)=\mathrm {Ext} ^{i}(\mathbb {Z} ,M)}$

${\displaystyle L_{i}:=\sum _{(g_{0},\ldots ,g_{i})\in G}\mathbb {Z} (g_{0},\ldots ,g_{i})}$
${\displaystyle g(g_{0},\ldots ,g_{i})=(gg_{0},\ldots ,gg_{i})}$
${\displaystyle d(g_{0},\ldots ,g_{i})=\sum _{j=0}^{i}(g_{0},\ldots ,{\hat {g}}_{j},\ldots ,g_{i})}$

${\displaystyle L_{0}\to \mathbb {Z} }$${\displaystyle g_{0}\mapsto 1}$ 給出。

• ${\displaystyle K^{i}}$ 的元素為 ${\displaystyle G^{i}\to M}$ 之函數。
• ${\displaystyle (df)(g_{1},\ldots ,g_{i+1})=g_{1}f(g_{2},\ldots ,g_{i+1})+\sum _{j=1}^{i}(-1)^{j}f(g_{1},\ldots ,g_{j}g_{j+1},\ldots ,g_{i+1})+(-1)^{i+1}f(g_{1},\ldots ,g_{i})}$

## 例子

${\displaystyle H^{1}(G,M)={\dfrac {\{f:G\to M|\forall g,g',\;f(gg')=gf(g')+f(g)\}}{\{f:G\to M:\exists m\,\forall g,\;f(g)=gm-m\}}}}$

${\displaystyle H^{2}(G,M)={\dfrac {\{f:G^{2}\to M|gf(g',g'')-f(gg',g'')+f(g,g'g'')-f(g,g')=0\}}{\{f:G^{2}\to M:\exists h:G\to M,f(g,g')=gh(g')-h(gg')+h(g)\}}}}$

## 群同調

${\displaystyle M\to M_{G}:=M/DM=\mathbb {Z} \otimes _{\mathbb {Z} [G]}M}$

${\displaystyle H_{i}(G,M)\simeq \mathrm {Tor} _{i}^{\mathbb {Z} [G]}(\mathbb {Z} ,M)}$

## 非阿貝爾群上同調

${\displaystyle H^{0}(G,A):=A^{G}=\{a\in A|\forall g\in G,\;g(a)=a\}}$
${\displaystyle H^{1}(G,A):={\dfrac {\{a_{s}:G\to A|\forall s,t\in G,\;a_{st}=a_{s}s(a_{t})\}}{\{b_{s}:G\to A|\exists a,b_{s}=a^{-1}s(a)\}}}}$

${\displaystyle 1\to A\to B\to C\to 1}$${\displaystyle G}$-群的短正合序列，則有長正合序列

${\displaystyle 1\to A^{G}\to B^{G}\to C^{G}\to H^{1}(G,A)\to H^{1}(G,B)\to H^{1}(G,C)}$

${\displaystyle A}$落在 ${\displaystyle B}$ 的中心，此序列右端可再加一項 ${\displaystyle H^{1}(G,C)\to H^{2}(G,A)}$

## 性質

### Res 與 Cor

${\displaystyle f:H\to G}$ 為群同態，則可將任一 ${\displaystyle G}$-模透過 ${\displaystyle f}$ 視為 ${\displaystyle H}$-模，此運算導出上同調之間的映射

${\displaystyle H^{\bullet }(G,M)\to H^{\bullet }(H,M)}$

${\displaystyle N_{G/H}:M^{H}\to M^{G},\quad N_{G/H}(m):=\sum _{g\in G/H}gm}$

### 中心擴張

${\displaystyle M}$ 是平凡的 ${\displaystyle G}$ 模（即 ${\displaystyle \forall g\in G,\;gm=m}$），則 ${\displaystyle H^{2}(G,M)}$ 中的元素一一對應於 ${\displaystyle G}$${\displaystyle M}$中心擴張的等價類

${\displaystyle 0\to M\to E{\stackrel {p}{\to }}G\to 1}$

### 譜序列

${\displaystyle N\subset G}$${\displaystyle G}$正規子群，則有下述譜序列

${\displaystyle H^{p}(G/N,H^{q}(N,A))\implies H^{p+q}(G,A).\,}$