群の cohomology
group cohomology$ H^n(G,M)
$ H^0(G,M)\cong M^G\cong{\rm Hom}_{\Z[G]}(\Z,M)
$ M^G=\{x|x\in M,g\in G,gx-x=0\}不變式の空閒 群作用$ G\times M\to Mである。$ (gh)x=g(hx),$ 1x=x $ Gの要素$ n個の組の成す集合を$ G^n:=G\times\dots\times G=\{(g_1,\dots,g_n)|g_1\in G,\dots g_n\in G\}と書く $ G^1\cong G
寫像の集合$ C^n(G,M):={\rm Hom}_{\bf Set}(G^n,M)=\{\phi:G^n\to M\} 加法$ (\phi_1+\phi_2)(g_1,\dots,g_n):=\phi_1(g_1,\dots,g_n)+\phi_2(g_1,\dots,g_n)
單位元$ 0_{C^n}:=(\_\mapsto 0_M)
逆元$ (-\phi)(g_1,\dots,g_n):=-(\phi(g_1,\dots,g_n))
$ C^0=\{\{*\}\to M\}\cong M
$ C^1={\rm Hom}_{\bf Set}(G,M)
餘鎖複體$ C^0\xrightarrow{d^1}\dots\xrightarrow{d^n}C^n\xrightarrow{d^{n+1}}C^{n+1}\xrightarrow{d^{n+2}}\dots,$ d^n;d^{n+1}=\_\mapsto 0を定義する 餘境界作用素$ d^{n+1}:C^n\to C^{n+1},$ (d^{n+1}(\phi))(g_1,\dots,g_{n+1}):=g_1\phi(g_2,\dots,g_{n+1})+\sum_{i=1}^n(-1)^i\phi(g_1,\dots,g_{i-1},g_i g_{i+1},g_{i+2},\dots,g_n)+(-1)^{n+1}\phi(g_1,\dots,g_n)
$ d^{n+1}(\phi_1+\phi_2)=d^{n+1}(\phi_1)+d^{n+1}(\phi_2)
$ d^{n+1}(-\phi)=-d^{n+1}(\phi)
$ (d^n;d^{n+1})(g_1,\dots,g_n)=0
低次元のものを計算すると
$ (d^1(\phi))(g)=g\phi(*)-\phi(*)
$ x:=\phi(*)と書くと$ (d^1(\phi))(g)=gx-x
$ (d^2(\phi))(g_1,g_2)=g_1\phi(g_2)-\phi(g_1 g_2)+\phi(g_1)
$ (d^3(\phi))(g_1,g_2,g_3)=g_1\phi(g_2,g_3)-\phi(g_1,g_2 g_3)+\phi(g_1 g_2,g_3)-\phi(g_1,g_2)
$ Z^0=\{f|f\in\{\{*\}\to M\},d^1(f)=0\}\cong\{x|x\in M,g\in G,gx-x=0\}=\{x|x\in M,g\in G,gx=x\}=M^G不變式の空閒 $ H^0=Z^0/B^0=Z^0/0=Z^0\cong M^G
$ Gが$ Mに自明に作用する場合、$ H^1(G,M)={\bf Grp}(G,M)