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\title{Lecture 4-26: Group cohomology}
\date{April 26, 2024}
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I will pick up from the definition of the cohomology groups $H^n(G,A)$ of a finite group $G$ with coefficients in a $G$-module $A$, given last time.  Such groups, like complex representations, are interesting in their own right; they also yield important purely group-theoretic information.  
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Note that $G$-modules $A$ have a very different flavor from representations of $G$ over a field $K$.  In particular, since free $\mathbb Z$-modules are never semisimple our focus will not be on irreducible modules but rather on the way that the submodule $A^G$ of $G$-invariants in $A$ (consisting of all $a\in A$ fixed by every $g\in G$) sits inside $A$.
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As mentioned last time, we define $H^n(G,A)$, the \alert{$n$th cohomology group of $G$ with coefficients in $A$}, to be the Ext group Ext$_{\mathbb ZG}^n(\mathbb Z,A)$, where $\mathbb Z$ is the trivial $\mathbb ZG$-module (on which every $g\in G$ acts trivially).  See p. 800 of DF.  Thus we can compute these groups using any convenient projective resolution of $\mathbb Z$.  For example, if $G$ is cyclic of order $m$ with generator $\sigma$, then we have a resolution
\[
\cdots\rightarrow\mathbb ZG\rightarrow\mathbb ZG\rightarrow\cdots\rightarrow\mathbb ZG\rightarrow\mathbb Z\rightarrow 0
\]
\noindent where the maps from $\mathbb ZG$ to itself alternate between multiplication by $\sigma-1$ and multiplication by $N=1+\sigma+\dots+\sigma^{n-1}$, and the map from $\mathbb ZG$ to $\mathbb Z$ is the \alert{augmentation map} sending $\sum_G z_gg$ to $\sum z_g$.
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Arguing as in the computation of Ext$_{\mathbb Z/m\mathbb Z}^n(\mathbb Z/d\mathbb Z,D)$ on April 22, we see that
\[
H^n(G,A)=\begin{cases}
A^G & \text{ if }n=0\\ 
A^G/NA & \text{ if }n\text{ even},n\ge2\\
{}_NA/(\sigma-1)A & \text{ if }n\text { odd},n\ge1
\end{cases}
\]
\noindent where ${}_NA$ is the subgroup of $A$ sent to 0 by $N$.  In particular, if $G$ acts trivially on $A$, then $Na=ma$ for all $a\in A$, so that $H^0(G,A)=A,H^n(G,A)=A/mA$ for even $n\ge2$, while $H^n(G,A)={}_mA$, the elements of $A$ sent to 0 by $m$, for $n$ odd.
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For more complicated groups $G$ the standard projective resolution of $\mathbb Z$ arising in this week's HW (called the \alert{bar resolution}) gives a uniform recipe for computing $H^n(G,A)$.   Since this is a free resolution it is easy to compute $\hom_{\mathbb ZG}(F_n,A)$ for all $A$.  We are then led to the following direct definition of $H^n(G,A)$ (making no reference to Ext groups or projective resolutions).  
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Denote by $C^n(G,A)$ the set of all maps from $G^n$ to $A$ for $n>0$ and set $C^0(G,A)=A$.  The elements of $C^n(G,A)$ are called {\sl homogeneous $n$-cochains of $G$ (with values in $A$)}.  For $n\ge0$ define $d_n$, the {\sl $n$th coboundary homomorphism} from $C^n(G,A)$ to $C^{n+1}(G,A)$ via
\begin{align*}
d_n(f)(g_1,\ldots,g_{n+1}) =\,&g_1\cdot f(g_2,\ldots,g_{n+1})\\
                                              &+\sum\limits_{i=1}^n (-1)^if(g_1,\ldots,g_{i-1},g_ig_{i+1},\ldots,g_{n+1})\\
                                              &+(-1)^{n+1}f(g_1,\ldots,g_n)
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One can then check (with some effort) that the composition $d_nd_{n-1}$ is 0, so that it makes sense to form the quotient $Z^n(G,A)/B^n(G,A)$ of the kernel $Z^n(G,A)$ of $d_n$ (the \alert{$n$-cocycles}) by the image $B^n(G,A)$ of $d_{n-1}$ (the \alert{$n$-coboundaries)}.  This quotient is then equal to $H^n(G,A)$.
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This formulation is frankly not all that useful for computing $H^n(G,A)$; the point of it is instead to interpret these groups.  We will see that in particular we can use the first and second cohomology groups to compute group-theoretic information.  For now we just observe that \alert{if $mA=0$ for some integer $m$, then $mZ^n(G,A)=0$ for all $n$, so that $mH^n(G,A)=0$ also}.  Corresponding to the long exact sequence for Ext groups arising from a short exact sequence of $R$-modules we get
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\begin{alertblock}{Theorem 21, DF, p. 802}
Given a short exact sequence $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ of $G$-modules we get a long exact sequence
\[
0\rightarrow A^G\rightarrow B^G\rightarrow C^G\rightarrow H^1(G,A)\rightarrow H^1(G,B)\rightarrow H^1(G,C)\rightarrow\cdots
\]
\noindent of abelian groups.
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Here the connecting homomorphisms $\delta_n:H^n(G,C)\rightarrow H^{n+1}(G,A)$ are those of the long exact sequence for Ext.
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Next we introduce the analogue of the induced representation construction.
\begin{alertblock}{Definition, DF, p. 803}
If $H$ is a subgroup of $G$ and $A$ is an $H$-module, then we define the {\sl coinduced module} to be $M=M_H^G(A)=\hom_{\mathbb ZH}(\mathbb ZG,A)$, where the action of $G$ is given by $g\cdot f(x) = f(xg)$ if $f\in M,g\in G$.
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We also have $M\cong\mathbb ZG\otimes_{\mathbb ZH}A$ (since $G$ is finite; but see next week's HW assignment for a counterexample in general), so $M$ is also often called the module induced from $A$.
\begin{alertblock}{Proposition:  Shapiro's Lemma, DF, p. 804}
For any subgroup $H$ of $G$ and $H$-module $A$ we have $H^n(G,M_H^G(A))\cong H^n(H,A)$.
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\begin{proof}
Set $M=M_H^G(A)$.  The bar resolution of $\mathbb Z$ over $G$ is a free resolution which remains free over $H$, since $\mathbb ZG$ is a free $\mathbb ZH$-module.  Taking $G$-homomorphisms into $M$ we get a cochain complex whose cohomology groups are the groups $H^n(G.M)$.  But now for any $G$-module $P$ there is a natural isomorphism $\Phi:\hom_{\mathbb ZG}(P,\hom_{\mathbb ZH}(\mathbb ZG,A))\cong\hom_{\mathbb ZH} (P,A)$ of abelian groups defined by $\Phi(f)(p) = f(p)(1)$.  Since this isomorphism commutes with the coboundary maps it induces an isomorphism between the corresponding cohomology groups, as desired.
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This is the analogue of Frobenius Reciprocity for coinduced modules.
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