Cohomology of groups

Historically, the earliest theory of a cohomology of algebras.

With every pair $(G,A)$, where $G$ is a group and $A$ a left $G$- module (that is, a module over the integral group ring $\mathbf{Z}G$), there is associated a sequence of Abelian groups $H^n(G,A)$, called the cohomology groups of $G$ with coefficients in $A$. The number $n$, which runs over the non-negative integers, is called the dimension of $H^n(G,A)$. The cohomology groups of groups are important invariants containing information both on the group $G$ and on the module $A$.

By definition, $H^0(G,A)$ is ${\mathrm{Hom}}_G(\mathbf{Z},A)\simeq A^G$, where $A^G$ is the submodule of $G$- invariant elements in $A$. The groups $H^n(G,A)$, $n>1$, are defined as the values of the $n$- th derived functor of the functor $A\mapsto H^0(G,A)$. Let

$$\cdots {\to }^{d_n}{P}_n{\to }^{d_n-1}{P}_{n-1}\to \cdots \to P_0\to \mathbf{Z}\to 0$$

be some projective resolution of the trivial $G$- module $\mathbf{Z}$ in the category of $G$- modules, that is, an exact sequence in which every $P_i$ is a projective $\mathbf{Z}G$- module. Then $H^n(G,A)$ is the $n$- th cohomology group of the complex

$$0\to \underset{G}{\mathrm{Hom}}(P_0,A){\to }^{{d_0}^{\prime }}\underset{G}{\mathrm{Hom}}(P_1,A)\to \dots ,$$

where $d_n^{\prime }$ is induced by $d_n$, that is, $H^n(G,A)=\mathrm{Ker}{d}_n^{\prime }/\mathrm{Im}{d}_{n-1}^{\prime }$.

The homology groups of a group are defined using the dual construction, in which ${\mathrm{Hom}}_G$ is replaced everywhere by ${\otimes }_G$.

The set of functors $A\mapsto H^n(G,A)$, $n=0,1\dots$ is a cohomological functor (see Homology functor; Cohomology functor) on the category of left $G$- modules.

A module of the form $B=\mathrm{Hom}(\mathbf{Z}[G],X)$, where $X$ is an Abelian group and $G$ acts on $B$ according to the formula

$$(g\varphi )(t)=\varphi (tg),\varphi \in B,t\in \mathbf{Z}G,$$

is said to be co-induced. If $A$ is injective or co-induced, then $H^n(G,A)=0$ for $n\ge 1$. Every module $A$ is isomorphic to a submodule of a co-induced module $B$. The exact homology sequence for the sequence

$$0\to A\to B\to B/A\to 0$$

then defines isomorphisms $H^n(G,B/A)\simeq H^{n+1}(G,A)$, $n\ge 1$, and an exact sequence

$$B^G\to (B/A{)}^G\to H^1(G,A)\to 0.$$

Therefore, the computation of the $(n+1)$- dimensional cohomology group of $A$ reduces to calculating the $n$- dimensional cohomology group of $B/A$. This device is called dimension shifting.

Dimension shifting enables one to give an axiomatic definition of cohomology groups, namely, they can be defined as a sequence of functors $A\mapsto H^n(G,A)$ from the category of $G$- modules into the category of Abelian groups forming a cohomological functor and satisfying the condition that $H^n(G,B)=0$, $n\ge 1$, for every co-induced module $B$.

The groups $H^n(G,A)$ can also be defined as equivalence classes of exact sequences of $G$- modules of the form

$$0\to A\to M_1\to \cdots \to M_n\to \mathbf{Z}\to 0$$

with respect to a suitably defined equivalence relation (see [1], Chapt. 3, 4).

To compute the cohomology groups, the standard resolution of the trivial $G$- module $\mathbf{Z}$ is generally used, in which $P_n=\mathbf{Z}[G^{n+1}]$ and, for $(g_0\dots g_n)\in G^{n+1}$,

$$d_n(g_0\dots g_n)=\sum _{i=0}^n(-1{)}^i(g_0\dots {\hat{g}}_i\dots g_n),$$

where the symbol $\hat{}$ over $g_i$ means that the term $g_i$ is omitted. The cochains in ${\mathrm{Hom}}_G(P_n,A)$ are the functions $f(g_0\dots g_n)$ for which $gf(g_0\dots g_n)=f(g{g}_0\dots g{g}_n)$. Changing variables according to the rules $g_0=1$, $g_1=h_1$, $g_2=h_1{h}_2\dots g_n=h_1\dots h_n$, one can go over to inhomogeneous cochains $f(h_1\dots h_n)$. The coboundary operation then acts as follows:

$$d^{\prime }f(h_1\dots h_{n+1})=h_{1}f(h_2\dots h_{n+1})+$$

$$+\sum _{i=1}^n(-1{)}^{i}f(h_1\dots h_i{h}_{i+1}\dots h_{n+1})+$$

$$+(-1{)}^{n+1}f(h_1\dots h_n).$$

For example, a one-dimensional cocycle is a function $f:G\to A$ for which $f(g_1{g}_2)=g_{1}f(g_2)+f(g_1)$ for all $g_1,g_2\in G$, and a coboundary is a function of the form $f(g)=ga-a$ for some $a\in A$. A one-dimensional cocycle is also said to be a crossed homomorphism and a one-dimensional coboundary a trivial crossed homomorphism. When $G$ acts trivially on $A$, crossed homomorphisms are just ordinary homomorphisms and all the trivial crossed homomorphisms are 0, that is, $H^1(G,A)=\mathrm{Hom}(G,A)$ in this case.

The elements of $H^1(G,A)$ can be interpreted as the $A$- conjugacy classes of sections $G\to F$ in the exact sequence $1\to A\to F\to G\to 1$, where $F$ is the semi-direct product of $G$ and $A$. The elements of $H^2(G,A)$ can be interpreted as classes of extensions of $A$ by $G$. Finally, $H^3(G,A)$ can be interpreted as obstructions to extensions of non-Abelian groups $H$ with centre $A$ by $G$( see [1]). For $n>3$, there are no analogous interpretations known (1978) for the groups $H^n(G,A)$.

If $H$ is a subgroup of $G$, then restriction of cocycles from $G$ to $H$ defines functorial restriction homomorphisms for all $n$:

$$\mathrm{res}:H^n(G,A)\to H^n(H,A).$$

For $n=0$, $\mathrm{res}$ is just the imbedding $A^G\subset A^H$. If $G/H$ is some quotient group of $G$, then lifting cocycles from $G/H$ to $G$ induces the functorial inflation homomorphism

$$inf:H^n(G/H,A^H)\to H^n(G,A).$$

Let $\varphi :G^{\prime }\to G$ be a homomorphism. Then every $G$- module $A$ can be regarded as a $G^{\prime }$- module by setting $g^{\prime }a=\varphi (g^{\prime })a$ for $g^{\prime }\in G^{\prime }$. Combining the mappings $\mathrm{res}$ and $inf$ gives mappings $H^n(G^{\prime },A)\to H^n(G,A)$. In this sense $H^{\ast }(G,A)$ is a contravariant functor of $G$. If $\mathrm{\Pi }$ is a group of automorphisms of $G$, then $H^n(G,A)$ can be given the structure of a $\mathrm{\Pi }$- module. For example, if $H$ is a normal subgroup of $G$, the groups $H^n(H,A)$ can be equipped with a natural $G/H$- module structure. This is possible thanks to the fact that inner automorphisms of $G$ induce the identity mapping on the $H^n(G,A)$. In particular, for a normal subgroup $H$ in $G$, $\mathrm{Im}\mathrm{res}\subset H^n(H,A{)}^{G/H}$.

Let $H$ be a subgroup of finite index in the group $G$. Using the norm map $N_{G/H}:A^H\to A^G$, one can use dimension shifting to define the functorial co-restriction mappings for all $n$:

$$\mathrm{cores}:H^n(H,A)\to H^n(G,A).$$

These satisfy $\mathrm{cores}\cdot \mathrm{res}=(G:H)$.

If $H$ is a normal subgroup of $G$ then there exists the Lyndon spectral sequence with second term $E_2^{p,q}=H^p(G/H,H^q(H,A))$ converging to the cohomology $H^n(G,A)$( see [1], Chapt. 11). In small dimensions it leads to the exact sequence

$$0\to H^1(G/H,A^H)\stackrel{inf}{\to }{H}^1(G,A)\stackrel{\mathrm{res}}{\to }{H}^1(H,A{)}^{G/H}\stackrel{\mathrm{tr}}{\to }$$

$$\stackrel{\mathrm{tr}}{\to }{H}^2(G/H,A^H)\stackrel{inf}{\to }{H}^2(G,A),$$

where $\mathrm{tr}$ is the transgression mapping.

For a finite group $G$, the norm map $N_G:A\to A$ induces the mapping ${\hat{N}}_G:H_0(G,A)\to H^0(G,A)$, where $H_0(G,A)=A/J_{G}A$ and $J_G$ is the ideal of $\mathbf{Z}G$ generated by the elements of the form $g-1$, $g\in G$. The mapping $N_G$ can be used to unite the exact cohomology and homology sequences. More exactly, one can define modified cohomology groups (also called Tate cohomology groups) $\hat{H}{}^n(G,A)$ for all $n$. Here

$$\hat{H}{}^n(G,A)=H^n(G,A)\text{ for }n\ge 1,$$

$$\hat{H}{}^n(G,A)=H_{-n-1}(G,A)\text{ for }n\le -1,$$

$$\hat{H}{}^{-}1(G,A)=\mathrm{Ker}{\hat{N}}_G\text{ and }{\hat{H}}_0(G,A)=\mathrm{Coker}{\hat{N}}_G.$$

For these cohomology groups there exists an exact cohomology sequence that is infinite in both directions. A $G$- module $A$ is said to be cohomologically trivial if $\hat{H}{}^n(H,A)=0$ for all $n$ and all subgroups $H\subseteq G$. A module $A$ is cohomologically trivial if and only if there is an $i$ such that $\hat{H}{}^i(H,A)=0$ and $\hat{H}{}^{i+1}(H,A)=0$ for every subgroup $H\subseteq G$. Every module $A$ is a submodule or a quotient module of a cohomologically trivial module, and this allows one to use dimension shifting both to raise and to lower the dimension. In particular, dimension shifting enables one to define $\mathrm{res}$ and $\mathrm{cores}$( but not $inf$) for all integral $n$. For a finitely-generated $G$- module $A$ the groups $\hat{H}{}^n(G,A)$ are finite.

The groups $\hat{H}{}^n(G,A)$ are annihilated on multiplication by the order of $G$, and the mapping $\hat{H}(G,A)\to {\oplus }_p\hat{H}{}^n(G_p,A)$, induced by restrictions, is a monomorphism, where now $G_p$ is a Sylow $p$- subgroup (cf. Sylow subgroup) of $G$. A number of problems concerning the cohomology of finite groups can be reduced in this way to the consideration of the cohomology of $p$- groups. The cohomology of cyclic groups has period 2, that is, $\hat{H}{}^n(G,A)\simeq \hat{H}{}^{n+2}(G,A)$ for all $n$.

For arbitrary integers $m$ and $n$ there is defined a mapping

$$\hat{H}{}^n(G,A)\otimes \hat{H}{}^m(G,B)\to \hat{H}{}^{n+m}(G,A\otimes B),$$

(called $\cup$- product, cup-product), where the tensor product of $A$ and $B$ is viewed as a $G$- module. In the special case where $A$ is a ring and the operations in $G$ are automorphisms, the $\cup$- product turns ${\oplus }_n\hat{H}{}^n(G,A)$ into a graded ring. The duality theorem for $\cup$- products asserts that, for every divisible Abelian group $C$ and every $G$- module $A$, the $\cup$- product

$$\hat{H}{}^n(G,A)\otimes \hat{H}{}^{-n-1}(G,\mathrm{Hom}(A,C))\to \hat{H}{}^{-}1(G,C)$$

defines a group isomorphism between $\hat{H}{}^n(G,A)$ and $\mathrm{Hom}(\hat{H}{}^{-n-1}(G,\mathrm{Hom}(A,C)),\hat{H}{}^{-}1(G,C))$( see [2]). The $\cup$- product is also defined for infinite groups $G$ provided that $n,m>0$.

Many problems lead to the necessity of considering the cohomology of a topological group $G$ acting continuously on a topological module $A$. In particular, if $G$ is a profinite group (the case nearest to that of finite groups) and $A$ is a discrete Abelian group that is a continuous $G$- module, one can consider the cohomology groups of $G$ with coefficients in $A$, computed in terms of continuous cochains [5]. These groups can also be defined as the limit $\underrightarrow{lim}{H}^n(G/U,A^U)$ with respect to the inflation mapping, where $U$ runs over all open normal subgroups of $G$. This cohomology has all the usual properties of the cohomology of finite groups. If $G$ is a pro- $p$- group, the dimension over $\mathbf{Z}/p\mathbf{Z}$ of the first and second cohomology groups with coefficients in $\mathbf{Z}/p\mathbf{Z}$ are interpreted as the minimum number of generators and relations (between these generators) of $G$, respectively.

See [6] for different variants of continuous cohomology, and also for certain other types of cohomology groups. See Non-Abelian cohomology for cohomology with a non-Abelian coefficient group.

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The norm map $N_{G/H}:A^H\to A^G$ is defined as follows. Let $g_1\dots g_k$ be a set of representatives of $G/H$ in $G$. Then $N_{G/H}(a)=g_{1}a+\cdots +g_{k}a$ in $A^G$. For a definition of the transgression relation in general spectral sequences cf. Spectral sequence; for the particular case of group cohomology, where this gives a relation, sometimes called connection, between $H^n(G,A)$ and $H^{n+1}(G/H,A^H)$ for all $n>0$, cf. also [a1], Chapt. 11, Par. 9.

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