Changes

Group cohomology of elementary abelian group of prime-square order

, 21:34, 24 October 2011
Over the integers
{{group family-specific information|
group family = elementary abelian group of prime-square order|
information type = group cohomology|
connective = of}}

Suppose $p$ is a [[prime number]]. We are interested in the [[elementary abelian group of prime-square order]] $E_{p^2} = (\mathbb{Z}/p\mathbb{Z})^2 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}$.
===Over the integers===
The homology groups below can be computed using the homology groups for the [[group of prime order]] (see [[group cohomology of finite cyclic groups]]) and combining it with the [[Kunneth formula for group homology]]. $H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{(q + 3)/2} & \qquad q = 1,3,5,\dots \\ (\mathbb{Z}/p\mathbb{Z})^{q/2}, & \qquad q = 2,4,6,\dots \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.$ The even and odd cases can be combined giving the following alternative description: $H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{q/2 + 3(1 - (-1)^q)/4}, & \qquad q > 0 \\ \mathbb{Z}, & \qquad q = 0 \\\end{array}\right.$
The first few homology groups are given below:
The homology groups with coefficients in an abelian group $M$ are given as follows:
$H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/pM)^{(q+3)/2} \oplus (\operatorname{Ann}_M(p))^{(q-1)/2}, & \qquad q = 1,3,5,\dots\\ (M/2MpM)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0 \\\end{array}\right.$
Here, $M/pM$ is the quotient of $M$ by $2M pM = \{ px \mid x \in M \}$ and $\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}$.
These homology groups can be computed in terms of the homology groups over integers using the [[universal coefficients theorem for group homology]].
| $M$ is a finitely generated abelian group || all isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}$ where $r$ is the rank for the $p$-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$ || all isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + sq/2}$ where $r$ is the rank for the $p$-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$
|}

==Cohomology groups for trivial group action==
$H^q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{(q-1)/2}, & q = 1,3,5,\dots \\ (\mathbb{Z}/p\mathbb{Z})^{(q+2)/2}, & q = 2,4,6,\dots \\ \mathbb{Z}, & q = 0 \\\end{array}\right.$

The odd and even cases can be combined as follows:

$H^q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/p\mathbb{Z})^{q/2 + 1/4 + 3(-1)^q/4}, & q > 0\\ \mathbb{Z}, & q = 0 \\\end{array}\right.$
The first few cohomology groups are given below:
These can be deduced from the homology groups with coefficients in the integers using the [[dual universal coefficients theorem for group cohomology]].

===Important case types for abelian groups===
===Important case types for abelian groups===
| $M$ is uniquely $p$-divisible, i.e., every element of $M$ can be divided by <matH>p[/itex] uniquely. This includes the case that $M$ is a field of characteristic not 2. || all zero groups || all zero groups
|-
| $M$ is $p$-torsion-free, i.e., no nonzero element of $M$ multiplies by $p$ to give zero. || $(M/pM)^{(q-31)/2}$ || $(M/pM)^{(q+2)/2}$
|-
| $M$ is $p$-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$ || $(\operatorname{Ann}_M(p))^{(q+3)/2}$ || $(\operatorname{Ann}_M(p))^{q/2}$
| $M$ is a finitely generated abelian group || all isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q - 1)/2}$ where $r$ is the rank for the $p$-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$ || all isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2}$ where $r$ is the rank for the $p$-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$
|}

==Tate cohomology groups for trivial group action==

{{fillin}}

==Growth of ranks of cohomology groups==

===Over the integers===

With the exception of the zeroth homology group and cohomology group, the homology groups and cohomology groups over the integers are all elementary abelian $p$-groups.

For the homology groups, the rank (i.e., dimension as a vector space over the field of $p$ elements) is a function of $q$ that is a sum of a linear function (of slope 1/2) and a periodic function (of period 2). The same is true for the cohomology groups, although the precise description of the periodic function differs.

* For homology groups, choosing the periodic function so as to have mean zero, we get that the linear function is $q \mapsto q/2 + 3/4$ and the periodic function is $3(-1)^{q+1}/4$.
* For cohomology groups, choosing the periodic function so as to have mean zero, we get that the linear function is $q \mapsto q/2 + 1/4$ and the periodic function is $3(-1)^q/4$.

Note that:

* The intercept for the cohomology groups is 1/4, as opposed to the intercept of 3/4 for the homology groups. This is explained by the somewhat slower start of cohomology groups on account of $H^1$ being torsion-free.
* The periodic parts for homology groups and cohomology groups are negatives of each other, indicating an opposing pattern that is explained by looking at the [[dual universal coefficients theorem for group cohomology]].

===Over the prime field===

If we take coefficients in the prime field $\mathbb{F}_p$, then the ranks of the homology and cohomology groups both grow as linear functions of $q$. The linear function in ''both'' cases is $q \mapsto q + 1$. Note that in this case, the homology groups and cohomology groups are vector spaces over $\mathbb{F}_p$ and the cohomology group is the vector space dual of the homology group.

Note that there is no periodic part when we are working over the prime field.