Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$

Question

I have two questions.

$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ring, $$H(G,{\mathbb F}_p)\cong\begin{cases}S(V)&p=2\\ \Lambda (V)\otimes S(V)&p>2\end{cases}.$$ Moreover, if $p=2$ then $V={\rm Hom}(G,{\mathbb F}_p)$ identifies as $H^1(G,{\mathbb F}_p)$; if $p>2$ then $V$ from $\Lambda (V)$ identifies as $H^1(G,{\mathbb F}_p)$, while $V$ from $S(V)$ identifies with the image of $V=H^1(G,{\mathbb F}_p)$ via the Bockstein boundary map $\beta :H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$, which happens to be injective.

An alternative description is $$H(G,{\mathbb F}_p)\cong\begin{cases}{\mathbb F}_2[x_1,\ldots,x_r]&p=2\\ \Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]&p>2\end{cases},$$ where $x_1,\ldots,x_r$ are a basis of $V$ and $y_i=\beta (x_i)$.

These results are proved via Kunneth formula.

My question is where I can find these results so that I can quote them. I saw them in a paper and in a book, but with no reference given. It seems that people regard them as "common knowledge". In the paper I mentioned the authors simply said "Recall that...", as if everybody knows this, but some need to be reminded in case they forgot.

$\bf 2.$ The second question is whether there are explicit formulas for these isomorphisms in the literature.

If $p=2$, then the isomorphism ${\mathbb F}_2[x_1,\ldots,x_r]\to H(G,{\mathbb F}_2)$ is given by $x_{i_1}\cdots x_{i_n}\mapsto x_{i_1}\cup\cdots\cup x_{i_n}\in H^n(G,{\mathbb F}_2)$.

If $p>2$, then the isomorphism $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H(G,{\mathbb F}_p)$ is given by $x_{i_1}\wedge\cdots\wedge x_{i_s}\otimes y_{j_1}\cdots y_{j_t}\mapsto x_{i_1}\cup\cdots\cup x_{i_s}\cup y_{j_1}\cup\cdots\cup y_{j_t}\in H^{s+2t}(G,{\mathbb F}_p)$.

How about the reverse isomorphisms? Did anybody see anything published regarding this problem?

I did obtained explicit formulas for the reverse isomorphisms, where the elements of $H(G,{\mathbb F}_p)$ are written in terms of normalized cocycles. However, I don't know wether these results are new. Somebody might have thought about them before.

Answer 1

They're essentially exercises, compute it for $r=1$ and then invoke Künneth, and I'd expect every book to include it: Adem--Milgram's classic book for example (Corollary II.4.3 and Theorem II.4.4).

You can look at the identical homology case in Brown's classic book (Theorem V.6.6) and in particular, his description in Section V.5.3 shows how to build your reverse isomorphism (starting from a free resolution of the group module $\mathbb{F}_pG$).

Answer 2

I'm almost done with writing the paper. Here is the result.

If someone saw anything similar, then please let me know.

We have a basis $s_1,\ldots,s_r$ of $G$ over ${\mathbb F}_p$ and a basis $x_1,\ldots,x_r$ of $V=G^*=H^1(G,{\mathbb F}_p)$, which is dual to $s_1,\ldots,s_r$. Recall that $y_i=\beta (x_i)$ where $\beta :H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$ is the Bockstein boundary map.

The easier to state is the result in the case $p=2$.

$\bf Theorem~1$ If $p=2$ the isomorphism $H^*(G,{\mathbb F}_2)\to{\mathbb F}_2[x_1,\ldots,x_r]$ is given by $$[a]\mapsto\sum_{1\leq i_1,\ldots,i_n\leq r}a(s_{i_1},\ldots,s_{i_n})X_{i_1}\cdots X_{i_n}$$ for every $a\in Z^n(G,{\mathbb F}_2)$.

The case $p>2$ is more complicated and it requires some extra definitions.

First note that $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]$ has a structure of graded algebra which makes the isomorphism $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H^*(G,{\mathbb F}_p)$ an isomorphism of graded algebras. For every $n\geq 0$ the homogeneous component of degree $n$ is $$(\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r])^n=\bigoplus_{2k+l=n}\Lambda^l(x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]^k.$$ (Here ${\mathbb F}_p[y_1,\ldots,y_r]^k$ denotes the homogeneous polynomials of degree $k$.)

For every $1\leq i\leq r$ and $m\geq 0$ we define $x_i^{(m)}\in (\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r])^m$ by $$x_i^{(m)}=\begin{cases}1\otimes y_i^k&m=2k\\ x_i\otimes y_i^k&m=2k+1\end{cases}.$$

We have a basis of $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]$ made of all products $x_{i_1}\wedge\cdots\wedge x_{i_l}\otimes y_1^{k_1}\cdots y_r^{k_r}$ with $1\leq i_1<\cdots <i_l<r$ and $k_1,\ldots,k_r\geq 0$.

For $1\leq i\leq r$ we put $l_i=1$ if $i\in\{ i_1,\ldots,i_l\}$ and $l_i=0$ otherwise. Then $$x_{i_1}\wedge\cdots\wedge x_{i_l}\otimes y_1^{k_1}\cdots y_r^{k_r}=x_1^{(n_1)}\cdots x_r^{(n_r)},$$ where $n_i=2k_i+l_i$.

It follows that $x_1^{(n_1)}\cdots x_r^{(n_r)}$, with $n_1,\ldots,n_r\geq 0$, are a basis of $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]$.

We consider the action of the symmetric group $S_n$ on $C^n(G,{\mathbb F}_p)$ given by $$\sigma a(u_1,\ldots,u_n)=sgn (\sigma)a(u_{\sigma^{-1}(1)},\ldots,u_{\sigma^{-1}(n)})~\forall u_1,\ldots,u_n\in G.$$ If $n_1,\ldots,n_r\geq 0$ with $n_1+\cdots +n_r=n$ then we denote by $Sh(n_1,\ldots,n_r)$ the set of all $(n_1,\ldots,n_r)$-shuffles. $$Sh(n_1,\ldots,n_r)=\{\sigma\in S_n\, :\,\sigma (h)<\sigma (h+1)\,\forall h,\, h\neq n_1+\cdots +n_i\,\forall 1\leq i\leq r-1\}.$$ The condition from the defintion of $Sh(n_1,\ldots,n_r)$ also writes as $\sigma (n_1+\cdots +n_{i-1}+1)<\cdots <\sigma (n_1+\cdots +n_r)$ $\forall 1\leq i\leq r$.

If $1\leq i\leq r$, $m\geq 0$, $k=[m/2]$ and $q_1,\ldots,q_k$ are nonnegative integers, then we define $s_{i,m,q_1,\ldots,q_k}\in G^m$ by $$s_{i,m,q_1,\ldots,q_k}=\begin{cases}(s_i^{q_1},s_i,\ldots s_i^{q_k},s_i)&m=2k\\ (s_i,s_i^{q_1},s_i,\ldots s_i^{q_k},s_i)&m=2k+1\end{cases}.$$

$\bf Theorem~2$ If $p>2$ the isomorphism $\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H^*(G,{\mathbb F}_p)$ is given by $$[a]\mapsto\sum_{n_1+\cdots +n_r=n}c_{n_1,\ldots,n_r}x_1^{(n_1)}\cdots x_r^{(n_r)}$$ for every $a\in Z^n(G,{\mathbb F}_p)$, where $$\begin{aligned} c_{n_1,\ldots,n_r}=(-1)^{\frac{l(l-1)}2}\sum_{\sigma\in Sh(n_1,\ldots,n_r)}\sum_{1\leq q_{i,j}\leq p-1}\sigma a(s_{1,n_1,q_{1,1},\ldots,q_{1,k_1}},\ldots,s_{r,n_r,q_{r,1},\ldots,q_{r,k_r}}), \end{aligned}$$ with $l=|\{ i\,\mid\, 1\leq i\leq r,\, n_i\text{ is odd}\}|$ and $k_i=[n_i/2]$.

Here by the sum $\sum_{1\leq q_{i,j}\leq p-1}$ we mean that every variable $q_{i,j}$, with $1\leq i\leq r$ and $1\leq j\leq k_i$, takes values between $1$ and $p-1$.

Also $(s_{1,n_1,q_{1,1},\ldots,q_{1,k_1}},\ldots,s_{r,n_r,q_{r,1},\ldots,q_{r,k_r}})\in G^n$ is the concatenation of the sequences $s_{i,n_i,q_{i,1},\ldots,q_{i,k_i}}$ for $1\leq i\leq r$, of lengths $n_1,\ldots,n_r$.

Answer 3

I posted an article with this result on arXiv: https://arxiv.org/abs/2005.11868

Before sending it to be published, I want to make sure it is original. If anybody saw a similar result somewhere, then please let me know.

Also please let me know if you saw somewhere the so-called ${\mathcal I}$-cochains I introduced in the first section. (Perhaps with other name, other notation.) I have already asked about them on mathoverflow, but didn't get any answer. An alternative description of normalized cochains in terms of tensor powers of the augmented ideal