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

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.

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 $$\mathbb{F}_pG$$).

• Thank you for the answer. Corollary II.4.3 and Theorem II.4.4 in Adem--Milgram is exactly what I needed. Funny thing, this is the book I was mentioning for having the result without a reference. I saw the result for $p>3$ at the beginning of section III.3. – Constantin-Nicolae Beli Mar 23 at 16:12
• It does answer the first question. However, for the second question, Section V.5.3 in Brown doesn't deal with the case $G\cong{\mathbb F}_p$, but with the case when $G$ is cyclic. So I guess I should go ahead with writing a small paper on the explicit formula for the isomorphism. (Provided I don't find out that the result already exists. Maybe somebody saw it somewhere.) I'm not, by far, an expert in cohomology. I only took a one semester reading course from Brown's book about 20 years ago and then forgot everything. Only recently I needed some cohomology in my work. – Constantin-Nicolae Beli Mar 23 at 16:37
• Sorry, I meant ${\mathbb F}_p^r$. Forgot to add the exponent $r$. I'll try to read more carefully that section in Brown to see if it leads to explicit closed formulas for the isomorphisms. I'm using a different approach. – Constantin-Nicolae Beli Mar 23 at 16:44

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 $$1\leq i_1<\cdots and $$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 $$n_1,\ldots,n_r$$.

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

• I am a little confused by your paper. As Chris Gerig pointed out in his answer, this isomorphism is well-known and can be found in many classic references. So I don't understand your question about originality... – R. van Dobben de Bruyn May 26 at 20:39
• Maybe he was a little hasty. Later he wrote a comment, but it disappeared, I guess he deleted it. In Brown V.5.3, which he suggested I should look at, $G$ is cyclic. I need $G={\mathbb F}_p^r$. Even if $r=1$, when $G$ is cyclic, V.5.3 only produces a reverse isomorphism if we describe the elements of $H^*(G,{\mathbb F}_p)$ in terms of that special resolution of period 2 for the cyclic groups. I need it in terms of the normalized bar resolution. To go from one to the other one needs an explicit homotopy, which is technically difficult. Even if you mange to do this, how about the case $r>1$? – Constantin-Nicolae Beli May 26 at 22:45
• I'm not commenting on originality, but $r>1$ follows from $r=1$ as I mentioned, and even Brown writes (for the proof of his Theorem V.6.6): "If G is cyclic, this follows easily from our earlier computations [V.5.3].The general case can now be deduced by using the Künneth formula and direct limits as in the proof of [Theorem] 6.4". – Chris Gerig May 26 at 23:12
• @Constantin-NicolaeBeli Ah, I understand a little better what you're trying to do. The computation of $H_*$ relies on the shuffle product, which on the cohomology level presumably translates to some Hopf algebra structure. This is implicitly present in your results as well, but shouldn't there be a direct way to deduce this from Brown Thm. V.6.6? (To be clear, what Chris Gerig is saying is that dualising the isomorphism on homology of V.6.6 gives a map in the direction you want on cohomology.) – R. van Dobben de Bruyn May 26 at 23:38
• Unfortunately, you are talking about things I'm not familiar with. I never worked with $H_*$ and at this moment I can't even state its definition. I too noticed the same action of the symmetric group and the shuffle permutations, which appear in my formulas. Very likely it's not a mere coincidence. If there is some duality between $H^*$ and $H_*$, then this might be used to determine my coefficients $c_{n_1,\ldots,n_r}$. In my paper I only use the basic properties of $H^*$ and the existence of the isomorphism whose inverse I want to find. (Corollary II.4.3 and Theorem II.4.4 in Adem--Milgram.) – Constantin-Nicolae Beli May 27 at 0:04