Explicit formulas for the cohomology of the elementary abelian $p$-groups

TL;DR

This paper provides explicit formulas for the inverse of the isomorphism between the cohomology ring of an elementary abelian p-group and its algebraic presentation, using normalized cochains and an alternative description of cochain complexes.

Contribution

It introduces explicit formulas for the inverse isomorphism in the cohomology ring of elementary abelian p-groups, enhancing computational understanding.

Findings

Explicit inverse formulas for cohomology ring isomorphisms

Use of normalized cochains and alternative descriptions

Applicable for all primes p

Abstract

Let $G$ be an elementary abelian $p$-group, $G\cong{\mathbb F}_p^r$ and let $s_1,\ldots,s_r$ be a basis of $G$ over ${\mathbb F}_p$. Let $V$ be the dual of $G$, $V={\rm Hom}(G,{\mathbb F}_p)=H^1(G,{\mathbb F}_p)$. Let $x_1,\ldots,x_r$ be the basis of $V$ over ${\mathbb F}_p$ which is dual to the basis $s_1,\ldots,s_r$ of $G$. For $1\leq i\leq r$ we denote by $y_i=\beta (x_i)\in H^2(G,{\mathbb F}_p)$, where $\beta :H^1(G,{\mathbb F}_p)\to H^2(G,{\mathbb F}_p)$ is the connecting Bockstein map. The ring $(H^*(G,{\mathbb F}_p),+,\cup )$ satisfies $$H^*(G,{\mathbb F}_p)\cong\begin{cases}{\mathbb F}_p[x_1,\ldots,x_r]&p=2\\ \Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]&p>2\end{cases}.$$ When $p=2$ the isomorphism $\tau :{\mathbb F}_p[x_1,\ldots,x_r]\to H^*(G,{\mathbb F}_p)$ is given by $x_{i_1}\cdots x_{i_n}\mapsto x_{i_1}\cup\cdots\cup x_{i_n}\in H^n(G,{\mathbb F}_p)$. When $p>3$ the isomorphism $\tau :\Lambda (x_1,\ldots,x_r)\otimes{\mathbb F}_p[y_1,\ldots,y_r]\to H^*(G,{\mathbb F}_p)$ is given by $x_{i_1}\wedge\cdots\wedge x_{i_l}\otimes y_{j_1}\cdots y_{j_k}\mapsto x_{i_1}\cup\cdots\cup x_{i_l}\cup y_{j_1}\cup\cdots\cup y_{j_k}\in H^{2k+l}(G,{\mathbb F}_p)$. In this paper we give explicit formulas for the inverse isomorphism $\tau^{-1}$. The elements of $H^*(G,{\mathbb F}_p)$ are written in terms of normalized cochains. During the proof we use an alternative way to describe the normalized cochains. Namely, for every $G$-module $M$ we have $C^n(G,M)\cong{\rm Hom}(T^n({\mathcal I}),M)$, where ${\mathcal I}$ is the augmented ideal of $G$, ${\mathcal I}=\ker\varepsilon :{\mathbb Z}[G]\to{\mathbb Z}$.