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

Abstract

Let $G$ be an elementary abelian $p$-group, $G\cong {\mathbb{F}}_p^r$ and let $s_1,\dots ,s_r$ be a basis of $G$ over ${\mathbb{F}}_p$. Let $V$ be the dual of $G$, $V=\mathrm{Hom}(G,{\mathbb{F}}_p)=H^1(G,{\mathbb{F}}_p)$. Let $x_1,\dots ,x_r$ be the basis of $V$ over ${\mathbb{F}}_p$ which is dual to the basis $s_1,\dots ,s_r$ of $G$. For $1\le i\le r$ 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 $(H^{\ast }(G,{\mathbb{F}}_p),+,\cup )$ satisfies $$H^{\ast }(G,{\mathbb{F}}_p)\cong \{\begin{array}{ll}{\mathbb{F}}_p[x_1,\dots ,x_r]& p=2\\ \Lambda (x_1,\dots ,x_r)\otimes {\mathbb{F}}_p[y_1,\dots ,y_r]& p>2\end{array}.$$ When $p=2$ the isomorphism $\tau :{\mathbb{F}}_p[x_1,\dots ,x_r]\to H^{\ast }(G,{\mathbb{F}}_p)$ 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 $\tau :\Lambda (x_1,\dots ,x_r)\otimes {\mathbb{F}}_p[y_1,\dots ,y_r]\to H^{\ast }(G,{\mathbb{F}}_p)$ 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 $H^{\ast }(G,{\mathbb{F}}_p)$ 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 $C^n(G,M)\cong \mathrm{Hom}(T^n(\mathcal{I}),M)$, where $\mathcal{I}$ is the augmented ideal of $G$, $\mathcal{I}=\ker \epsilon :\mathbb{Z}[G]\to \mathbb{Z}$.