Essential cohomology for elementary abelian p-groups

TL;DR

This paper characterizes the essential cohomology classes of elementary abelian p-groups for odd primes, showing they form a Steenrod closure and are freely generated over Mui invariants.

Contribution

It identifies the structure of the essential ideal as the Steenrod closure of a top class and as a free module over Mui invariants, providing new algebraic insights.

Findings

Essential classes form the Steenrod closure of the top exterior class.

The essential ideal is a free module over Mui invariants.

Provides explicit description of the essential cohomology structure.

Abstract

For an odd prime p the cohomology ring of an elementary abelian p-group is polynomial tensor exterior. We show that the ideal of essential classes is the Steenrod closure of the class generating the top exterior power. As a module over the polynomial algebra, the essential ideal is free on the set of Mui invariants.