On the generators of the polynomial algebra as a module over the Steenrod algebra

Abstract

Let $P_k:={\mathbb{F}}_2[x_1,x_2,\dots ,x_k]$ be the polynomial algebra over the prime field of two elements, ${\mathbb{F}}_2$, in $k$ variables $x_1,x_2,\dots ,x_k$, each of degree 1. We are interested in the Peterson hit problem of finding a minimal set of generators for $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we study the hit problem in degree $(k-1)(2^d-1)$ with $d$ a positive integer. Our result implies the one of Mothebe [4,5].

9 pages. The detailed proof of Theorem 3.10 of this paper was accepted for publication in Acta Mathematica Vietnamica, available online at arXiv:1502.05569

Cited by in corpus (6)

• On the dimension of $H^{\ast }(({\mathbb{Z}}_2{)}^{\times t},{\mathbb{Z}}_2)$ as a module over Steenrod ring
• The squaring operation and the hit problem for the polynomial algebra in a type of generic degree
• On Singer's conjecture for the fifth algebraic transfer
• On the algebraic transfers of ranks 4 and 6 at generic degrees
• The admissible monomial bases for the polynomial algebra of five variables in some types of generic degrees
• A note on the non-trivial elements in the cohomology groups of the Steenrod algebra