A note on the hit problem for the Steenrod algebra in some generic degrees

Nguyen Khac Tin

Published 2021-06-20Version 1

Let $\mathcal P_n=\oplus_{d\geqslant 0} (\mathcal P_n)_d \cong \mathbb Z_2[x_1,x_2,\ldots ,x_n]$ be the graded polynomial algebra over the prime field of two elements $\mathbb Z/2$, in $n$ generators $x_1, x_2, \ldots , x_n$, each of degree 1. Being the mod-2 cohomology of the classifying space $B(\mathbb Z/2)^n$, the algebra $\mathcal P_n$ is a module over the mod-2 Steenrod algebra $\mathcal A$. We study the {\it hit problem}, set up by Frank Peterson, of finding a minimal set of generators for the polynomial algebra $\mathcal P_{n},$ viewed as a module over the mod-2 Steenrod algebra $\mathcal{A}$. The purpose of this paper is to continue our study of the hit problem by developing a result in \cite{ph31} for $\mathcal P_n$ in the generic degree $d_t=r(2^t-1)+s.2^t$ with $r=n=5,\ s=13,$ and $t$ an arbitrary positive integer. Note that in the case $t=1, d_1=5(2^1-1)+13.2^1=31,$ this problem has been studied by Phuc \cite{ph31}. Moreover, base on these results, we get the dimension result for the graded polynomial algebra in the generic degree $m=(n-1).(2^{s+4}-1)+67.2^{s+4}$ with $s$ an arbitrary positive integer, and in the case $n=6.$