Relations among the kernels and images of {S}teenrod squares acting on right mathcal{A}-modules

In this note, we examine the right action of the Steenrod algebra $\mathcal{A}$ on the homology groups $H_*(BV_s, \F_2)$, where $V_s = \F_2^s$. We find a relationship between the intersection of kernels of $Sq^{2^i}$ and the intersection of images of $Sq^{2^{i+1}-1}$, which can be generalized to arbitrary right $\mathcal{A}$-modules. While it is easy to show that $\bigcap_{i=0}^{k} \mathrm{im}\,Sq^{2^{i+1}-1} \subseteq \bigcap_{i = 0}^k \mathrm{ker}\,Sq^{2^i}$ for any given $k \geq 0$, the reverse inclusion need not be true. We develop the machinery of homotopy systems and null subspaces in order to address the natural question of when the reverse inclusion can be expected. In the second half of the paper, we discuss some counter-examples to the reverse inclusion, for small values of $k$, that exist in $H_*(BV_s, \F_2)$.