Relations among the kernels and images of Steenrod squares acting on right A-modules

In this note, we examine the right action of the Steenrod algebra A on the homology groups H∗(BVs,F2), where Vs=F2s. We find a relationship between the intersection of kernels of Sq2i and the intersection of images of Sq2i+1−1, which can be generalized to arbitrary right A-modules. While it is easy to show that ⋂i=0kimSq2i+1−1⊆⋂i=0kkerSq2i for any given k≥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∗(BVs,F2).