Toward Quillen stratification for the odd primary Steenrod algebra

Fix an odd prime $p$ and let $A$ be the mod $p$ Steenrod algebra. There is a normal sub-Hopf algebra $D$ of $A$, and we show that the cohomology of $A$ is $F$-isomorphic to the subalgebra of the cohomology of $D$ which is invariant under the $A$-action. We also show that the sub-Hopf algebras of $A$ with bounded profile function detect nilpotence in cohomology.