Graded $p$-polar rings and the homology of $\Omega^n\Sigma^nX$


As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the free affine $p$-adic group scheme functor, as well as the free formal group functor, defined on $k$-algebras for a perfect field $k$ of characteristic $p$, factors through $p$-polar $k$-algebras. It follows that the same is true for any affine $p$-adic or formal group functor, in particular for the functor of $p$-typical Witt vectors. As an application, we show that the homology of the free $E_n$-algebra $H^*(\Omega^n\Sigma^n X;\mathbf{F}_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf{F}_p)$ in a functorial way.