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

TL;DR

This paper introduces graded p-polar rings as a new algebraic structure, demonstrating their role in understanding the homology of iterated loop-suspension spaces and their relation to p-typical Witt vectors.

Contribution

It defines graded p-polar rings, shows their relevance to affine p-adic and formal group schemes, and connects these structures to the homology of free E_n-algebras.

Findings

The free affine p-adic group scheme functor factors through p-polar k-algebras.

The homology of free E_n-algebras depends functorially on the p-polar structure of cohomology.

p-polar rings generalize previous ungraded structures to a graded setting.

Abstract

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.