Graded $p$-polar rings and the homology of $Ω^nΣ^nX$

TL;DR

The paper develops graded analogues of $p$-polar algebras and proves that free affine $p$-adic group schemes and free formal groups factor through graded $p$-polar algebras over a perfect field of characteristic $p$. It then constructs graded Witt and co-Witt vector theories and establishes graded Dieudonné correspondences to analyze unipotent and connected graded Hopf structures, enabling a functorial description of homological invariants. A key topological consequence is that the Hopf algebra $H^*(Ω^nΣ^n X; extbf{F}_p)$ depends only on the $p$-polar structure of $H^*(X; extbf{F}_p)$, linking graded polar data to iterated loop-space homology. Together, these results unify algebraic and topological perspectives on graded $p$-polar structures, providing refined control over the homology of free $E_n$-algebras and their loop-space cohomology via Witt–Dieudonné theory.

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 $H^*(Ω^nΣ^n X;\mathbf F_p)$, as a Hopf algebra, only depends on the $p$-polar structure of $H^*(X;\mathbf F_p)$ in a functorial way.

Theorems & Definitions (75)

• Definition
• Theorem 1.1
• Corollary 1.2
• proof
• Definition
• Lemma 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7