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 En-algebra H∗(ΩnΣnX; Fp), as a Hopf algebra, only depends on the p-polar structure of H∗(X; Fp) in a functorial way.

We show that the functor of p-typical co-Witt vectors on commutative algebras over a perfect field k of characteristic p is defined on, and in fact only depends on, a weaker structure than that of a k-algebra. We call this structure a p-polar k-algebra. By extension, the functors of points for any p-adic affine commutative group scheme and for any formal group are defined on, and only depend on, p-polar structures. In terms of abelian Hopf algebras, we show that a cofree cocommutative Hopf algebra can be defined on any p-polar k-algebra P, and it agrees with the cofree commutative Hopf algebra on a commutative k-algebra A if P is the p-polar algebra underlying A; a dual result holds for free commutative Hopf algebras on finite k-coalgebras.

The Joker is an important finite cyclic module over the mod-2 Steenrod algebra A. We show that the Joker, its first two iterated Steenrod doubles, and their linear duals are realizable by spaces of as low a dimension as the instability condition of modules over the Steenrod algebra permits. This continues and concludes prior work by the first author and yields a complete characterization of which versions of Jokers are realizable by spaces or spectra and which are not. The constructions involve sporadic phenomena in homotopy theory (2-compact groups, topological modular forms) and may be of independent interest.

In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G(1)circle times G(2) of two affine abelian group schemes G(1), G(2) over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G(1)circle times G(2). The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G(1)circle times G(2) explicitly. using Dieudonne theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.

Unstable operations in a generalized cohomology theory E give rise to a functor from the category of algebras over E_* to itself which is a colimit of representable functors and a comonoid with respect to composition of such functors. In this paper I set up a framework for studying the algebra of such functors, which I call formal plethories, in the case where E_* is a Prüfer ring. I show that the "logarithmic" functors of primitives and indecomposables give linear approximations of formal plethories by bimonoids in the 2-monoidal category of bimodules over a ring.

In this paper we study tensor products of affine abelian group schemes over a perfect field k. We first prove that the tensor product G_1 ⊗ G_2 of two affine abelian group schemes G_1,G_2  over a perfect field k exists. We then describe the multiplicative and unipotent part of the group scheme G_1 ⊗G_2. The multiplicative part is described in terms of Galois modules over the absolute Galois group of k. We describe the unipotent part of G_1 ⊗ G_2 explicitly, using Dieudonn\'e theory in positive characteristic. We relate these constructions to previously studied tensor products of formal group schemes.