Moduli space of filtered λ‐ringstructures over a filtered ring

Abstract

Motivated by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered λ-ring structures over a filtered ring from a purely algebraic point of view. From a global perspective, we first show that this set has a canonical topology compatible with the filtration on the given filtered ring. For power series rings Rwhere R is between Z and Q, with the x-adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered λ-ring structures over R [[x]] is canonically isomorphic to the set of ring maps from some "universal" ring U to R. From a local perspective, we demonstrate the existence of uncountably many mutually non-isomorphic filtered λ-ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial and powers series rings over torsionfree Q-algebras.

On λ‐rings and topological realization

“…Now it follows from (5.1.3) and [11,Theorem 4.4] that (ψ p S • σ)(x) = (σ • ψ p T )(x) for all primes p. (It is here that we are using the hypothesis a p,1 = 0 for all p.) That is, σ : S → T is a filtered λ-ring isomorphism, as claimed. It is clear that Theorem 1.8 also follows from this claim.…”

“…= Corollary 4.1.2 in[11]). There is a bijection between Λ(Z[x]/(x 2 )) and the set of sequences (b p ) indexed by the primes in which the component b p is divisible by p. The filtered λ-ring structure corresponding to (b p ) has Adams operations ψ p (x) = b p x.…”

Cohomology of λ-rings

“…For more discussions about λ-rings, consult Atiyah and Tall [1] or Knutson [6]. The author's articles [9,10] contain some recent results on λ-rings which might also be of interest to the reader.…”

Binomial rings, integer-valued polynomials, and λ-rings

“…Thus, a binomial ring is in a precise sense the simplest kind of λ-ring. More recently, Yau [17] used binomial rings to prove the existence of a filtered ring admitting an uncountable family of mutually non-isomorphic filtered λ-ring structures; his example is the ring B[ε]/(ε 2 ) in the ε-adic filtration, where B is any binomial ring. In their study of plethystic algebra, which they used to generalize the Witt vector construction, Borger and Wieland [3] used the "binomial plethory" as an illuminating example of a Z-plethory.…”