Moduli space of filtered λ‐ringstructures over a filtered ring

Abstract

Motivated in part 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 R[[x]], where 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 nonisomorphic filtered λ‐ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over ℚ‐algebras.