Moduli space of filtered λ -ringstructures over a filtered ring

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.