Moduli space of filteredλ-ringstructures over a filtered ring

摘要

Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filteredλ-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 ringsR[[x]], whereRis between?and?, with thex-adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filteredλ-ring structures overR[[x]]is canonically isomorphic to the set of ring maps from some ?universal? ringUtoR. From a local perspective, we demonstrate the existence of uncountably many mutually nonisomorphic filteredλ-ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over?-algebras.