Moduli space of filtered \(\lambda\)-ring structures over a filtered ring.
Summary: Motivated in part by recent works on the genus of classifying spaces of compact Lie groups, here we study the set of filtered \(\lambda\)-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 \(\mathbb{Z}\) and \(\mathbb{Q}\), with the \(x\)-adic filtration, we mimic the construction of the Lazard ring in formal group theory and show that the set of filtered \(\lambda\)-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 \(\lambda\)-ring structures over some filtered rings, including rings of dual numbers over binomial domains, (truncated) polynomial, and power series rings over \(\mathbb{Q}\)-algebras.
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