Computation of the classifying ring of formal modules
Andrew Salch
TL;DR
The paper develops $U$-homology as a computational tool for the classifying ring $L^A$ of one-dimensional formal $A$-modules and proves that $U$-homology vanishes in positive degrees, reducing obstructions to $L^A$ being polynomial. It shows that, after suitable localization, $L^A$ decomposes as a tensor product of suspended Rees or symmetric algebras on the ideals $I^A_n=(\nu(n),a^n-a)$, enabling explicit presentations even when $L^A$ is not a polynomial algebra. The main results yield closed-form descriptions for $L^A$ when $A$ is a number ring or a group ring, including complete computations for $A$ the ring of integers in $\mathbb{Q}(\sqrt[4]{-18})$ and for $A=\mathbb{Z}[C_m]$ after inverting $m$, together with corollaries on lifting formal $A$-modules and extending formal buds. These computations provide concrete presentations of the moduli stack of formal $A$-modules and have potential implications for both arithmetic geometry and stable homotopy theory. Overall, the machinery bridges number-theoretic structure and topological moduli via explicit algebraic decompositions of $L^A$.
Abstract
In this paper, we develop general machinery for computing the classifying ring $L^A$ of one-dimensional formal $A$-modules, for various commutative rings $A$. We then apply the machinery to obtain calculations of $L^A$ for various number rings and cyclic group rings $A$. This includes the first full calculations of the ring $L^A$ in cases in which it fails to be a polynomial algebra. We also derive consequences for the solvability of some lifting and extension problems.