Structure and cohomology of moduli of formal modules

TL;DR

This paper investigates the structure and cohomology of moduli stacks of formal modules over rings, providing explicit calculations and connections to algebraic invariants, with implications for stable homotopy groups and algebraic geometry.

Contribution

It establishes structural results for the classifying ring of formal modules and computes flat cohomology groups of moduli stacks, linking them to algebraic invariants like the delta-invariant.

Findings

Generator of H^1_{fl} matches that of the stable homotopy group of spheres.

Order of certain cohomology groups equals 2^{N_1}, related to the 2-local zeta-function.

Cohomology relates to the delta-invariant and syzygetic ideals in commutative algebra.

Abstract

Given a commutative ring $A$, a "formal $A$-module" is a formal group equipped with an action of $A$. There exists a classifying ring $L^A$ of formal $A$-modules. This paper proves structural results about $L^A$ and about the moduli stack $\mathcal{M}_{fmA}$ of formal $A$-modules. We use these structural results to aid in explicit calculations of flat cohomology groups of $\mathcal{M}_{fmA}^{2-buds}$, the moduli stack of formal $A$-module $2$-buds. For example, we find that a generator of the group $H^1_{fl}(\mathcal{M}_{fm\mathbb{Z}}; \omega)$, which also generates (via the Adams-Novikov spectral sequence) the first stable homotopy group of spheres, also yields a generator of the $A$-module $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega)$ for any torsion-free Noetherian commutative ring $A$. We show that the order of the $A$-modules $H^1_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega)$ and $H^2_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ are each equal to $2^{N_1}$, where $N_1$ is the leading coefficient in the $2$-local zeta-function of $Spec A$. We also find that the cohomology of $\mathcal{M}_{fmA}^{2-buds}$ is closely connected to the delta-invariant and syzygetic ideals studied in commutative algebra: $H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ is the delta-invariant of the largest ideal of $A$ which is in the kernel of every ring homomorphism $A\rightarrow \mathbb{F}_2$, and consequently $H^0_{fl}(\mathcal{M}_{fmA}^{2-buds}; \omega\otimes \omega)$ vanishes if and only if $A$ is a ring in which that ideal is syzygetic.